Topological Phases, Boson Mode, Immiscibility Window and Structural

Topological Phases, Boson mode, Immiscibility window and Structural

Groupings in Ba-Borate and Ba-Borosilicate glasses

A dissertation submitted to

Division of Research and Advanced Studies

University of Cincinnati

In partial fulfillment of the requirements for the degree of

Doctor of Philosophy (Ph.D.)

In the Department of Electrical Engineering and Computing Systems

Of the College of Engineering and Applied Sciences

October 2015

Chad Holbrook

M.S., University of Cincinnati, 2007

B.S., Northern Kentucky University, 2003

Committee Chair: Punit Boolchand, Ph.D.

Abstract

In a dry ambient,(BaO)x(B2O3)100-x (a pseudo-binary glass system) were synthesized over a wide

composition range, 0 mol% < x < 40 mol% , by utilizing induction melting precursors. These

high quality glasses were comprehensively examined in Modulated DSC, Raman Scattering,

Infrared reflectance experiments. Raman Scattering experiments and the analysis of the

symmetric stretch of intra-ring Boron-Oxygen (BO) bonds (A1’) of characteristic “mixed-rings”, permits the identification of Medium Range Structure (MRS) which form in the titled glasses.

These modes consist of a triad of modes (705 cm-1, 740 cm-1 and 770 cm-1), and their scattering

strengths display a positive correlation to the nucleation of characteristic structural groupings

(SGs); analogous to structural groupings found in the corresponding crystalline phases of

Barium-Tetraborate (x = 20 mol%), and Barium-Diborate (x = 33 mol%). Identification of the

SG’s permit an understanding of the extended range structure apparent in these modified borate

glasses. Furthermore, a microscopic understanding of the Immiscibility range in the titled glasses

in the 0 mol% < x < 15 mol% range, can be traced to the deficiency of Barium that prohibits

nucleation of the Barium-Tetraborate species.

(BaO)x[32(B2O3)68(SiO2)]100-x (pseudo-ternary glasses) were synthesized and their Glass

Transition Temperature (Tg(x)), molar volume (Vm(x)), and Raman Scattering were examined as

a function of modifier (BaO) content in the 25 mol% < x < 48 mol% range. Three distinct

regimes of behavior were observed: (1) At low x, 24 mol% < x < 29 mol% range, the modifier

largely polymerizes the backbone, and Tg(x) increases. This is a feature that we identify with the stressed-rigid elastic phase. (2)At high x, 32 mol% < x < 48 mol% range, the modifier depolymerizes the network by creating non-bridging oxygen (NBO) atoms; in this regime, Tg(x)

ii decreases and networks are viewed to be in the flexible elastic phase. (3) In the narrow intermediate x regime, 29 mol% < x < 32 mol% range, Tg(x) shows a broad global maximum

(nearly x-independent), Vm(x) displays a global minimum, and Raman-modes (scattering- strengths and frequencies) become x-independent. These are features that we associate with the isostatically rigid elastic phase (also called the intermediate phase). In this phase, medium range structures adapt as revealed by the counting of Lagrangian bonding constraints and Raman Mode

Scattering strengths.

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Acknowledgements

Pursuing my dreams as a Ph.D. student has been a rewarding, enriching and enlightening

experience. I appreciate the time and dedication the committee has devoted to ensure that I contribute a quality Ph.D. Dissertation. This document represents the culmination of years of

hard work made possible due to the support provided to me by personal and professional

relationships, and I would like to acknowledge these important influences that have fostered and

inspired me to achieve my educational goals.

First, I would like to thank Dr. Punit Boolchand for the amazing person that you are and for

being my mentor for the past 10 years. It is apparent that you truly care about your students and

their academic and professional careers. Over the years, I have grown to consider you a part of

my family, and without your experience and direction, it would’ve been impossible to navigate

the complexities of glass science. It has been inspiring to witness your approach to your career

and education, and I will always use this as a guide to my own.

Secondly, I would like to thank Dr. Jonathan Goldstein for considering my support on research

efforts conducted at the Wright Patterson Air Force Base. Dr. Goldstein, I appreciate your

encouragement, your time, and wisdom that you have devoted to me. I have had the pleasure to

know you since my Master’s thesis defense in 2003 and look forward to a continued professional

relationship. Thank you for the time you have spent in helping me develop my experimental

achievements.

Next, I would like to thank Dr. John Derov. I appreciate you welcoming me into our Branch on

base. I enjoy listening to your wisdom regarding the Physics and the Mathematics of Science. I

consider you not only a colleague, but a friend. Thank you for your interest in my work.

I would also like to thank Dr. Peter Kosel for the interesting hallway conversations about

Science and course curriculum. I appreciate your individualized approach to teaching and your enthusiasm of the material. I wish more Professors would demonstrate such pride in their work.

Thank you for your insightful discussion during my Ph.D. proposal.

Furthermore, I would like to thank Dr. Marc Cahay for taking the time to guide me through my

Master’s Thesis and now Ph.D. Dissertation. I really appreciate you serving on the committee

and appreciate all of the one-on-one conversations that we have had in the past.

I would like to thank Dr. Bernard Goodman, Dr. Wayne Bresser and Dr. Andrew Czaja for

your help and support.

I would also like to thank the students of the past and the present at the University of Cincinnati:

Ping Chen, Deassy Novita, Sriram Ravindren, Vignarooban Kandasamy, Shibalik

Chakraborty, Kapila Gunasekera, Ralph Chebir, Aaron Welton, Sriram Dash, Somendu

Chakraborty, and Chandi Mohanty.

I would like to thank my base supervisor, Mark G. Schmitt, for your support of my Ph.D.

aspirations. Thank you for your encouragement, trust, and belief that you have had in me.

I would like to thank my wife, Paola. You are an amazing mother and you have provided me

with so much emotional support. I realize you have lost many valuable hours of sleep in order to

listen to my concerns of whether or not I would fall short of my personal goals. Thank you for

reassuring me of my abilities and allowing me to appreciate the journey.

I would also like to thank my parents, Danny and Sally Holbrook, brother, Jacob Holbrook,

and sister-in-law, Kelly Holbrook, for your words of encouragement and constant support. I

truly admire the way that you conduct your personal and professional lives and use it as a model

for my own.

Lastly, I would like to thank the Air Force for their support and the support of NSF grant

DMR-08-53957.

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Table of Contents

CHAPTER 1 INTRODUCTION 1

1.1 Background 1

1.1.1 Topological Constraint Theory 1

1.1.2 Revolutionary Change in Topological Constraint Theory 3

1.2 Relevance of Borate and Silicate Glasses 4

1.3 Findings 8

CHAPTER 2 SAMPLE SYNTHESIS 11

2.1 Barium Borates 11

2.2 Barium Borosilicates 14

CHAPTER 3 THERMAL CHARACTERIZATION 16

3.1 Background for Thermal Characterization Techniques 16

3.1.1 The Dynamic Glass Transition 16

3.1.2 Modulated DSC 18

3.1.3 Frequency Correction 21

3.2 M-DSC Experiments 22

3.3 M-DSC Results 23

3.4 M-DSC Discussion 24

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CHAPTER 4 DENSITY MEASUREMENTS 26

4.1 Background for Density Measurements 26

4.2 Density Experiments of Barium Borate Glasses 26

CHAPTER 5 OPTICAL CHARACTERIZATION 28

5.1 Background for Optical characterization Techniques 28

5.1.1 Requirements for Raman Active Modes 28

5.1.2 New Picture of Light Interaction 30

5.1.3 Crossing Phenomena of Two Coupled Modes 30

5.1.4 Structure 33

5.1.5 Considerations of Symmetry in Crystalline Materials 34

5.1.6 Symmetry Analysis of Different Symmetry sets of the Meta-borate Crystalline

Compound 36

5.1.7 Factor Group Analysis 37

5.1.8 Site Group Analysis 38

5.1.9 Symmetry coordinates, Internal coordinates and Displacement Configurations 40

5.1.10 G-Matrix Example for Familiarity 42

5.1.11 Introduction of a Modifier to the Base Material 44

5.1.12 Mixed Ring Breathing Modes 45

5.1.13 The Raman Line Shape Profile 46

5.2 IR/Raman Experiments of modified Barium Borate Glasses 48

5.2.1 Raman Spectroscopy Experiments 48

5.2.2 Compositioinal Trends of Barium Borate glasses 50

5.2.3 Observations at the Low Frequency Regime Boson and Lattice Vibrations

(Extended Range Structure) 53

5.2.4 Observations of BR and Mixed Rings (Medium-Range Structure): Intra-Ring B-O

Bonds 53

5.2.5 Observations at the High Frequency Regime: Extra-Ring B-O bonds 53

5.2.6 Mixed Rings and the Intermediate Phase 55

5.2.7 Quantification of Raman Vibrational Mode Characteristics Through Line-shape

De-convolution 57

5.2.8 Polarized Raman Experiments 59

5.2.9 Results of Barium Borates 60

5.2.10 FTIR Results 65

CHAPTER 6 DISCUSSION 69

6.1 Topological Phases of Ba-Borate Glasses 69

6.2 Raman Scattering and aspects of Glass-structure 73

6.2.1 Raman scattering as a probe of local, medium-range-structure and extended-

range-structure 73

6.2.2 The nature of structural groupings (SGs) contributing to the mixed ring modes

observed in Ba Borates 79

6.3 IR reflectance a quantitative probe of B4/B3 content of BaO modified B2O3 86

6.4 Microscopic origin of the Immiscibility range in the Equilibrium Phase diagram

of the BaO-B2O3 binary 92

6.5 Glass Network dimensionality considerations and the origin of the Boson mode in

Borate glasses 95

6.6 Boson mode and the Stress and Rigidity transitions in Ba-borate glasses 100

CHAPTER 7 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK 102

7.1 Conclusions on (BaO)x(B2O3)100-x binary glass system 102

7.2 Conclusions on (BaO)x[32 (B2O3) 68 (SiO2)]100-x pseudo-ternary glasses 103

7.3 Suggestions for future work 104

CHAPTER 8 APPENDIX THE PUBLISHED WORK ON THE PSEUDO-TERNARY

SYSTEM (BAO)X[(B2O3)32(SIO2)68]100-X 106

LIST OF FIGURES

FIGURE 2-1: PSEUDO-BINARY PHASE DIAGRAM OF THE SYSTEM BAO-B2O3. THE BINARY PHASE DIAGRAM SHOWS

THERMODYNAMIC PHASES; PHASES ARE IN EQUILIBRIUM AND ARE DELINEATED BY SOLID LINES [39]...... 13

FIGURE 3-1: THE BLACK CURVE IN THIS FIGURE REPRESENTS THE TYPICAL HEAT FLOW AS WE PROGRESS FROM SUB-

TG TO A TEMPERATURE ABOVE THE MELTING POINT OF A GLASS. THE TEAL CURVE SHOWS THE GENERAL

PROPERTY OF THE SOLID (E.G. ENTHALPY, VOLUME, AND ENTROPY) OF A MELT WHEN SUPER-COOLED OR

QUENCHED FROM THE LIQUID TO A GLASSY STATE. THE CHARACTERISTIC TRANSITION TEMPERATURES,

PROGRESSING FROM LEFT TO RIGHT ARE THE GLASS, CRYSTALLIZATION AND MELT TEMPERATURES...... 17

FIGURE 3-2: CHARACTERISTIC TEMPERATURE BEHAVIOR OF THE ENTROPY OF PROTOTYPICAL GLASSES. THE

ZSTANDARD TEMPERATURE REFERENCES SUCH AS THE MELTING POINT T M THE GOLDSTEIN TEMPERATURE TX,

GLASS TRANSITION TEMPERATURE (T G), AND KAUZMANN T K TEMPERATURE ARE DISPLAYED. THE TRAJECTORY

OF THE GLASSY STATE SHOWN IN COLOR IS DEPENDENT ON QUENCH TEMPERATURE OR EXPERIMENTAL

TEMPERATURE AT WHICH THE MEASUREMENT TAKES PLACE[40]...... 18

FIGURE 3-3: THE TOTAL HEAT FLOW IN RED IS CALCULATED AS THE AVERAGE BEHAVIOR OF THE MODULATED HEAT

FLOW SHOWN IN BLACK...... 19

FIGURE 3-4: THE TOTAL HEAT FLOW CAN BE BROKEN DOWN IN TO TWO SEPARATE COMPONENTS. THE OVERSHOOT

MARKED BY THE GREEN ARROW AND LINE INDICATES THE PART OF HEAT FLOW THAT IS ASSOCIATED WITH

LATENT HEAT AND CONTRIBUTES TO THE NON-REVERSING HEAT FLOW...... 21

FIGURE 3-5: STANDARD HEAT-FLOW SIGNALS OBTAINED FROM M-DSC HEATING AND COOLING EXPERIMENTS...... 22

FIGURE 3-6: THIS IS A TYPICAL MODULATED TEMPERATURE PROFILE WHICH IS PROGRAMMED FOR ALL M-DSC

MEASUREMENTS...... 23

FIGURE 3-7: COMPOSITIONAL DEPENDENCE OF THE GLASS TRANSITION TEMPERATURE AND CHANGE OF ENTHALPY

OF RELAXATION OF BARIUM BORATE GLASSES...... 24

FIGURE 3-8: A) OXIDE NETWORK DISPLAY POLYHEDRAL ARE CONNECTED BY BRIDGING OXYGEN; B) MODIFIED

NETWORKS ABOVE THE PERCOLATION THRESHOLD 16% BY VOLUME OF MODIFIER FORM CONTINUOUS

CHANNELS FOR SELF-DIFFUSION. NOTE THIS FIGURE WAS TAKEN FROM [43] AND IS USED FOR ILLUSTRATION

PURPOSES ...... 25

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FIGURE 4-1: ILLUSTRATION OF VOLUMETRIC EXPERIMENTS USED TO DETERMINE THE DENSITY OF SUBMERGED

VOLUMES...... 27

FIGURE 4-2: MOLAR VOLUME OF BARIUM BORATE GLASS OBTAINED FROM DENSITY EXPERIMENT...... 27

FIGURE 5-1: SYMMETRIC AND ASYMMETRIC BRANCHES OF A COUPLED RESONATOR[45]. THIS ILLUSTRATES THE

SAME BASIC PHENOMENA THAT DESCRIBE THE COUPLING OF THE PHOTON AND PHONONS, WHICH IN TURN,

DESCRIBES THE POLARITONS - THIS DEPICTS THE NEW PICTURE OF LIGHT PROPAGATION IN RAMAN ACTIVE

MATERIALS...... 32

FIGURE 5-2: ENERGY DISPERSION CURVE DISPLAYING THE RELATIONSHIP BETWEEN LO- POLARITONS, LO-PHONON,

AND TO-POLARITONS IN CRYSTALLINE AND POLYCRYSTALLINE GAP [46]...... 32

FIGURE 5-3: DIAGRAMMATIC ILLUSTRATION OF THE RELATIONSHIP OF VIBRATIONAL SPECIES WITH RESPECT TO

LOCAL AND EXTENDED RANGE ORDER OF THE METABORATE CRYSTAL STRUCTURE [48]...... 36

FIGURE 5-4: THIS IS AN ILLUSTRATION OF THE INTERNAL COORDINATES PROVIDED BY BRIL ET. AL. [36] OF THE

BOROXYL RING; INTERNAL COORDINATES ARE USED TO SIMPLIFY THE DESCRIPTION OF NORMAL MODES.

INTERNAL COORDINATES CONSOLIDATES THE DISPLACEMENT COORDINATES (DESCRIBED IN BY CARTESIAN

COORDINATES) USING GEOMETRIC ARGUMENTS PROVIDING A MORE COMPACT DESCRIPTION...... 41

FIGURE 5-5: ILLUSTRATION OF COORDINATE BASIS INDEPENDENT TO EACH ATOM OF A MOLECULE WITH C2V

SYMMETRY SUCH AS H2O...... 42

3 2 FIGURE 5-6 MIXED RINGS OF VARIOUS STOICHIOMETRIC COMPOUNDS HAVE VARYING RATIOS OF SP /SP SPECIES;

3 2 THE RATIO OF SP :SP DETERMINE THE VIBRATIONAL FREQUENCY OF THE SYMMETRIC STRETCH OF INTRA-RING

B-O BONDS. DEMONSTRATED BY VIGNAROOBAN [17] IN, THE SODIUM BORATE GLASSES MODES AT 770, 740,

-1 AND 705 CM ARE SYMMETRIC STRETCH OF MIXED RINGS IN THE SODIUM TRI-BORATE, DI-BORATE AND TRI-

SODIUM PENTA-BORATE STRUCTURAL GROUPINGS...... 45

FIGURE 5-7: NON-LINEAR BEHAVIOR OF THE A1’ VIBRATION WITH RESPECT TO VARIOUS BORATE COMPOUNDS. THIS

FIGURE WAS TAKEN FROM VIGNAROOBAN [17]...... 46

FIGURE 5-8: FUNCTION OF THE DIFFRACTION GRATINGS IN THE T64000 TRIPLE ADDITIVE/SUBTRACTIVE

MONOCHROMATIC RAMAN SYSTEM ILLUSTRATED IN [50]...... 49

FIGURE 5-9: RAMAN SPECTRA OF THE NARROW BAND REPRESENTATIVE OF THE ENERGY-SPECTRUM FROM A

SINGULAR TRANSITION IN A MERCURY (HG) LAMP. THIS NARROW ATOMIC TRANSITION IN HG-ATOM PERMITS

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INCREASED PRECISION FOR ALIGNMENT AND THE CALIBRATION OF THE SPECTROMETER AND

FOREMONOCHROMATOR IN OUR T64000 SYSTEM...... 50

FIGURE 5-10: GLOBAL PERSPECTIVE OF RAMAN-LINE SHAPES FOR THE SYSTEM (BAO)X(B2O3)100-X SYSTEM.

NOTICE THE RICHNESS OF VIBRATIONAL INFORMATION CONTAINED IN THE RAMAN SPECTRA...... 51

FIGURE 5-11: INTENSITY GRAPH OF RAMAN DATA SHOWING THE GROWTH/REDUCTION OF A MODE AROUND 1500,

-1 CORRELATION OF THE 770 AND 808 CM MODE...... 52

FIGURE 5-12: DISPLAYED IS AN ENHANCEMENT OF LOW, MID, AND HIGH RANGE VIBRATIONAL BANDS THAT PROVIDE

INFORMATION ABOUT DIFFERENT STRUCTURAL COMPONENTS OF NETWORK GLASSES...... 55

FIGURE 5-13: THE LEFT PANEL (A) IS THE MID-RANGE SPECTRUM OF SODIUM BORATE [17] GLASS AND ON THE RIGHT

PANEL (B) IS THE MID-RANGE SPECTRA OF BARIUM BORATE GLASS; BOTH PROVIDE INSIGHT INTO MEDIUM RANGE

STRUCTURE. NOTE THAT THE RED STAR INDICATES COMPOSITIONS OF STAINED GLASSES...... 56

FIGURE 5-14: DECONVOLUTION OF THE X = 15 MOL% AND 30 MOL% ([BAO]X[B2O3]100-X) LINE SHAPE IN RAMAN

EXPERIMENT...... 58

FIGURE 5-15: SCATTERED COMPONENTS OF AN ISOTROPIC AND ANISOTROPIC MOLECULE; THIS ILLUSTRATION WAS

BORROWED FROM [44]...... 59

FIGURE 5-16: POLARIZATION MEASUREMENTS PREFORMED ON THE X = 15 MOL%. THE BLACK CURVE IS THE

SPECTRA WITH OF VERTICALLY POLARIZED BACKSCATTERED LIGHT AND THE BLUE REPRESENTS THE SPECTRA

OBTAINED CAPTURING THE “HORIZONTALLY” POLARIZED LIGHT. THIS IS DIRECT EVIDENCE OF THE EXISTENCE

OF THIS TRIAD OF MODES WHOSE REDUCTION OF INTENSITY INDICATE A SYMMETRIC STRETCH MUCH LIKE THE

BOROXYL RING (BR) DEMONSTRATED BY GALEENER AND THORPE [53, 54]...... 60

FIGURE 5-17: QUANTIFICATION OF GROWTH/REDUCTION OF VIBRATIONAL FREQUENCIES ASSOCIATED WITH THE

BOROXYL RING, TETRA-, DI-, AND TRI- BORATE MIXED RINGS...... 61

-1 FIGURE 5-18: THE 808 CM MODE DEMONSTRATES RED-SHIFTING AFTER 15 MOL% OF BAO HAS BEEN ADDED TO THE

BASE MATERIAL. THIS SUGGESTS THAT THE BR REMAINS UNMODIFIED UNTIL WE REACH 15 MOL% OF BAO. ... 62

-1 FIGURE 5-19: THE 770 CM MODE, ASSOCIATED WITH THE TETRA-BORATE GROUP, INITIALLY BLUE-SHIFTS AND

THEN RED-SHIFTS AFTER REACHING THE STRESS THRESHOLD 24 MOL% OF BAO, DELINEATED BY THE

INTERMEDIATE PHASE...... 62

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-1 FIGURE 5-20: THE 740 CM MODE ASSOCIATED WITH THE DI-BORATE SHOWS A BROAD MAXIMUM IN THE

INTERMEDIATE PHASE 24 MOL%≤ X ≤32 MOL% WHERE X INDICATES MOL OF BAO...... 63

-1 FIGURE 5-21: THE 705 CM MODE SHOWS A MAXIMUM IN THE INTERMEDIATE PHASE OF COMPOSITIONAL SPACE. ... 63

FIGURE 5-22: THE TOP PANEL IS THE GLOBAL VIEW OF THE NORMALIZED SCATTERING STRENGTH OF THE BOSON

MODE. THE BOTTOM PANEL INDICATES THE MONOTONIC REDUCTION WHICH MAY BE DUE TO THE LOSS OF 2D

- CHARACTER UPON THE CREATION OF BO4 UNITS, OF THE BOSON MODE SCATTERING STRENGTH IN THE

STRESSED RIGID REGION OF COMPOSITIONAL SPACE...... 64

FIGURE 5-23: LEFT - THE PRESENT RESULTS FROM IR –REFLECTANCE MEASUREMENTS; RIGHT - IR DATA FROM

-1 - YIANNOPOULOUS [25]. NOTE THAT THE PEAK AROUND 1600 CM IS INDICATIVE OF B R3 AND B R2O

TRIANGULAR UNITS NOT FOUND IN THE WORK OF YIANNOPOULOS [25]...... ∅ ∅ 66

FIGURE 5-24: IN RED ARE THE RATIO OF 4-COORDINATED BORON TO 3-COORDINATED BORON (B4/B3) INDICATED

BY YIANNOPOULOS [25]. IN BLUE IS THE B4/B3 RATIO OF THE PRESENT WORK. AR IS THE SYMBOL SUGGESTED

BY YIANNOPOULOS [25] TO INDICATE THAT THESE RATIOS WERE OBTAINED BY TAKING THE “AREA” UNDER THE

-1 -1 SPECTRAL BANDS 800-1200 CM (B4) AND 1200-1600 CM (B3)...... 67

FIGURE 5-25: THE NUMBER OF FOUR COORDINATED BORON (N4) DERIVED FROM FTIR DATA OF THE PRESENT WORK

(BLUE) AND YIANNOPOULOS [25] (RED). NOTICE THAT THE PRESENT WORK FOLLOWS THE MEAN FIELD

PREDICTION. THE RED MAY BE LOWER DUE TO THE FACT YIANNOPOULOS USED TWO DATA POINTS FROM NMR

DATA TO SCALE THE FTIR DATA, WHERE AS IN THE PRESENT WORK I DISCOVERED THREE SIMILAR

COMPOSITIONS AND THUS USED THREE DATA POINTS...... 68

FIGURE 6-1: GLASS TRANSITION TEMPERATURE FOR THE PRESENT EXPERIMENTAL STUDIES (BLUE) AND THE WORK

OF LOWER ET AL. [63]. NOTICE THAT THE PRESENT WORK SHOWS HIGHER TG’S AT LOWER MOLAR

CONCENTRATIONS OF BAO...... 72

FIGURE 6-2: RAMAN SPECTRA OF THE X = 25 MOL% SAMPLE DISPLAYING VIBRATIONAL MODES MANIFESTING FROM

LOCAL BO4 STRUCTURES, MEDIUM RANGE STRUCTURE (SYMMETRIC STRETCH OF MIXED RING TETRA.-B), AND

EXTENDED RANGE STRUCTURE SUCH AS THE BOSON MODE...... 74

FIGURE 6-3: VIBRATIONAL REGIME OF THE SYMMETRIC STRETCH OF THE BOROXYL AND MIXED RINGS. THIS FIGURE

SHOWS THE GROWTH/REDUCTION, RED-SHIFTING OF MIXED RINGS AS WE PROGRESS UP THE VERTICAL AXIS

WHICH CORRESPONDS TO THE INCREASE OF MODIFIER CONTENT (MOL% BAO OR NA2O). THE LEFT PANEL(A) IS

THE WORK ON SODIUM BORATES VOINAROOBAN [17] AND THE PANEL (B) ON THE RIGHT IS THE CURRENT WORK

ON BARIUM BORATES...... 75

FIGURE 6-4: POLARIZATION MEASUREMENTS PREFORMED ON THE X = 15 MOL%. THE BLACK CURVE IS THE SPECTRA

OF VERTICAL POLARIZATION AND THE BLUE REPRESENTS THE SPECTRA OBTAINED CAPTURING THE

“HORIZONTALLY” POLARIZED LIGHT. THIS IS DIRECT EVIDENCE OF THE EXISTENCE OF THE TRIAD OF MODES

WHOSE REDUCTION OF INTENSITY INDICATES A SYMMETRIC STRETCH MUCH LIKE THE BOROXYL RING (BR) AND

DEMONSTRATED BY GALEENER AND THORPE [53, 54]...... 77

FIGURE 6-5: ILLUSTRATION OF THE BOROXYL RING WHICH CONSTITUTES THE PLANAR STRUCTURE OF THE BASE

GLASS. THE BLACK ARROWS INDICATE THE VECTOR DISPLACEMENT OF THE OXYGEN ATOMS DURING A

-1 SYMMETRIC STRETCH EXCITATION CONTRIBUTING TO THE 808 CM RAMAN ACTIVE MODE...... 78

FIGURE 6-6: NON-LINEAR BEHAVIOR (SLOPE CHANGES AT 60 MOL%) OF THE A1’ SYMMETRIC STRETCH VIBRATION

IN VARIOUS BORATE SGS. THIS FIGURE WAS ADAPTED FROM VIGNAROOBAN’S THESIS [17]. THE RED DATA

POINTS ARE THE SG PRESENT IN THE BA-BORATES (PRESENT EFFORT) WHILE THE BLUE DATA POINTS ARE ON

NA-BORATES...... 79

FIGURE 6-7: QUANTIFICATION OF GROWTH/REDUCTION OF FRACTIONAL MODE SCATTERING STRENGTHS

ASSOCIATED WITH THE BOROXYL RING, TETRA-, DI-, AND TRI- BORATE SG IN NA-MODIFIED (A) AND BA-

MODIFIED GLASSES (B)...... 81

FIGURE 6-8: CRYSTALLINE STRUCTURE OF BARIUM TETRA-BORATE[34]; THE GREEN DASHED LINES REPRESENT

- DIRECTION OF CONSTRAINT BETWEEN THE BARIUM CATION AND THE TWO [BO4] ANIONS...... 82

FIGURE 6-9: THIS SG FORMS THE EXTENDED RANGE STRUCTURE OF CRYSTALLINE BA-DIBORATE [33]...... 83

FIGURE 6-10: LEFT IS THE PRESENT RESULTS FROM IR –REFLECTANCE MEASUREMENTS, RIGHT-IR DATA FROM

-1 YIANNOPOULOUS [25]. NOTE THAT THE PEAK AROUND 1600 CM IS INDICATIVE OF TRIANGULAR UNITS OF B 3

- AND B 2O ...... ∅88

FIGURE 6-11:∅ IR RESPONSE DEDUCED B MODIFIED FRACTION, B4/B3 PLOTTED AS A FUNCTION OF BAO MOLAR

CONTENT IN THE PRESENT GLASSES (BLUE CIRCLES) AND IN THE GLASSES OF YANNOPOULOS ET AL. [25] (RED

CIRCLES)...... 90

-1 FIGURE 6-12: FTIR ABSORPTION DATA TAKEN FROM VIGNAROOBAN [17]. THE PEAK AROUND 3200 CM GROWS IN

THE WET SAMPLE...... 90

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FIGURE 6-13: FULL RANGE OF ABSORPTION DATA FOR THE X = 15 MOL% AND X =20 MOL% SHOWING ABSENCE OF

B-OH ABSORPTION PEAK...... 91

FIGURE 6-14: EQUILIBRIUM PHASE DIAGRAM OF THE B2O3-BAO BINARY[39] SHOWING THAT THREE DISTINCT

CRYSTALLINE COMPOUNDS FORM AT X = 20 MOL% (BA-TETRABORATE) , X = 33 MOL% (BA-DIBORATE) AND X =

50 MOL% (BA-META BORATE). IN ADDITION, AN IMMISCIBLE RANGE EXISTS BETWEEN X = 0 AND X = 15 MOL%

OF BAO WHERE MELTS DO NOT ALLOY...... 92

FIGURE 6-15: THE LEFT FIGURE SHOWS AN ATTEMPT TO SYNTHESIZE A GLASS AT 4 MOL% OF BAO , WHILE THE

RIGHT FIGURE SHOWS A SIMILAR ATEMPT AT X = 15 MOL% OF BAO. NOTE THAT A HOMOGENOES GLASS FORMS

X = 15 MOL% OF BAO, BUT A STAINED AND HETEROGENEOUS GLASS FORMS AT X = 4 MOL%. THE

HETEROGENOUS PRODUCT AT X = 4 MOL% IS COMPOSED OF BAO INCLUSIONS IN A B2O3 GLASS...... 93

FIGURE 6-16: TOP LEFT (A) AND RIGHT (C) SHOW THE GLOBAL TREND IN SCATTERING STRENGTH OF THE BOSON

MODE FOR THE PRESENT WORK ON (BAO)X(B2O3)100-X AND THE WORK CONDUCTED BY BARANOV ET AL. [71]

ON (NA2O)X(B2O3)100-X, RESPECTIVELY. THE BOTTOM PANELS SHOW THE BOSON MODE IN THE STRESSED–

RIGID REGIME CORRELATING AMONG BOTH SYSTEMS. MOREOVER, THE BR MODE SHOWS CORRELATIVE

REDUCTION IN THE PRESENT WORK...... 98

FIGURE 6-17: CORRELATION OF THE BOROXYL MODE AND THE BOSON MODE IN SODIUM BORATES FROM THE WORK

OF VIGNAROOBAN [17]. NOTICE THE SIMILARITIES IN THE SLOPE TO THE PRESENT WORK AND THAT OF

BARANOV...... 99

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List of Acronyms

IP - Intermediate Phase

BBG - Barium Borate Glass

M-DSC - modulated Differential Scanning Calorimetry

NBO – Non Bridging Oxygen

FWHM – Full Width at Half Maximum

BR – Boroxyl Ring

RW – Reversibility Window

FTIR – Fourier Transform Infrared Spectroscopy

CCD – Charged Coupled Device

TCT – Topological Constraint Theory

NMR – Nuclear Magnetic Resonance Spectroscopy

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Chapter 1 Introduction

1.1 Background

Glassmaking can be traced back to 3500 BC in Eastern Mesopotamia [1], and glass as such, has

been part of human civilization ever since. Glasses have been used for both decorative and

functional purposes. For instance, early glassmaking was used for the production of jewelry and

currency [1]. However, more modern production of glass have been used for applications such

as kitchen appliances, which we find in non-stick enameling and see-through cookware[2], or as

robust displays, such as robust cell phone displays[3].

Glasses are exotic physical manifestations of super-cooled liquids which maintain an off- equilibrium phase of matter. These materials pose a formidable challenge for physical explanation as they demonstrate exponentially complex relaxation (a result of broken ergodicity) and, due to their random nature, an infinite unit. According to American Physicist and Nobel laureate, Phillip Anderson, the nature of the glass transition may be “The deepest and most interesting unsolved problem in solid state theory” [4].

1.1.1 Topological Constraint Theory

The 1980’s marked the beginning of modern-day glass science when physicist Jim C. Phillips and Mike Thorpe introduced the world to a topological theory based on valence force field constraints [5, 6]. In short, this theory describes a glass network globally in terms of characteristic local structures via near-neighbor interactions which are composed of bond- stretching and bond-bending constraints.

This theory enumerates mechanical constraints due to two-body (bond-stretching) and three-

body (bond-bending) forces by calculating the independent number of bonds and bond angles.

For instance, let’s consider a network which contains N atoms. In this group, the ith atom has

𝑖𝑖 near-neighbors (due to its valence number) which can accommodate - bonds. For 3𝑟𝑟- dimensional systems we have the following: 𝜎𝜎

= 2 𝑟𝑟𝑖𝑖 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆ℎ𝑖𝑖𝑖𝑖𝑖𝑖 = 2 3

𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑟𝑟𝑖𝑖 − 1 ( ) = 𝑁𝑁 + 2 3 2 𝑟𝑟𝑖𝑖 𝑛𝑛𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 �� 𝑟𝑟𝑖𝑖 − � 𝑁𝑁 𝑖𝑖=1

The work of Phillips and Thorpe [7] predicted a singular elastic phase transition when nc equals

the degrees of freedom per atom (or number orthogonal directions which make up a basis that

completely describe displacements of an atom). In a 3-D network, an atom has three independent coordinates, x,y and z, that describe translational motion, and nc = 3. This phase transition describes the effect of constraints on a flexible network, prevalent at nc < 3, which

spontaneously becomes rigid when nc > 3. Thorpe demonstrated that when rigidity percolates (at

the nc = 3 transition), the number of zero-frequency (floppy) modes vanish as described in a mean-field picture. Moreover, at this rigidity-percolation threshold, constituent atoms of the network structure are optimally constrained, and can be pictured in the same way as nodes of trusses that make a bridge. In fact, this work was inspired by the stability criterion applicable to macro-structures as described by JC Maxwell [8] in a paper published in 1864.

Often, when using topological constraint theory, the network connectivity is described in terms

of the mean coordination number , which can be viewed as the average number of bonds per

atom. When the average number of constraints/atom is equal to 2.40, a 3-D disordered network

is optimally or “isostatically” constrained, which marks the rigidity percolation threshold (or

elastic phase boundary). When the average connectivity is less than 2.40, the network is

described as “flexible”, and when greater than 2.40, the network becomes “stressed-rigid”.

3 = 3 = + 2 2 〈𝑟𝑟〉 𝑁𝑁 𝑛𝑛𝑐𝑐 〈𝑟𝑟〉 − 𝑁𝑁 5 12 = 6 = = 2.4 2 5 〈𝑟𝑟〉 ⇒ 〈𝑟𝑟〉 1.1.2 Revolutionary Change in Topological Constraint Theory

Cincinnati, after being invested in Topological Constraint Theory since the 1980’s, would make

a monumental discovery [9] that would forever change the way glass scientists describe the

elastic threshold predicted by TCT.

Research at the University of Cincinnati utilized state of the art M-DSC technology and Raman

spectroscopy to examine the nature of elastic threshold of chalcogenide glasses. Most

interesting, the first piece of work conducted on Silicon-Selenide uncovered the true nature of

the elastic phase-space of glasses [10]. This work discovered that an isostatically-constrained state occurs over a finite range of compositions, known as the “Intermediate Phase”, and not at a singular composition as predicted by TCT. Ever since this monumental discovery, work done at the University of Cincinnati has leveraged the “Intermediate Phase” to delineate the elastic phase space and understand nature of glasses in terms of three distinct topological phases which are the

“flexible”, “intermediate”, and “stressed-rigid” phases.

1.2 Relevance of Borate and Silicate Glasses

Borate-, Silicate-based glasses (or a mixture of the two) provide industrial grade glasses that find

application as optical glass [11], nuclear waste management [11], and ion-exchanged

strengthened glasses that provide robust display applications [3].

Oxide glasses consist of [12-16] a base material that function as a network former such as (SiO2,

B2O3, P2O5, GeO2). The base material is modified by either an alkali-oxide (Na2O, K2O) and/or alkaline-earth oxide (CaO, BaO), whereas, a simple modified borate could consist of sodium borate (Na2O)x(B2O3)100-x [17]or (BaO)x(B2O3)100-x[18]. A wide variety of multi-

component glasses exist in academia and industry. One example of an alkaline-earth silicate

alloyed with an alkaline-earth Borate would be the system, [BaO]x [68 (SiO2) 32(B2O3)]100-x. It

is widely known that physical properties such as Tg, thermal expansion, and molar volumes in

modified borate glasses, vary non-monotonically (display maxima or minima) upon the addition

of the modifier (a behavior that is known as the Borate anomaly). Extensive work done in

characterization of Borate glasses by methods such as X-ray pair-distribution functions

(diffraction) [19, 20], NMR [21-24], Raman Scattering [14, 25], density [25], and calorimetric measurements display these anomalies. However, it is also fair to say that the “molecular origin” of this anomaly remains largely speculative. Recently, Vignarooban [17, 18] comprehensively investigated the sodium-borate glass system utilizing Thermal, Optical, and Electrical methods.

Utilizing these techniques, he was able to show that the maximum of Tg , which occurs near 33

mol% of Na2O, occurs in an isostatically-rigid elastic-phase. What is more, this phase actually

separates stressed-rigid glasses, formed at x < 20 mol% of soda, from flexible ones formed at x >

40 mol% of soda. These results, for the first time, provided a molecular origin of the so-called

“Borate anomaly”.

Rigidity Theory [26] provides the basis for the existence of the three elastic phases that describe the nature of glassy networks. Flexible networks have floppy modes [27] which function as a new degree of freedom and relaxation channels for networks as they evolve or age in time. In such networks, bond-stretching and bond-bending constraints together (nc) do not exhaust the available degrees of freedom, nf (3 for 3D networks). The difference (nf - nc) gives the count of floppy modes/atom. On the other hand, Stressed-rigid networks possess nc>nf and are viewed as over-constrained. In such networks, stress creating “redundant bonds” steadily restricts the network of atoms and the system gets “stuck”. Such networks relax as well but usually at high temperatures and therefore also display aging. Intermediate phase (IP) networks are composed of networks that are optimally constrained (nc = nf) and have remarkable properties. These networks are stress-free[28], weakly aging,[29] form compacted networks (lowest molar volumes), and can adapt by rearranging in a multitude of energetically equivalent configurations. Networks such as these have attracted much attention in device applications [30]

(MOSFET) and large scale applications such as window glass [31] and flat panel displays [32].

In our quest to understand glassy solids, we question why it is so important to identify the three elastic phases. With the enormity of glass compositions that could possibly be synthesized in multi-component glasses, tuning physical properties for a given application by trial and error (in an Edisonian manner) would be time consuming, and therefore rather expensive. On the other hand, using Rigidity Theory, the powerful tool of enumerating bonding constraints, one can describe the general elastic-phase space and delineate regions of interest. The Topological approach to the design of glassy solids by tuning chemical compositions, according to their mechanical constraints, is no longer a dream. It has been demonstrated in the celebrated case of

Gorilla glass [3] by Corning in New York. This glass encases nearly billions of cell phones

which are used today.

The three generic elastic phases (flexible, isostatically rigid, and stressed-rigid) is expected to

exist in alkaline-earth borates and alkali-earth Boro-silicates, much as they do in alkali-borates

[17, 18]. The present Dissertation, to the best of our knowledge, is the maiden effort to look for

these phases. Thus, I have conducted a detailed examination of the pseudo-binary Ba-Borate

[(BaO)x(B2O3)1-x] and pseudo-ternary Ba-borosilicate [BaO]x [(SiO2)68(B2O3)32]1-x glasses over wide compositional ranges of x. My broad objective is to establish the three elastic phases in these two borate glass systems, as well as gain crucial insights into how aspects of the molecular structure of these species connect with their functionality or physical properties.

This work leverages previous experiments [17, 18] conducted on sodium borate glasses, crystalline reference compounds [33, 34] at select compositions, topological constraint theory[5,

6], and the findings of Molecular Dynamic (MD) simulation[35], to elucidate the three elastic phases and the profound importance of medium-range structure.

Three challenges exist when examining the molecular structure of oxide glasses. First, bulk oxide glasses must be dry – reducing water impurities as much as possibly is extremely important. Second, batch compositions must be homogeneous so that there is no measurable compositional variation across the batch synthesized. This requirement imposes limitations on the batch sizes and heating methods utilized to combine and react sample precursors. Finally, one has to identify the appropriate heat treatment, post melt-quench, to obtain bulk glasses that are free from stress and frozen by virtue of the quench method before undertaking physical measurements. In the present work, in order to produce homogeneous glasses, we have utilized

an induction melting process to react precursors. Pseudo-binary glasses, (BaO)x(B2O3)100-x,

have been utilized to understand the role of the alkaline-earth modifier in borate glasses. In the investigated Barium modified borosilicate glasses, (BaO)x(68 SiO2 32B2O3)100-x, splat or roller-

quenching was utilized at appropriate compositions to extend homogeneous glass formation in

the Barium Borate and Barium Borosilicate systems.

The general approach utilized for the production and proper characterization of dry homogenous

melts can be compactly described with a flow chart.

PROCESS FLOW CHART 1

Literature research

Bulk Glass Synthesis

Raman Scattering

No Homogeneous?

Yes

No Amorphous?

Yes

Mode ID, Compositional m-DSC FTIR Density/Volumetric Trends Meas.

Tg, ∆H , ∆C nr p No Low Variance? Continuity of No Trends Yes Yes Yes No Defined Physical/Chemical Reversibility Understanding of Network Glass Window?

1.3 Findings

I present findings reported for the first time. Which are as follows:

• Complete Raman Scattering data on the Ba-Borates and Ba-Borosilicates over wide

composition ranges were obtained and the observed Line shapes quantitatively analyzed

to decode the medium-range structure in barium borate and borosilicate glasses.

• The elastic phase diagram studied using calorimetric (reversibility window), volumetric

(mass density), and optical methods (Raman Scattering and IR reflectance), has been

mapped.

• Demonstrate synthesis of dry B2O3 utilizing Boric Acid as a precursor.

• Independently identify the location of the intermediate phase using optical and thermal

techniques.

Raman vibrational modes in crystalline compounds (i.e. phonons) are well defined and

enumerated by methods such as the G-F matrix method; furthermore, the G-F matrix method

[36] and other derivatives of structural symmetry, are developed in the Raman section of this

dissertation. Intense peaks present in crystalline compounds are also evident in glassy materials

but display peak broadening [37]. Through work conducted on sodium borates [17, 18], we have

remarkable insight into the evolution of Raman-line shapes and its application to structural

phenomena. figure 1-1 is an illustration of an important crystalline solid, Barium Tetra-borate

((BaO)20(B2O3)80), which forms in the present system under investigation; also displayed are eigenvectors of the vibrational modes of different constituent components of these structural compounds. The indicated vibrational modes are evident in the Raman line-shapes garnered in

this experimental effort and provide remarkable insight into structural manifestations upon the

addition of the barium modifier.

(2) Intra-Mixed- Ring for 770cm-1 Ba-Tetra-borate

FIGURE 1-1: The crystalline structure of Barium-Tetra-borate (BaO)20(B2O3)80 is shown[34]. Note the black line marks the boundary of the unit cell. In this cell the ratio of Barium to Boron is 4 Barium (green) to 16 Boron (light-pink), which is correct for one BaO to four formula units of B2O3, i.e. (BaO)20(B2O3)80. Also displayed (1) are the eigenvectors for the symmetric stretch (A1’) of the tetrahedral BØ4 (where indicates bridging oxygen) unit (center frequency near 500 cm-1) [17]and (2) the symmetric stretch of the mixed-intra-ring B-O bonds (center frequency near 770 cm-1[36]. ∅

As an example of Raman Scattering line-shape analysis, we provide in figure 1-2 the observed

spectrum at x = 15 mol% in the Ba-Borate glass. The richness of the line-shape permits, among

other things, to decode the intermediate range order from the intra- mixed ring modes observed

in the 600-900 cm-1 range. 9

BR (BaO)x(B2O3)100-x x = 15%

TetraB B--O- Stretch Extra-Ring Modes Intra-Ring Modes DiB Boson Mode Boson BO4(A1) BO4 (F2) X 5 Intensity (Arb. Units) Intensity

200 400 600 800 1000 1200 1400 1600 Raman Shift (cm-1)

FIGURE 1-2: Shows Raman scattering of a (BaO)x(B2O3)100-x glass, at x = 15 mol%. In the low frequency (0-200 cm-1) regime we observe the Boson mode. In the 200- 600 cm-1 we -1 observe bending modes of BO4 tetrahedra. In the 600- 900 cm range we observe the intra- mixed rings characteristic of the Intermediate range order. At the high frequency range, ( 100- -1 - 1600 cm ) coming from F2 modes of BO4 tetrahedra, B-O mode from extra-ring BO bonds bonded to the barium cation.

Chapter 2 Sample Synthesis

2.1 Barium Borates

Approach

Synthesis of glasses, due to their disordered state, has far too often hindered the type of control

practiced when synthesizing single crystals of the same stoichiometric compound. However,

under the direction of Dr. Boolchand at the University of Cincinnati, tight control of humidity,

precursor purity, and sample homogeneity are practiced to ensure glassy products which

demonstrate the true intrinsic physical properties of these glassy materials.

Homogeneity is continually tested with Raman and FT-Raman profiling. The intrinsic-nature of glass is monitored through calorimetric probes, optical probes, and volumetric measurements.

Sample compositions are chosen in an iterative fashion in order to map out properties such as the boundaries of the elastic phases, compositional dependence of molar volumes and the nature of

Tg, and to lower statistical variance of characterizing measurements. Remarkably, one finds

physical properties examined as a function of composition change almost “discontinuously” near

elastic thresholds provided the glass compositions are homogeneous and dry [38].

Starting Materials

Barium Carbonate (99.999% Sigma Aldrich) and Boric Acid (99.9% Sigma-Aldrich) were chosen instead of their oxide or anhydrous form in order to lower the synthesis temperature which permits a lower activation barrier and reduction of Boron evaporation. In order to ensure appropriate chemical activity (de-carbonation of precursors, reduction of the evaporation of boron and the reaction of starting materials), target temperatures for de-carbonation and

11 synthesis are chosen based on the pseudo-binary diagram. The BaO-B2O3 pseudo-binary phase diagram is displayed in figure 2-1 [39] phase diagram and provides a useful guide to liquidus temperatures, important in synthesis, and vary with composition .

Phase Diagram

The pseudo-binary phase diagram, figure 2-1, displays thermodynamically distinct phases in equilibrium as a function of temperature and composition [39]. This diagram shows pure B2O3 on the extreme right and pure BaO on the extreme left. Solids present at different regions are displayed by the stoichiometric compound listed. Regions of liquids, indicated by the letter L, are delineated with horizontal lines and the bold liquidus curves. It is important to note that in

o the region beyond 70 weight % B2O3, shaded green, at a temperature below 878 C, we form combinations of a liquid (L1) and Barium-Tetraborate solid (BaO)20(B2O3)80. Above this temperature, we enter a region of two immiscible liquids (L1+L2). This particular region is of interest to us because it coincides with a region where glass-forming tendency ceases to exist.

Additionally, Raman Scattering studies corroborate this and provide insights into the structural manifestations that limit glass-forming tendency in this psuedo-binary.

FIGURE 2-1: Pseudo-binary phase diagram of the system BaO-B2O3. The binary phase diagram shows thermodynamic phases; phases are in equilibrium and are delineated by solid lines [39].

Temperature Process

Sample precursors were combined and mixed in a Platinum Rhodium Crucible. Precursors were

then de-carbonated for 30 minutes. Following de-carbonation, sample melts were then synthesized at a temperature of 1200oC for several hours until the melt appeared to be homogeneous. Following synthesis, the temperature was increased to 1300oC to lower melt

viscosity and permit complete evacuation of the melt from the crucible. Samples of low modifier

content, 0 mol%

of Nitrogen gas; Nitrogen gas was utilized to lower the Relative Humidity (RH) of the quenching

ambient. The sample of the higher modifier content x = 40 mol% was quenched using a twin-

roller technique. The twin-roller facility, shown in figure 2-2, provides even pressure to the melt and permit maximum surface area contact with the heat sink used to quench the sample.

FIGURE 2-1: Image of a Platinum-Rhodium Crucible during heating with an induction coil. Notice at the bends we have bright bands indicating local maxima in current density and temperature.

2.2 Barium Borosilicates

Thirty gram size batches were synthesized using reagent-grade H3BO3 (Sigma-Aldrich, 99.8%),

BaCO3 (Sigma-Aldrich, 99:999% trace metal basis), and SiO2 (Sigma-Aldrich, 99:995% trace metal basis). The carbonate and acid based precursors were utilized to introduce targeted binary compounds due to their lower decomposition temperature, which facilitates a lower melt mixing temperature. Mechanically mixed powders were de-carbonated and melted in a Pt90Rh10

crucible for several hours utilizing an inductively heated furnace. Inductive heating promotes 14

sample homogenization through direct and localized heat. Here, heating occurs via eddy

currents in materials, in contrast to diffusion limited mixing in a conventional furnace which uses

either surface-contact or radiant heating. Following inductive heating, the samples were

equilibrated in a box furnace for several hours at a temperature of 1600oC and then quenched on a stainless steel plate with a constant flow of Nitrogen gas; Nitrogen gas was used to lower the

RH of the ambient air.

FIGURE 2-2: Image of the twin-roller quenching unit. Twin Rollers permit maximum sample surface area exposure and even pressure upon quenching. This permits accelerated quenching to freeze in liquid state and prevent crystallization.

Chapter 3 Thermal Characterization

3.1 Background for Thermal Characterization Techniques

3.1.1 The Dynamic Glass Transition

When measuring the constant pressure heat capacity, one notices a change when progressing

from a glassy to liquid state. Dynamically, this step in Cp demonstrates, at experimental time scales, that the glassy state is one of structural arrest or perceived loss in accessible degrees of freedom. In other words, the ergodicity time (time needed for the system to explore a fraction of the ambient phase space) is larger than our experimental time [40].

The glassy state can be reached by super-cooling a liquid (quenching) so that we by-pass crystallization. figure 3-1 shows the relationship of entropy, volume, and enthalpy along a temperature trajectory taken when proceeding from the liquid to the glassy state [41].

The trajectory of the teal curve is dependent on the rate at which we quench the glass or the experimental time which we use to measure the glass transition. figure 3-2 has been taken from

A Cavagna to illustrate this point [40]. When progressing from the red curve to the yellow curve, we lower the rate at which we cool the liquid. Slower quench rates permit enough time for kinetic events to occur such as allowing the system to lower its configurational-entropy which results in a lower glass transition temperature.

Tm Tx Liquid Tg

W. Kauzman 1948, Tg~2/3Tm

Glass Enthalpy, Volume, Entropy Volume, Enthalpy, Enthalpy, Volume, Entropy Volume, Enthalpy, Enthalpy, Volume, Entropy Volume, Enthalpy,

Crystal

Tg Tx Tm Temperature

FIGURE 3-1: The black curve in this figure represents the typical heat flow as we progress from sub-Tg to a temperature above the melting point of a glass. The teal curve shows the general property of the solid (e.g. Enthalpy, Volume, and Entropy) of a melt when super-cooled or quenched from the liquid to a glassy state. The characteristic transition temperatures, progressing from left to right are the glass, crystallization and melt temperatures. The glass transition is really not a transition at all but an artifact of observational time in experiment [40]. Due to its smooth characteristic, this transition is defined as a second order

transition (discontinuity in second derivative at the heat capacity event). First order transitions

are transitions of states that have infinite slope at the inflection point (discontinuity in first

derivative) which is identified as the transition temperature.

high quench rate

Tg Tg low quench rate

FIGURE 3-2: Characteristic temperature behavior of the entropy of prototypical glasses. The zstandard temperature references such as the melting point Tm the Goldstein temperature Tx, glass transition temperature (Tg), and Kauzmann Tk temperature are displayed. The trajectory of the glassy state shown in color is dependent on quench temperature or experimental temperature at which the measurement takes place[40].

Demonstrated by the Sulfide- and Selenide- work conducted at the University of Cincinnati, the

sharpness of this transition is directly related to structural correlations in the glassy network.

3.1.2 Modulated DSC

Modulated Differential Calorimetry measures the heat profile of a glass sample by comparing

it’s responsivity for heat flow to that of a empty reference pan. However, M-DSC provides an

additional sophistication via modulation of heat flow along constant temperature ramp, and

permits decomposition of the total heat flow into its kinetic and heat capacity components. M-

DSC essentially provides for two separate heating experiments; one that tracks the instantaneous

modulated heat flow, and the other that tracks the average heat flow.

= + ( , ) 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝐶𝐶𝑝𝑝 𝑓𝑓 𝑇𝑇 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑 ∶ 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑑𝑑𝑑𝑑 : .

𝑝𝑝 𝐶𝐶 𝑇𝑇ℎ(𝑒𝑒 𝐻𝐻, 𝐻𝐻𝐻𝐻𝐻𝐻): 𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝐶𝐶 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑤𝑤ℎ 𝑖𝑖𝑖𝑖ℎ 𝑖𝑖𝑖𝑖 𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ 𝑒𝑒 (𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ). ℎ𝑒𝑒𝑒𝑒𝑒𝑒

𝑓𝑓 𝑇𝑇 𝑡𝑡 𝑇𝑇ℎ𝑒𝑒 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 − 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ℎ𝑒𝑒𝑒𝑒𝑒𝑒

L. Thomas of TA Instruments suggest[42], the total heat flow is calculated by determining the

average of the modulated heat flow signal using Fast Fourier Transform (matrix method of

discrete Fourier transform components which transform time space into frequency space) of the

modulated signal sine wave which is sampled every .1 second [42].

FIGURE 3-3: The Total heat flow in red is calculated as the average behavior of the modulated heat flow shown in black.

The Reversing Heat flow signal is calculated from the same FT-based averaging procedure

performed on the modulated heat flow to calculate the amplitude of heat flow and heating rate.

𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑅𝑅𝑅𝑅𝑅𝑅 𝐶𝐶𝑝𝑝 𝐾𝐾𝐶𝐶𝑝𝑝𝑅𝑅𝑅𝑅𝑅𝑅 = 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 ( )

𝑝𝑝 𝐾𝐾𝐶𝐶 𝑅𝑅𝑅𝑅𝑅𝑅 𝐶𝐶 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶= 𝐶𝐶 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶× 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅

𝑝𝑝 Finally, the Non-reversing𝑅𝑅𝑅𝑅𝑅𝑅 𝐻𝐻 𝐻𝐻𝐻𝐻𝐻𝐻heat𝐹𝐹 𝐹𝐹𝐹𝐹flow𝐹𝐹 (the𝑅𝑅𝑅𝑅𝑅𝑅 remaining𝐶𝐶 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 component𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 of𝑅𝑅 𝑅𝑅𝑅𝑅𝑅𝑅heat) is obtained after

subtracting the Reversing Heat flow component. The peak in the endotherm occurs at the point

of transition in our second-order phase change. This endotherm provides a qualitative measure

of the amount of heat required for configurational-change or change in the phase of the material.

It has been demonstrated through Monte Carlo Simulations and Kingsley Harmonics that this

overshoot is a direct measure of the hysteresis of the glass during heating and cooling cycles.

Glasses in the intermediate phase possess a relatively low overshoot, therefore, it is reasonable to say that they have a non-hysteretic behavior and are thermally reversing as the transition of the

total heat flow tracks that of the reversing heat flow. Glasses in the intermediate phase, show

little to no overshoot in the total heat flow and suggest a state that experiences little

configurational change when approaching the liquid. For illustration purposes, an example of

the endotherm which contributes to the non-reversing heat flow is shown on the x = 17 mol%

sample in figure 3-4.

FIGURE 3-4: The total heat flow can be broken down in to two separate components. The overshoot marked by the green arrow and line indicates the part of heat flow that is associated with latent heat and contributes to the non-reversing heat flow.

3.1.3 Frequency Correction

The inflection point which marks the location of the glass transition is a function of modulation frequency. For instance, at higher frequencies, one will measure a higher glass transition temperature relative to lower frequency measurements. This change is a direct result of the shorter observational time (higher heating/shorter sampling) allowed for the sensing equipment to measure noticeable kinetic events in the specimen. Frequency correction in M-DSC experiments is made by conducting both heating and cooling cycles during each sample run. The heating scan of figure 3-5 displays a kinetic up-shift of the reversing heat flow to higher temperatures due to the finite modulation frequency used. The non-reversing heat flow is thus greater (or polluted) than it should be (1.19 cal/g). Upon cooling, the reversing heat flow shifts to a lower temperatures, yielding an exothermic non-reversing heat flow, in essence yielding the correction term (0.88 cal/g). Thus, the frequency corrected non-reversing heat flow is given as

(1.19 0.88 = .31 cal/g) for the glass sample of figure 3-5.

− 21

FIGURE 3-5: Standard heat-flow signals obtained from M-DSC heating and cooling experiments.

3.2 M-DSC Experiments

M-DSC experiments were conducted using either the Q2000 or Q2920 M-DSC developed by TA

Instruments. Sample mass of 15-20 mg were used to accentuate the glass transitions. The

o modulation of temperature, ±1P C with a modulation period of 100 seconds, were used with a

constant temperature heat ramp of 3oC/min; refer to figure 3-6.

High purity Nitrogen purge gas flowed at 80 cc/min flow rate, was used to maintain a dry and

isothermal environment. Gold pans, which are most suitable for high temperature

measurements, were used for all samples with the exception of the base material. For the base

material, Aluminum hermetically sealed lids and pans were used to conduct these thermal

experiments.

FIGURE 3-6: This is a typical modulated temperature profile which is programmed for all M- DSC measurements.

3.3 M-DSC Results

Figure 3-7 shows a summary of the data obtained from M-DSC experiments on Barium Borate glasses. The top panel shows the frequency corrected glass transition temperature for all samples and displays a non-monotonic behavior. The bottom panel displays the frequency corrected non- reversing change in enthalpy of relaxation ( ) for the same samples and displays a global

𝑛𝑛𝑛𝑛 minimum. Note that the maximum in the ∆glass𝐻𝐻 transition temperature occurs broadly in the

center of the minima.

∆𝐻𝐻𝑛𝑛𝑛𝑛

(Na2O)x(B2O3)100-

FIGURE 3-7: Compositional dependence of the glass transition temperature and change of enthalpy of relaxation of barium borate glasses. 3.4 M-DSC Discussion

The glass transition temperature is broadly recognized as a measure of glass-network global

connectivity. Upon the addition of the network modifier (Na+ or Ba2+), global connectivity

increases through the production of BO4 units. Upon the addition of the network modifier, the creation of -units eventually slows down and the modifier predominately produces Non-

4 Bridging Oxygen𝐵𝐵∅ (NBO).

The observed term in Barium Borates demonstrates a Gaussian like minimum, with the

𝑛𝑛𝑛𝑛 high slope points∆𝐻𝐻 serving to delineate the three elastic phases. One may notice that the sodium

borates of Vignarooban [17, 18] show a first-order rigidity- (x= 20 mol%) and stress- (x = 40

mol%) elastic phase transitions. The differences in the profiles between Na- and Ba-

∆𝐻𝐻𝑛𝑛𝑛𝑛 24

borates are striking. These results are due to structural differences of these systems – I comment

on this issue next.

Diffusion of alkali metals in oxides occurs via migration of the cation by displacements over

short-range distances, i.e. local interstitial sites. The cationic components form continuous

channels as the modifier is increased, as displayed in figure 3-8. The modifier is highly mobile compared to the network formers and possesses a diffusivity that is Arrenhius (thermally activated). Since the cation dominates the mechanism of diffusivity, the kinetics of the glass is dictated by the cation diffusivity. Therefore, since the mass of Ba (137 amu) far exceeds that of

Na (23 amu), it is reasonable for the sodium borate glass to have first order thresholds (faster kinetics) and the barium-borates to have second-order thresholds (slower kinetics).

FIGURE 3-8: a) Oxide network display polyhedral are connected by bridging oxygen; b) modified networks above the percolation threshold 16% by volume of modifier form continuous channels for self-diffusion. Note this figure was taken from [43] and is used for illustration purposes

Chapter 4 Density Measurements

4.1 Background for Density Measurements

Archimedes was able to demonstrate that the weight of liquid displaced by a mass placed in the liquid is equivalent to the buoyancy force exerted on the mass. This principle allows us to determine the density and the molar volume of fully submerged objects indirectly through weight measurements in air and in a liquid.

= 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠−𝑎𝑎𝑎𝑎𝑎𝑎 𝜌𝜌𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝜌𝜌𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠−𝑎𝑎𝑎𝑎𝑎𝑎 − 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠−𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 4.2 Density Experiments of Barium Borate Glasses

Density experiments were conducted with a Mettler Toledo Digital Scale Model College B 154.

The density of the liquid which is displaced (ethyl alcohol) is determined utilizing the measured mass and recorded density of a either Silicon or Germanium. The calculated density of the liquid is verified by using the density derived for the liquid to calculate the density of a second standard.

, , = . , , 𝑚𝑚𝐺𝐺𝐺𝐺 𝑆𝑆𝑆𝑆−𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠−𝑎𝑎𝑎𝑎𝑎𝑎 − 𝑚𝑚𝐺𝐺𝐺𝐺 𝑆𝑆𝑆𝑆−𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠−𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 − 𝜌𝜌𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝜌𝜌𝐿𝐿𝐿𝐿𝐿𝐿 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑜𝑜𝑜𝑜 𝐺𝐺𝐺𝐺 𝑆𝑆𝑖𝑖 𝑚𝑚𝐺𝐺𝐺𝐺 𝑆𝑆𝑆𝑆−𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠−𝑎𝑎𝑎𝑎𝑎𝑎

Three samples of at least 100 mg or more were used to ensure an instrumental uncertainty of

0.25% in density. Samples were first weighed in the air on a mil sized quartz fiber suspended from the pan balance. Afterwards, the sample was unloaded from the quartz fiber and the fiber

26 was submerged in a beaker containing ethyl alcohol. To properly dismiss the buoyant force on the fiber, the digital scale was then tared. Finally, the sample was then loaded onto the fiber and completely submerged in the alcohol to determine the apparent weight of the sample.

FIGURE 4-1: Illustration of volumetric experiments used to determine the density of submerged volumes.

/mol) 30 3

24 MolarVolume (cm 22

20 15 20 25 30

Ba Content x(%)

FIGURE 4-2: Molar volume of barium borate glass obtained from density experiment.

Chapter 5 Optical Characterization

This section starts with a brief review of important concepts and background which are fundamental for understanding Raman/Fourier Transform IR (FTIR) experiments. First, I will describe light and matter interaction from a classical point of view and make some basic arguments about requirements for Raman/IR activity. Next, I will discuss how symmetry aspects of stoichiometric crystals are important, and demonstrate how these compounds provide a diagnostic tool for Raman and IR experiments in glasses. Lastly, this section will conclude with experimental observations and important findings of the present dissertation.

5.1 Background for Optical characterization Techniques

5.1.1 Requirements for Raman Active Modes

Classically, when we place a material body in an external electric field, induced dipole moments are created. These dipole moments absorb energy from the oscillating electric field and emit light in all directions. Raman active modes, as demonstrated below, require a change in polarizability-tensor (which may be looked at as the “deformability” of the electron cloud of the molecule) [44].

The relationship of the dipole moment to the electric field is given by the following equation:

= Equation 5-1 𝝁𝝁��⃗ 𝜶𝜶�𝑬𝑬�⃗ We assume that the electric field is oscillating and therefore has the following form:

= ( ) Equation 5-2 𝑬𝑬��⃗ 𝑬𝑬𝟎𝟎 𝒄𝒄𝒄𝒄𝒄𝒄 𝟐𝟐𝝅𝝅𝝅𝝅𝝅𝝅

Normal vibrational modes are modes of oscillation in which harmonic behavior is demonstrated;

all atoms go through their equilibrium position at the same time and vibrate with the same

frequency. Normal modes in di- and poly- atomic molecules cause changes in electron clouds during compression extension of the molecular species.

To characterize the functional dependence of polarizability with regard to the normal coordinate,

we may expand it using a Taylor Series expansion.

= + + + 𝟐𝟐 𝝏𝝏𝝏𝝏 𝟏𝟏 𝝏𝝏 𝜶𝜶 𝟐𝟐 Equation 5-3 𝜶𝜶 𝜶𝜶𝟎𝟎 𝑸𝑸 𝟐𝟐 𝑸𝑸 ⋯ 𝝏𝝏𝝏𝝏 𝟐𝟐 𝝏𝝏𝑸𝑸 The normal coordinates of the individual atoms that make up the molecule vibrate at frequency

′.

𝜈𝜈 = ( ) Equation 5-4 𝑸𝑸 𝑸𝑸𝟎𝟎 𝐜𝐜𝐜𝐜𝐜𝐜 𝟐𝟐𝝅𝝅𝝅𝝅′𝒕𝒕 Now, substituting this into Equation 5-1 we have:

= = + ( ) ( ) 𝝏𝝏𝝏𝝏 �𝝁𝝁�⃗ 𝜶𝜶𝑬𝑬��⃗ �𝜶𝜶𝟎𝟎 𝑸𝑸𝟎𝟎 𝐜𝐜𝐜𝐜𝐜𝐜 𝟐𝟐𝝅𝝅𝝅𝝅′𝒕𝒕 � 𝑬𝑬𝟎𝟎 𝒄𝒄𝒄𝒄𝒄𝒄 𝟐𝟐𝝅𝝅𝝅𝝅𝝅𝝅 𝝏𝝏𝝏𝝏 = ( ) + ( ) ( ) 𝝏𝝏𝝏𝝏 𝜶𝜶𝟎𝟎𝑬𝑬𝟎𝟎 𝒄𝒄𝒄𝒄𝒄𝒄 𝟐𝟐𝝅𝝅𝝅𝝅𝝅𝝅 𝑬𝑬𝟎𝟎𝑸𝑸𝟎𝟎 𝒄𝒄𝒄𝒄𝒄𝒄 𝟐𝟐𝝅𝝅𝝅𝝅𝝅𝝅 𝐜𝐜𝐜𝐜𝐜𝐜 𝟐𝟐𝝅𝝅𝝅𝝅′𝒕𝒕 𝝏𝝏𝝏𝝏 = ( ) + [ ( ) + ( + ) ] 𝟏𝟏 𝝏𝝏𝝏𝝏 Equation �𝝁𝝁�5⃗-5 𝜶𝜶𝟎𝟎𝑬𝑬𝟎𝟎 𝒄𝒄𝒄𝒄𝒄𝒄 𝟐𝟐𝝅𝝅𝝅𝝅𝝅𝝅 𝑬𝑬𝟎𝟎𝑸𝑸𝟎𝟎 𝐜𝐜𝐜𝐜𝐜𝐜 𝟐𝟐𝝅𝝅 𝝂𝝂 − 𝝂𝝂′ 𝒕𝒕 𝐜𝐜𝐜𝐜𝐜𝐜 𝟐𝟐𝝅𝝅 𝝂𝝂 𝝂𝝂′ 𝒕𝒕 𝟐𝟐 𝝏𝝏𝝏𝝏 The above equation suggest the dipole moment varies with respect to three frequencies,

namely he Rayleigh , Stokes (ν-ν'), and Anti-Stokes (ν+ν'), which are the components that

relate to𝑡𝑡 Rayleigh, Stokes𝜈𝜈 and anti-stokes respectively.

5.1.2 New Picture of Light Interaction

Classically, the light impinging on the surface of a sample was described as penetrating the

sample and directly interacting with the polarization modes of the solid; for example, TO

phonons could absorb a portion of the light [44].

Second-quantization (the quantization of force fields) provides a new framework to describe

particles and their force fields by which they interact. Polaritons are quasi-particles which are composed of phonon and photon interactions and are utilized to describe light-matter interaction.

In this new description, light and TO phonons are coupled into a new set of normal modes known as polaritons. In the next section, I will explore the crossing phenomena of two coupled modes which can be experimentally demonstrated in Raman Scattering and provide a pedagogical example of the “New Picture” of light/matter interaction.

5.1.3 Crossing Phenomena of Two Coupled Modes

In order to easily identify the coupling effect between the photon and phonon modes in the solid, let’s examine a more general case of two lossless resonators and demonstrate how this leads to two separate dispersion branches.

To begin, let’s consider the amplitudes of two lossless resonators with time dependence [45]

(exp( ) and exp( )):

𝑗𝑗𝜔𝜔1𝑡𝑡 𝑗𝑗𝜔𝜔2𝑡𝑡 = 𝑑𝑑𝑎𝑎1 𝑗𝑗𝜔𝜔1𝑎𝑎1 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑎𝑎2 𝑗𝑗𝜔𝜔2𝑎𝑎2 𝑑𝑑𝑑𝑑

Demonstrated by Haus and Haung [45], when coupling is weak, we have the following form:

= + 𝑑𝑑𝑎𝑎1 𝑗𝑗𝜔𝜔1𝑎𝑎1 𝑗𝑗𝜅𝜅12𝑎𝑎2 𝑑𝑑𝑑𝑑 = + 𝑑𝑑𝑎𝑎2 𝑗𝑗𝜔𝜔2𝑎𝑎2 𝑗𝑗𝜅𝜅21𝑎𝑎1 𝑑𝑑𝑑𝑑 Where, as shown by Haus et. al., = = . Notice that the above equations are the same ∗ 12 21 as the uncoupled system when the 𝜅𝜅coupling𝜅𝜅 terms𝜅𝜅 ( and ) are zero.

𝜅𝜅12𝑎𝑎2 𝜅𝜅21𝑎𝑎1 Now, assuming the coupled system has a normal mode with time dependence ( ( )):

𝑒𝑒𝑒𝑒𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗 = +

𝑗𝑗𝑗𝑗𝑎𝑎1 𝑗𝑗𝜔𝜔1𝑎𝑎1 𝑗𝑗𝑗𝑗𝑎𝑎2 = +

𝑗𝑗𝑗𝑗𝑎𝑎2 𝑗𝑗𝜔𝜔2𝑎𝑎2 𝑗𝑗𝑗𝑗𝑎𝑎2 ( ) + = 0

𝜔𝜔1 − 𝜔𝜔 𝑎𝑎1 𝜅𝜅𝑎𝑎2 ( ) + = 0

𝜔𝜔2 − 𝜔𝜔 𝑎𝑎2 𝜅𝜅𝑎𝑎1 ( ) = ( )( ) ( ) 𝜔𝜔1 − 𝜔𝜔 𝜅𝜅 2 � � 𝜔𝜔1 − 𝜔𝜔 𝜔𝜔2 − 𝜔𝜔 − 𝜅𝜅 𝜅𝜅 𝜔𝜔2 − 𝜔𝜔 ( )( ) = ( + ) + ( ) = 0 2 2 2 𝜔𝜔1 − 𝜔𝜔 𝜔𝜔2 − 𝜔𝜔 − 𝜅𝜅 𝜔𝜔 − 𝜔𝜔 𝜔𝜔1 𝜔𝜔2 𝜔𝜔1𝜔𝜔2 − 𝜅𝜅 ( + ) = ± + | | 2 2 2 𝜔𝜔1 𝜔𝜔2 𝜔𝜔1 − 𝜔𝜔2 2 𝜔𝜔 �� 𝜅𝜅 This ideal coupling of two resonators displays two solutions, namely: A symmetrical lower branch and an asymmetrical upper branch displayed in figure 5-1 taken from Haus and Haung

[45].

FIGURE 5-1: Symmetric and asymmetric branches of a coupled resonator[45]. This illustrates the same basic phenomena that describe the coupling of the photon and phonons, which in turn, describes the polaritons - this depicts the new picture of light propagation in Raman active materials.

FIGURE 5-2: Energy dispersion curve displaying the relationship between LO- Polaritons, LO- phonon, and TO-polaritons in crystalline and polycrystalline GaP [46].

Raman experiments of GaP also demonstrate this behavior and provide evidence to the accuracy

of the “New picture” of light-matter interaction [46].

5.1.4 Structure

Crystalline compounds have long range order; unit cells that can traverse space and completely

describe the position of atoms in the extended network. The certainty of atomic positions in

crystalline compounds provides aspects of symmetry that can be leveraged to provide a physical

description of their macroscopic properties. Amorphous compounds do not possess such

extended order, but as indicated by low variance in local structure, the inability to detect phase

separation in electron-microscopy experiments, cannot have order on a scale greater than 100

[37]. However, the strong peaks evident in crystalline compounds also appear in the amorphousÅ

compound but appear broadened and diminished [37]. With this in mind, it is important to

consider aspects of the crystalline structure to provide a basis of understanding in the amorphous

materials.

Following the same understanding as with previous experiments conducted on Sodium Borate

glasses, I will utilize the knowledge-base of crystalline compounds to help de-convolute complex

Raman spectra which will provide insight into the local, medium-range, and extended range

structure that describe the glassy network and the molecular origin of the elastic phases. In order

to quickly appreciate the role of symmetry in crystalline materials, I will first develop an

example used in Burns [47] to illustrate the symmetry-related restrictions imposed on

macroscopic properties important to Raman activity, such as the Polarizability tensor.

5.1.5 Considerations of Symmetry in Crystalline Materials

According to Neumann’s Principle, macroscopic physical properties have at least the symmetry of the point group for which they belong [47]. Through the method of direct inspection one can quickly see the restrictions imposed by symmetry elements of a point group [47].

Gerald burns [47] demonstrates that the C2 symmetry of a system has profound implications on general tensor quantities like the polarizability tensor shown below.

= = 𝛼𝛼11 𝛼𝛼12 𝛼𝛼13 𝐸𝐸1 21 22 23 𝑃𝑃 𝛼𝛼𝛼𝛼 �𝛼𝛼 𝛼𝛼 𝛼𝛼 � �𝐸𝐸2� 𝛼𝛼31 𝛼𝛼32 𝛼𝛼33 𝐸𝐸3 The restrictions imposed by requiring a crystalline structure to have a C2 symmetry element can be identified by considering its transformation upon a general point ( , , and ). The definition of the transformation and its transformation-matrix is also shown below.𝑥𝑥 𝑦𝑦 𝑧𝑧

Example:

Note: The C2 operation is a proper rotation of (2 2 [ ] = 180 ) about the principal axis (highest symmetry axis) of the crystal system𝜋𝜋⁄ 𝑟𝑟 𝑎𝑎under𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 investigation.𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

C2 represented as a Rotation matrix about the [001] is as follows:

′ cos( ) sin( ) 0 1 0 0 ′ = sin( ) cos( ) 0 = 0 1 0 𝑥𝑥′ 0𝜋𝜋 − 0 𝜋𝜋 1 𝑥𝑥 −0 0 1 𝑥𝑥 �𝑦𝑦 � � 𝜋𝜋 𝜋𝜋 � �𝑦𝑦� � − � �𝑦𝑦� 𝑧𝑧 𝑧𝑧 𝑧𝑧 Indicated in the above transformation matrix, the C2 operation transforms the general point

, , into point – , , .

𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥 −𝑦𝑦 𝑧𝑧

Transformation of Cartesian coordinates and the suggested symmetry transformation

matrices ( ′ = ) are mathematical constructs which should not alter the description of

𝑖𝑖𝑖𝑖 macroscopic𝑥𝑥 properties𝑎𝑎 𝑥𝑥 upon transformation.

Tensor quantities transform in the same manner as Cartesian coordinates of equivalent rank. The

Polarizability tensor (rank/dimension equal to 2) requires an xi and an xj coordinate. Therefore,

we need to consider the multiples of Cartesian coordinates such as x2,y2,z2, xy,xz, yz.

Now, when we apply the C2 operation, we can make some useful observations.

; ; ; ; ; 2 2 2 2 2 2 𝑥𝑥 → 𝑥𝑥 𝑦𝑦 → 𝑦𝑦 𝑧𝑧 → 𝑧𝑧 𝑥𝑥𝑥𝑥 → 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 → −𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 → −𝑦𝑦𝑦𝑦 Note that we require a symmetry operation to leave general equivalent points (general points in

the crystal which are equivalent to the symmetry of the crystal) unchanged and since

; leave these points changed which would also indicate a change of the tensor𝑥𝑥𝑥𝑥 →

elements−𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 → −=𝑦𝑦𝑦𝑦 = = = 0

𝑎𝑎13 𝑎𝑎31 𝑎𝑎23 𝑎𝑎32 Symmetry of crystalline compounds can also be used to understand the number and symmetry of

vibrational modes that are Raman/IR active. In order to understand the nature of vibrational modes that we observe in IR/Raman experiments, we look to the crystalline counterpart for reference. Next, I will consider important stoichiometric compounds formed in these crystalline modified borate glasses, provide Factor Group analysis to ascertain the number of vibrational modes present, and demonstrate how mixed rings, which are constituent components of these structural groupings, provide insight into elastic phase boundaries that delineate the Intermediate

Phase (IP)[17].

5.1.6 Symmetry Analysis of Different Symmetry sets of the Meta-borate Crystalline

Compound

The meta-borate crystal provides a great pedagogical tool for understanding the number and difference in magnitude of vibrational modes present in the Raman spectra of modified borates.

The differences in bonding mechanisms allow us to quantify various bonding strengths which

break the crystalline compound into various components. Considering the fact that strong peaks

which are present in crystalline compounds are also present in the glassy systems, we can

properly assign vibrational modes in our Raman spectra of our borate glasses by utilizing the

stoichiometric crystal. figure 5-3 shows the correlation of different groups, D3h, D3 and D3h,

- which describe the isolated B3O6 ion, the unit cell, and the site symmetry, respectively, of the metaborate crystal. Breaking down the crystal structure into its components allows us to describe the vibrational modes of the lattice and its molecular constituents independently.

Moreover, the weaker ionic bonds form along the vertices of B3O6 units, the symmetric stretch of intra-ring breathing modes appear decoupled from the network and make identification of the vibrational mode obvious.

Free ion Site

𝟑𝟑𝟑𝟑1 1 Unit Cell 𝑫𝑫 𝟑𝟑𝟑𝟑 2′ 1 𝑫𝑫 𝐴𝐴 𝟑𝟑 1𝑔𝑔 ′ 𝑫𝑫2 𝐴𝐴 2 𝐴𝐴 𝐴𝐴 𝑢𝑢 (Factor Group) 1 𝐴𝐴 𝐸𝐸′ 𝐴𝐴 2𝑔𝑔 ′′2 𝐴𝐴 𝐸𝐸 𝐴𝐴′′ 𝐴𝐴 𝑢𝑢 𝐴𝐴 𝐸𝐸𝑔𝑔 𝐸𝐸′′ 𝑢𝑢 𝐸𝐸

FIGURE 5-3: Diagrammatic illustration of the relationship of vibrational species with respect to local and extended range order of the metaborate crystal structure [48].

5.1.7 Factor Group Analysis

Factor groups are isomorphic with the crystalline point group. Additionally, Factor Group

analysis refers to the use of the unit cell symmetry species (point group) to characterize the

normal vibrational modes of the crystal.

The meta-borate crystalline compound consists of two formula units of Na3B3O6 per primitive

unit cell (N =24 atoms) and belongs to space group no. 167 R-3c. Simple counting of possible degrees of freedom (3N), suggests that there are 72 independent translational modes.

Referring to Table 1 we see that the 72 modes (which is taken from [36] ) are distributed as follows:

(6) + 2 + 3 + + 2 + 3

𝑁𝑁𝑁𝑁 (6) 𝐴𝐴1𝑔𝑔+ 2 𝐴𝐴2𝑔𝑔+ 3 𝐸𝐸𝑔𝑔+ 𝐴𝐴1𝑢𝑢+ 2 𝐴𝐴2𝑢𝑢+ 3 𝐸𝐸𝑢𝑢

(𝐵𝐵12) + 2𝐴𝐴1𝑔𝑔 + 𝐴𝐴42𝑔𝑔 + 𝐸𝐸6𝑔𝑔 +𝐴𝐴21𝑢𝑢 +𝐴𝐴42𝑢𝑢 +𝐸𝐸6𝑢𝑢 Taken from [36] Total for Crystal: 𝑂𝑂 4 𝐴𝐴1𝑔𝑔+ 8 𝐴𝐴2𝑔𝑔+ 12 𝐸𝐸𝑔𝑔 + 4 𝐴𝐴1𝑢𝑢 + 8 𝐴𝐴2𝑢𝑢 + 12𝐸𝐸𝑢𝑢

𝐴𝐴1𝑔𝑔 𝐴𝐴2𝑔𝑔 𝐸𝐸𝑔𝑔 𝐴𝐴1𝑢𝑢 𝐴𝐴2𝑢𝑢 𝐸𝐸𝑢𝑢

TABLE 1 Space Group R3c, no. 167. Factor group is isomorphous with D3d [36]

Wyckoff pos. A1g A2g Eg A1u A2u Eu

Translation 2a 0 1 1 0 1 1

Rotation 2a 0 1 1 0 1 1

Translation 6e 1 2 3 1 2 3

Rotation 6e 1 2 3 1 2 3

However, studying the character table in Table 2, we can eliminate the modes that are not Raman/IR active.

TABLE 2 Character table of the point group D3d [36]

D3d E 2S6 2C3 i 3C2 3σd

A1g 1 1 1 1 1 1 + ,

𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 𝑧𝑧𝑧𝑧 A1u 1 -1 1 -1 1 -1 𝛼𝛼 𝛼𝛼 𝛼𝛼

A2g 1 1 1 1 -1 -1

𝑧𝑧 A2u 1 -1 1 -1 -1 1 𝑅𝑅

𝑧𝑧 Eg 2 -1 -1 2 0 0 𝑇𝑇, , , ( , )

𝑥𝑥 𝑦𝑦 𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 𝑧𝑧𝑧𝑧 Eu 2 1 -1 -2 -1 0 �𝑅𝑅 , 𝑅𝑅 � �𝛼𝛼 − 𝛼𝛼 𝛼𝛼 � 𝛼𝛼 𝛼𝛼

𝑥𝑥 𝑦𝑦 �𝑇𝑇 𝑇𝑇 �

It is important to identify three modes (one A2u and two Eu) that carry out the translation of the whole lattice and therefore are Raman/IR inactive. Furthermore, we see that A2g and A1u species are inactive so we are left with 57 modes of vibration distributed as follows:

/ : 4 + 12 + 7 + 11 = 57

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐼𝐼𝐼𝐼 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴1𝑔𝑔 𝐸𝐸𝑔𝑔 𝐴𝐴2𝑢𝑢 𝐸𝐸𝑢𝑢 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Note: The A modes represents a symmetric stretch, and the E modes represents a doubly degenerate (count for two modes), subscript u ungerade species and subscript g gerade species.

5.1.8 Site Group Analysis

The difference in bonding types at dissimilar crystalline sites allows us to distinguish lattice vibrations from intra-ring, or internal vibrations (vibrations localized to a specific lattice site).

This method of using different crystalline sites to breakdown the crystal into component-form and to distinguish vibrational species is known as site group analysis.

The internal bonds consist of covalent B-O bonds and are much stronger than the ionic bond

-3 + between the B3O6 anion and the sodium cation Na . Therefore, qualitatively, we can say that lattice vibrations will occur at lower frequencies (according to Bril [36], frequencies less than

-1 -3 250 cm ) than the internal vibrations. The B3O6 isolated ion has D3h point group symmetry and 9 atoms indicating 21 modes of vibration. When considering both rings, the ion has 42 modes of vibration. These modes are associated with internal vibrations, 72-42 =30, lattice vibrations. When we place the ions in the crystal at Wyckoff site a, we experience some symmetry collapse. In the crystal, locally, this collection of atoms possesses D3 point group

symmetry. Table 3 indicates that we have A1, E Raman active modes and A2 IR active modes of

vibration.

TABLE 3 Character table of the point group D3 [36]

D3d E 2C3 3C2

A1 1 1 1 + ,

𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 𝑧𝑧𝑧𝑧 A2 1 1 -1 ; 𝛼𝛼 𝛼𝛼 𝛼𝛼

𝑧𝑧 𝑧𝑧 E 2 -1 0 , 𝑇𝑇 ; 𝑅𝑅 , , , ( , )

𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 𝑧𝑧𝑧𝑧 �𝑇𝑇 𝑇𝑇 � �𝑅𝑅 𝑅𝑅 � �𝛼𝛼 − 𝛼𝛼 𝛼𝛼 � 𝛼𝛼 𝛼𝛼

The total lattice site modes can be calculated using the site locations of the factor group table.

( ): + 2 + 3 + + 2 + 3 + 𝑁𝑁𝑁𝑁 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ( ): 𝐴𝐴 1 𝑔𝑔 𝐴𝐴2𝑔𝑔 + 𝐸𝐸𝑔𝑔 𝐴𝐴 1 𝑢𝑢 + 𝐴𝐴2𝑢𝑢 + 𝐸𝐸𝑢𝑢 − 𝐵𝐵3𝑂𝑂6 𝑟𝑟 (𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟): + 𝐴𝐴2𝑔𝑔 + 𝐸𝐸𝑔𝑔 + 𝐴𝐴2𝑢𝑢 + 𝐸𝐸𝑢𝑢 Taken from [36] − 𝐵𝐵3𝑂𝑂6 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 : +𝐴𝐴42𝑔𝑔 +𝐸𝐸5𝑔𝑔 + +𝐴𝐴42𝑢𝑢 +𝐸𝐸5𝑢𝑢

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴1𝑔𝑔 𝐴𝐴2𝑔𝑔 𝐸𝐸𝑔𝑔 𝐴𝐴1𝑢𝑢 𝐴𝐴2𝑢𝑢 𝐸𝐸𝑢𝑢

Just as before, after subtracting the translations of the whole lattice (A2u and Eu) we have: ( ) / : + 5 + 3 + 4 = 27 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝐴𝐴1𝑔𝑔 𝐸𝐸𝑔𝑔 𝐴𝐴2𝑢𝑢 𝐸𝐸𝑢𝑢 Now, we are in the position𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 to calculate the total number of internal vibrations.

: 4 + 12 + 7 + 11

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝐴𝐴 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 : 𝐴𝐴1𝑔𝑔 + 5 𝐸𝐸𝑔𝑔+ 3 𝐴𝐴2𝑢𝑢+ 4 𝐸𝐸 𝑢𝑢

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 :− �3𝐴𝐴1𝑔𝑔 + 7𝐸𝐸𝑔𝑔 + 4𝐴𝐴2𝑢𝑢 + 7𝐸𝐸𝑢𝑢� = 32 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑛𝑛𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝐴𝐴1𝑔𝑔 𝐸𝐸𝑔𝑔 𝐴𝐴2𝑢𝑢 𝐸𝐸𝑢𝑢 Taken from [36]

𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 5.1.9 Symmetry coordinates, Internal coordinates and Displacement Configurations

In order to visualize the bond stretching, bending, and torsion modes of normal vibrational

modes, it is important to enumerate the symmetry species that we are investigating. An

illustration of the boroxyl ring with the internal coordinates indicated is found below.

7 1

7 𝛿𝛿 = 𝑟𝑟 , ,𝑆𝑆𝑆𝑆𝑆𝑆=𝑆𝑆𝑆𝑆𝑆𝑆ℎ𝑖𝑖𝑖𝑖𝑖𝑖 1 𝑟𝑟 1 2 𝛼𝛼 𝛽𝛽 𝛾𝛾 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝛾𝛾6 𝛾𝛾1 = 1 6 𝑟𝑟 𝛿𝛿 𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑤𝑤𝑤𝑤𝑤𝑤 𝑟𝑟 2 3 𝛼𝛼 1 5 2 𝛽𝛽 𝛽𝛽 6 3 2 3 𝑟𝑟 5 3 𝑟𝑟 2 9𝛾𝛾 𝛼𝛼 𝛼𝛼 𝛾𝛾 8 4 3 4 5 𝛽𝛽 9 𝑟𝑟 4 𝑟𝑟 8 3 𝛾𝛾 𝑟𝑟 𝑟𝑟 𝛾𝛾 2 Taken from [36]

𝛿𝛿 𝛿𝛿

FIGURE 5-4: This is an illustration of the internal coordinates provided by Bril et. al. [36] of the boroxyl ring; internal coordinates are used to simplify the description of normal modes. Internal coordinates consolidates the displacement coordinates (described in by Cartesian coordinates) using geometric arguments providing a more compact description.

Symmetry coordinates are linear combinations of internal coordinates. For instance,

s1=r1+r2+r3+r4+r5+r6 belongs to the A1’ species and refers to the stretching vibrations of the

intra-ring B-O bonds. This particular type of breathing mode is important for assignment of ring

breathing modes of stoichometric groups that contribute to the IP.

Armed with the symmetry coordinates and the internal coordinates, we can build a unitary matrix

of internal coordinates U, which is essentially an assignment of the vibrational modes of the

spectra. Utilizing these assignments, the inverse kinetic energy matrix G and potential energy

matrix F which is utilized in the GF-matrix method which is a method employed to calculate the

amplitudes and frequencies of various vibrational modes.

5.1.10 G-Matrix Example for Familiarity

The G –matrix is related to the kinetic energy of vibrational modes. These matrices can be derived from the normal coordinates and are related to the internal coordinates. To familiarize ourselves, let’s examine the simple case of H2O molecule; note that this example and matrices was taken from Turrell [48], but I have worked out some intermediate steps for completeness.

𝑧𝑧3

3 𝑦𝑦 𝟏𝟏 𝒆𝒆� 𝒆𝒆�𝟐𝟐 2 𝑧𝑧1 𝛼𝛼 𝑧𝑧

1 𝑦𝑦 𝑦𝑦2

FIGURE 5-5: Illustration of coordinate basis independent to each atom of a molecule with C2v symmetry such as H2O.

= sin cos + sin + cos 2 2 2 2 𝛼𝛼 𝛼𝛼 𝛼𝛼 𝛼𝛼 ∆𝑟𝑟1 − ∆𝑦𝑦1 − ∆𝑧𝑧1 ∆𝑦𝑦3 ∆𝑧𝑧3 = sin cos sin + cos 2 2 2 2 𝛼𝛼 𝛼𝛼 𝛼𝛼 𝛼𝛼 ∆𝑟𝑟2 ∆𝑦𝑦2 − ∆𝑧𝑧2 − ∆𝑦𝑦3 ∆𝑧𝑧3 = tan + tan −1 𝑦𝑦1 −1 𝑦𝑦2 𝛼𝛼 � � �− � 𝑧𝑧1 𝑧𝑧2 = tan ( ) = , = + + −1 𝜕𝜕𝜕𝜕 𝑦𝑦 𝜕𝜕𝜕𝜕 𝑧𝑧 𝛼𝛼 𝑦𝑦⁄𝑧𝑧 𝑎𝑎𝑎𝑎𝑎𝑎 � − 2 2 � 2 2 𝜕𝜕𝜕𝜕 𝑦𝑦 𝑧𝑧 𝑦𝑦 𝜕𝜕𝜕𝜕 𝑧𝑧 𝑧𝑧 𝑦𝑦 cos 2 sin 2 cos 2 cos 2 sin 2 = + + + 𝛼𝛼⁄ 𝛼𝛼⁄ 𝛼𝛼⁄ 𝛼𝛼⁄ 𝛼𝛼⁄ ∆𝛼𝛼 − ∆𝑦𝑦1 ∆𝑧𝑧1 − ∆𝑦𝑦2 ∆𝑦𝑦3 − ∆𝑧𝑧3 𝑟𝑟 cos 2 𝑟𝑟 sin 2 𝑟𝑟 𝑟𝑟 𝑟𝑟

𝛼𝛼⁄ 𝛼𝛼⁄ − ∆𝑦𝑦3 − ∆𝑧𝑧3 𝑟𝑟 𝑟𝑟 42

cos 2 sin 2 cos 2 2 sin 2 = + 𝛼𝛼⁄ 𝛼𝛼⁄ 𝛼𝛼⁄ 𝛼𝛼⁄ ∆𝛼𝛼 − ∆𝑦𝑦1 ∆𝑧𝑧1 − ∆𝑦𝑦2 − ∆𝑧𝑧3 𝑟𝑟 𝑟𝑟 𝑟𝑟 𝑟𝑟 The directional cosines and the Cartesian coordinate displacements can be represented in matrix

form with matrices and respectively.

𝐵𝐵 𝜉𝜉

∆𝑥𝑥1 1 0 0 0 0 0 ∆𝑦𝑦 ⎛ 1⎞ = = 0 0 0 0 0 ∆𝑧𝑧 ⎜ 2⎟ 0 −/𝑠𝑠 −/𝑐𝑐 0 / / 0 0𝑠𝑠 2𝑐𝑐 / ⎜∆𝑥𝑥 ⎟ 𝑆𝑆 𝐵𝐵𝐵𝐵 � � 𝑠𝑠 −𝑐𝑐 � −𝑠𝑠 𝑐𝑐 � ∆𝑦𝑦2 ⎜ 2⎟ −𝑐𝑐 𝑟𝑟 𝑠𝑠 𝑟𝑟 𝑐𝑐 𝑟𝑟 𝑠𝑠 𝑟𝑟 − 𝑠𝑠 𝑟𝑟 ⎜∆𝑧𝑧 ⎟ 3 ⎜∆𝑥𝑥 ⎟ ∆𝑦𝑦3 ⎝∆𝑧𝑧3⎠ The Internal coordinates are written in Matrix form with respect to Cartesian coordinate

displacements, where = sin , = cos , and is the initial length of the H-O bond. We can 𝛼𝛼 𝛼𝛼 2 2 simplify the internal coordinate𝑠𝑠 matrix𝑐𝑐 by atomic𝑟𝑟 displacements in the direction of and

𝒆𝒆�𝟏𝟏 𝒆𝒆�𝟐𝟐 = sin cos = sin cos 2 2 2 2 𝛼𝛼 𝛼𝛼 𝛼𝛼 𝛼𝛼 𝒆𝒆�𝟏𝟏 − 𝒚𝒚�𝟏𝟏 − 𝒛𝒛�𝟏𝟏 𝒆𝒆�𝟐𝟐 𝒚𝒚�𝟐𝟐 − 𝒛𝒛�𝟐𝟐 = , = , = ∆𝑥𝑥1 ∆𝑥𝑥2 ∆𝑥𝑥3 𝝆𝝆𝟏𝟏 � � 𝝆𝝆𝟐𝟐 � � 𝝆𝝆𝟑𝟑 � � ∆𝑦𝑦1 ∆𝑦𝑦2 ∆𝑦𝑦3

−𝑠𝑠 −𝑐𝑐 𝒆𝒆�𝟏𝟏 � � � 𝑠𝑠 −𝑐𝑐 𝒆𝒆�𝟐𝟐 1 0 ( ) 2

0 1 ( + ) 2 𝒆𝒆�𝟐𝟐 − 𝒆𝒆�𝟏𝟏 ⁄ 𝑠𝑠 � � � − 𝒆𝒆�𝟏𝟏 𝒆𝒆�𝟐𝟐 ⁄ 𝑐𝑐 = ( ) 2 , = ( + ) 2

𝒚𝒚� 𝑒𝑒̂2 − 𝑒𝑒̂1 ⁄ 𝑠𝑠 𝒛𝒛� − 𝒆𝒆�𝟏𝟏 𝒆𝒆�𝟐𝟐 ⁄ 𝑐𝑐 ( + ) 1 + = = ( ) ( + ) 2 2 2 𝑐𝑐 𝑠𝑠 𝑐𝑐 𝒆𝒆�𝟐𝟐 − 𝒆𝒆�𝟏𝟏 𝑠𝑠 𝒆𝒆�𝟏𝟏 𝒆𝒆�𝟐𝟐 2 2 2 2 − 𝒚𝒚� 𝒛𝒛� − � � − � � � 𝑐𝑐 − 𝑠𝑠 𝒆𝒆�𝟏𝟏 − 𝑐𝑐 𝑠𝑠 𝒆𝒆�𝟐𝟐� 𝑟𝑟 𝑟𝑟 𝑟𝑟 𝑠𝑠 𝑟𝑟 𝑐𝑐43 𝑟𝑟𝑟𝑟𝑟𝑟

( cos ) + = sin 𝑐𝑐 𝑠𝑠 𝒆𝒆�𝟐𝟐 − 𝛼𝛼 𝒆𝒆�𝟏𝟏 − 𝒚𝒚� 𝒛𝒛� − 𝑟𝑟 𝑟𝑟 𝑟𝑟 𝛼𝛼 0 0 = = 𝟏𝟏 𝟏𝟏 ( 𝒆𝒆�cos ) ( cos ) ( + −)(𝒆𝒆�1 cos ) 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝝆𝝆 sin 𝒆𝒆�sin −𝒆𝒆� 𝟐𝟐 𝑆𝑆 𝒔𝒔 ∙ 𝝆𝝆 � 𝟐𝟐 1 � 𝟏𝟏 2 � 𝟏𝟏 2sin � ∙ �𝝆𝝆 � 𝒆𝒆� − 𝛼𝛼 𝒆𝒆� 𝒆𝒆� − 𝛼𝛼 𝒆𝒆� 𝒆𝒆� 𝒆𝒆� − 𝛼𝛼 𝟐𝟐 − − 𝝆𝝆 𝑟𝑟 𝛼𝛼 𝑟𝑟 𝛼𝛼 𝑟𝑟 𝛼𝛼 Finally the G-matrix can be derived by the following relationship using the first matrix on the

left and the diagonal-matrix of elemental mass m (shown below).

= −1 𝐺𝐺 𝑠𝑠 ∙ 𝑚𝑚 ∙ 𝑠𝑠 0 0 = 0 0 H 𝑚𝑚0 0 𝑚𝑚 � 𝑚𝑚H � 𝑚𝑚O

5.1.11 Introduction of a Modifier to the Base Material

2 The Boroxyl-ring consists of planar (two dimensional) B 3 units bonded via sp hybridized

orbitals and are the integral components of the base material.∅ Upon addition of a modifier sp3,

hybridization occurs and the network begins to take on a 3-dimensional character. B 3 Bonds of

the BR are converted to B 4 which create a mixed ring. Frank Galeener [53,54] ∅was able to

demonstrate that the BR ring∅ has a signature symmetric stretch vibration of 808 cm-1. Brill was

able to demonstrate that when rings are modified to include one B 4 species, that the π-bond

character of the ring is disturbed and the length of the intra-ring B-O ∅bonds increases resulting in the red shifting of the symmetric stretch ring-breathing mode to 770 cm-1 [36]. Work conducted

by Vignarooban [17] at the University of Cincinnati extended these ideas to include structural

3 2 groupings with various amounts of sp (B 4) to sp bonds (B 3) to demonstrate a

comprehensive analysis of structure that manifest∅ in modified borate glasses∅ [17].

5.1.12 Mixed Ring Breathing Modes

The work of Vignarooban provided a creative way to extend the ideas of Brill regarding ring breathing modes of modified rings. Figure 5-6 shows structural groupings of crystalline compounds identified as Sodium Triborate, Sodium Diborate, and Sodium Pentaborate. These stoichiometric crystals are present in the Sodium Borate system (NaO)x(B2O3)100-x and form

when the modifier is at a molar ratio of 25%, 33.3% and 37.5% respectively.

Sodium-Triborate Sodium - D iborate Tri - Sodium Penta borate x = 25%, Na:B = 1:3 x = 33.3%, Na:B = 2:4 x = 37.5%, Na:B = 3:5

FIGURE 5-6 Mixed Rings of various stoichiometric compounds have varying ratios of sp3/sp2 species; the ratio of sp3:sp2 determine the vibrational frequency of the symmetric stretch of intra- ring B-O bonds. Demonstrated by Vignarooban [17] in, the sodium borate glasses modes at 770, 740, and 705 cm-1 are symmetric stretch of mixed rings in the sodium tri-borate, di-borate and Tri-sodium Penta-borate structural groupings.

This work identified characteristic breathing modes in the 600 cm-1 to 900 cm-1 range and

intuitively recognized a relationship between the mode frequencies and the percentage of sp3

bonds, which are displayed in figure 5-7.

808 cm-1 (6 sp2, 0 sp3) 820 (0 %)

800 BR )

-1 770 cm-1 (4 sp2, 2 sp3) 780 (33%) 750 cm-1 2 3 PB (6 sp , 6 sp ) 760 Tri B (50 %) Tetra B 740 cm-1 2 3 740 DPB (4 sp , 6 sp ) (60 %) DB 720 705 cm-1 ModeFrequency (cm (4 sp2, 8 sp3) (66 %) 700 2 (B-O)sp = 1.37 A (ref. 1) Tri PB (B-O)sp3 = 1.43 A (ref. 1) 680 0 20 40 60 80 3 % of sp bonds

FIGURE 5-7: Non-linear behavior of the A1’ vibration with respect to various borate compounds. This figure was taken from Vignarooban [17].

5.1.13 The Raman Line Shape Profile

Quantum Theory states that energy of an isolated molecule is absorbed and emitted at a well- defined discrete frequency. However, molecules of a solid can be thought to interact with a bath of surrounding molecules which couple to each other, absorb and scatter a continuous spectrum of energy.

Interaction of molecules with their environment (bath of molecules) determines attenuation/relaxation of excited states and nature of vibrational coherence (constructive interference of different vibrating species). The time which is a measure of relaxation of an excited state is called the amplitude correlation time and the measure of the coherence

46 𝜏𝜏𝑎𝑎

lifetime is The relationship of these two timeframes determines how the detector perceives

𝑐𝑐 the vibrational𝜏𝜏 events and the characteristic shape of the Raman or IR profile.

Extreme Cases

Case i .)

𝝉𝝉𝒄𝒄 ≫ 𝝉𝝉𝒂𝒂 In this case, the excited vibrational state relaxes before the mode becomes incoherent. This

would best represent the case when the number of floppy modes is zero, just as in a solid. The

nature of the vibrational envelope tends to resemble a Gaussian profile [49].

Case ii .)

𝝉𝝉𝒂𝒂 ≫ 𝝉𝝉𝒄𝒄 In the second extreme case, dephasing of the vibrational species dominates. In this situation, the

anharmonic nature of the network dominates the behavior of vibrational modes. This is

synonymous with gasses where energy can be transferred by lossy-channels such as rotations and

collisions. The nature of the vibrational envelope tends to resemble a Lorentzian profile [49].

Voigt Profile

Glasses represent a conundrum where they are solids that can have the disorder of a liquid.

Glasses have a distribution of bond angles and may possess a larger degree of decoupling relative to their crystalline counterparts. The profiles tend to have a mixture of Gaussian and

Lorentzian character; in fact, it has been demonstrated that a convolution of the two provide a superior fit. The convolution of the Gaussian and Lorentzian profile is known as the Voigt profile and is used in this work to fit Raman spectra during the process of deconvolution. The

mathematical description for the Voigt profile is displayed below:

∞ ( ; , ) = ( ′: ) ( ′; ) ′ ∞

𝑉𝑉 𝑥𝑥 𝜎𝜎 𝛾𝛾 �− 𝐺𝐺 𝑥𝑥 𝜎𝜎 𝐿𝐿 𝑥𝑥 − 𝑥𝑥 𝛾𝛾 𝑑𝑑𝑑𝑑 ( ) ( : ) 2 2 −𝑥𝑥 ⁄22𝜎𝜎 𝑒𝑒 𝐺𝐺 𝑥𝑥 𝜎𝜎 ≡ 𝜎𝜎√ 𝜋𝜋 ( : ) ( + ) 𝛾𝛾 𝐿𝐿 𝑥𝑥 𝛾𝛾 ≡ 2 2 𝜋𝜋 𝑥𝑥 𝛾𝛾 5.2 IR/Raman Experiments of modified Barium Borate Glasses

Now that we have firm arguments to substantiate IR/Raman mode assignments, it is time to

investigate compositional trends of barium borates glasses. In this section, I will describe the

procedure and results of this experimental effort.

5.2.1 Raman Spectroscopy Experiments

Raman studies were performed with a T64000 triple dispersive Raman system, manufactured by

Horiba-Jobin-Yvon. This facility allows us to explore extended range spectra (capable of

capturing spectra as low as 4 cm-1) with incredibly high resolution (triple additive .15 cm-1). The

Raman system was operated in a Triple-subtractive mode (enables low frequency measurements) using 150 mW laser light power with a laser wavelength of 514.532 nm. Samples were mounted on a translation stage to allow for proper focus with confocal microscope using either a 100X- or

80X-objective.

Before Raman data was measured, the Raman system underwent a proper alignment of the

foremonochromator and spectrometer, to ensure that all spectral information is obtained and to

eliminate Rayleigh back-scatter which results from reflection on the internal mirrors.

The foremonochromator functions as a band-pass filter using two optical stages by using two separate diffraction gratings; the reason for this is to first disperse the light, and second, to collimate polychromatic light scattered from the sample.

The spectrometer disperses the light so that different wavelengths of the polychromatic light hit different pixels on 1024K Symphony CCD (cooled with liquid nitrogen LN2).

FIGURE 5-8: Function of the diffraction gratings in the T64000 Triple Additive/Subtractive monochromatic Raman System illustrated in [50].

The spectrometer was calibrated utilizing the narrow transition of a Mercury Lamp which ensures two things: First, that the spectrometer is centered on the appropriate frequency and second that the spectrometer is centered on the entrance slit to the Raman System.

All samples were handled in sealable mason jars to reduce sample exposure to room humidity

(RH ~ 50%), before and after measurements were taken.

-1 νc =1122.73 cm FWHM =2-3 cm-1

Col 1 vs Col 2 Intensity (Arb. Units) (Arb. Intensity

1080 1100 1120 1140 1160

-1 Raman Shift(cm ) FIGURE 5-9: Raman spectra of the narrow band representative of the energy-spectrum from a singular transition in a Mercury (Hg) Lamp. This narrow atomic transition in Hg-atom permits increased precision for alignment and the calibration of the spectrometer and foremonochromator in our T64000 system.

5.2.2 Compositioinal Trends of Barium Borate glasses

Raman Scattering experiments display rich line shapes and, as we can see in figure 5-10, the

spectra systematically evolves upon addition of the BaO modifier. The system (BaO)x(B2O3)100- x 15 mol%≤x≤40 mol% is shown with increasing modifier content as we progress up the vertical

axis. A few observations can be ascertained as we observe the landscape from the low to high

frequencies:

i.) The Boson mode near 50 cm-1 continues to be the strongest mode in the spectra.

ii.) A mode assigned to the symmetric stretch of the units increases upon the addition of the

4 network modifier. 𝐵𝐵∅ iii) The 770 cm-1 mode associated with mixed rings where 33% of the sp bonds are sp3 hybridized orbitals, grows at the expense of the BR mode at 808 cm-1.

iv.) A mode exists which is associated with the triply degenerate vibrations of the B 4 units.

This mode increases upon the addition of the modifier and the B-O- stretch mode percolates∅ at compositions above 32 mol%.

( ) ( ) 2 3 100 B — O- BO4(A1) Stretch 𝐵𝐵𝐵𝐵𝐵𝐵 𝑥𝑥 𝐵𝐵 𝑂𝑂 −𝑥𝑥 BO4(F2)

FIGURE 5-10: Global perspective of Raman-line shapes for the system (BaO)x(B2O3)100-x system. Notice the richness of vibrational information contained in the Raman spectra.

Boson mode

-1 150 cm ( ) -1 450 cm ( ) 𝑩𝑩𝑩𝑩𝟒𝟒 𝑨𝑨𝟏𝟏 ( ) Intra-ring modes -1 900 cm𝟒𝟒 𝟐𝟐 -1 𝑩𝑩𝑩𝑩 𝑭𝑭 1100 cm𝟒𝟒 𝟐𝟐 𝑩𝑩𝑩𝑩 𝑭𝑭 -1 − 1450𝑩𝑩 cm− 𝑶𝑶

% % %𝟒𝟒 𝟒𝟒 %𝟑𝟑 𝟑𝟑 %𝟑𝟑𝟑𝟑 %𝟑𝟑 𝟑𝟑 %𝟑𝟑 𝟑𝟑 % 𝟐𝟐𝟐𝟐 %𝟐𝟐 𝟐𝟐 %𝟐𝟐𝟐𝟐 %𝟐𝟐 𝟐𝟐 %𝟐𝟐𝟐𝟐 %𝟐𝟐𝟐𝟐 %𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏

FIGURE 5-11: Intensity graph of Raman data showing the growth/reduction of a mode around 1500, correlation of the 770 and 808 cm-1mode.

In order to visually see the evolution of modes, we stretch the vertical axis and focus our attention on low, medium, and high frequency bands.

5.2.3 Observations at the Low Frequency Regime Boson and Lattice Vibrations

(Extended Range Structure)

Initially, there is a broad mode, around 500 cm-1, that appears to grow and sharpen as we

increase the modifier content. There is an emergence of a broad mode around 200 cm-1 at x =32

mol% that grows as we increase the modifier. Finally, the Boson mode around 50 cm-1 appears

to be the most intense mode throughout all compositions.

5.2.4 Observations of BR and Mixed Rings (Medium-Range Structure): Intra-Ring B-O

Bonds

A mode is present, near 770 cm-1, grows in intensity and appears to red-shift as we increase the

modifier content. Furthermore, the mode near 770 cm-1 develops an asymmetric character upon

addition of the modifier content. Conversely, the mode associated with the BR, reduces in intensity and also appears to red-shift as we progress to higher modifier content.

5.2.5 Observations at the High Frequency Regime: Extra-Ring B-O bonds

Vibrational broad modes, around 950 cm-1 and 1100 cm-1, increase in intensity as we progress to

higher concentrations of our network modifier. A broad spectrum of modes exists in the 1200

cm-1-1600 cm-1 range. This band starts to accentuate a mode around 1450 cm-1 as we progress

to a higher modifier content 20 mol% ≥x.

Frequency (cm-1) Mode Assignment Comments Refs

Mode intensity grows upon addition of the Yiannopoulous, Phys. Stretching of the extra-ring 1450 - modifier and sharpen Chem. Glasses 2001 B-O bonds at compositions (Ref. 23) greater than Mode intensity grows Yiannopoulous, Phys. Scissor Mode of BO 1120 4 upon addition of the Chem. Glasses 2001 Tetrahedra modifier (Ref. 23) Mode intensity grows Yiannopoulous, Phys. Scissor Mode of BO 940 4 upon addition of the Chem. Glasses 2001 Tetrahedra modifier (Ref. 15) Galeener, PRB, vol Mode intensity Symmetric Stretch of intra- 22(8),1980 (Ref.52 ) reduces in strength 808 ring bonds in a Boroxyl Galeener and Thorpe, upon the addition of Ring (BR) PRB, vol 28(10), the modifier 1983 (Ref. 51) Symmetric Stretch of intra- Mode intensity grows ring bonds in a Mixed Ring 770 upon addition of the Current work constituent of a Tetra-borate modifier Structural Group Symmetric Stretch of intra- Mode intensity grows ring bonds in a Mixed Ring 750 upon addition of the Current work constituent of a Di-borate modifier Structural Group Symmetric Stretch of intra- Mode intensity grows ring bonds in a Mixed Ring 705 upon addition of the Current work constituent of a Meta-borate modifier Structural Group Symmetric Stretch Mode intensity Yiannopoulous, Phys. - 630 vibrations of BØ2O reduces upon addition Chem. Glasses 2001 triangular units of the modifier (Ref. 23) Mode intensity grows Symmetric Stretch upon addition of the Wen-Zhi Yao[51] and 475 vibrations of BO tetrahedra modifier and sharpen 4 current work units at compositions greater than Mode intensity grows 220 Lattice vibrations Current work after x = 36 mol% Boson peak related to the Appears to be the soft springs created by van most intense peak 50 Current work der Waals forces between though out all B2O3 planes compositions

FIGURE 5-12: Displayed is an enhancement of low, mid, and high range vibrational bands that provide information about different structural components of network glasses.

5.2.6 Mixed Rings and the Intermediate Phase

As indicated in the study of Sodium Borates, a medium range structure governs the elastic phase boundaries of the IP[17]. Through cluster combinatorics and constraint counting, the paper of

Micoulaut and Phillips [52] demonstrated that the width of the intermediate phase and the nature

of the phase transition depends on the medium range structure (relative ring fractions) of a

covalently bonded network[52]. The relative ring population can be studied by investigating the

symmetric stretch of intra-ring breathing modes. Thus, I will focus my attention on the medium

range frequency regime.

(b) (Na2O)x(B2O3)100-x

* *

FIGURE 5-13: The left panel (a) is the mid-range spectrum of Sodium Borate [17] glass and on the right panel (b) is the mid-range spectra of barium borate glass; both provide insight into medium range structure. Note that the red star indicates compositions of stained glasses.

5.2.7 Quantification of Raman Vibrational Mode Characteristics Through Line-shape

De-convolution

Glasses, due to their solid nature and degree of disorder (e.g. distribution of bond angles), have line shapes with a mixture of Lorentzian and Gaussian character. The Voigt profile is a convolution of the Gaussian and Lorentzian profile, and offers an improvement in fitting line shapes which have traditionally been conducted with Gaussian line shapes.

Deconvolution is the controlled process of fitting complex Raman line shapes with Voigt-peaks

to distinguish the contribution of different spectral bands; spectra bands are a result of initially

coherent oscillators that de-phase overtime due to the difference in coupling environments [49].

Borates have a large library of crystalline compounds which permit enumeration of vibrational frequencies with specific structural groupings. This allows one the ability to fit Raman line shapes intuitively with the appropriate number of peaks.

Deconvolution of the Barium Borates was carried out with Systat Software Peakfit. All spectra

were fitted with a minimal amount of Voigt-peaks. Initially, Voigt-peaks were fixed at their

respective frequencies and were allowed to vary in intensity and width. After iterations at a

fixed frequency, the peaks were allowed to vary in frequency, intensity, and width until the SW

settled on a solution.

When spectra contain well-resolved vibrational modes and reduced populations of proximal

modes, Raman line shapes are reasonably clean and easy to fit. For instance, consider the

Raman line shape of the x = 15 mol% sample shown in figure 5-14. This spectra, in the region

of 600-900 cm-1, consists of three distinct peaks centered at 806 cm-1, 774 cm-1 and 748 cm-1.

One is able to get a good fit, shown by the red dashed line.

Difficulty arises when adjacent peaks show subtle shoulders and create asymmetry about prominent peaks. Figure 5-14 shows a Raman line shape in the 600-900 cm-1 range of the x = 30

mol% sample. For spectra like the x = 30 mol%, a manual process of peak placement ensues.

Peak placement, which involves the scaling of peak intensity and width, is conducted manually,

at first, until a reasonable fit is obtained. Armed with the knowledge of enumerated vibrational,

modes that are present in the featureless regions of Raman Spectra, it is necessary to invoke a

manual fitting process as using the automated method available in fitting software tends to show no constraint on the variation of peak parameters (i.e. center-frequency, width, and intensity).

After a reasonable fit is ascertained, the software is allowed to vary parameters of specified peaks by fixing all parameters of shoulder peaks and varying the intensity, widths, and frequencies of more resolved peaks.

FIGURE 5-14: Deconvolution of the x = 15 mol% and 30 mol% ([BaO]x[B2O3]100-x) line shape in Raman experiment.

5.2.8 Polarized Raman Experiments

Graphically, the polarization for an isotropic and anisotropic molecule is shown in figure 5-15.

𝐼𝐼∥

⊥ 𝐼𝐼

FIGURE 5-15: Scattered components of an isotropic and anisotropic molecule; this illustration was borrowed from [44].

The isotropic (trace of the polarizability tensor) and the anisotropic components (off-diagonal) of the polarizability tensor can be used to relate the parallel and perpendicular components of the detected scattered light. When the ratio of intensity of the vertically polarized scattered light to the horizontally scattered light is at or below 75% [44], this indicates a symmetric character to the vibration.

BaOX(B2O3)100-X Un-PolarizedV-V X = 15%

V-H X 4 Intensity Units) (Arb.

700 720 740 760 780 800 820 840 860 -1 Raman Shift (cm )

FIGURE 5-16: Polarization measurements preformed on the x = 15 mol%. The black curve is the spectra with of vertically polarized backscattered light and the blue represents the spectra obtained capturing the “horizontally” polarized light. This is direct evidence of the existence of this triad of modes whose reduction of intensity indicate a symmetric stretch much like the Boroxyl Ring (BR) demonstrated by Galeener and Thorpe [53, 54].

5.2.9 Results of Barium Borates

Figure 5-17 shows the results of the deconvolution process. Four peaks were used to analyze the mixed ring vibrational modes in the 600-900 cm-1 region. These modes are associated with the symmetric stretch of the B-O intra-ring bonds representative of vibrational modes centered near

808 cm-1 , 770 cm-1, 740 cm-1, and 705 cm-1.

FIGURE 5-17: Quantification of growth/reduction of vibrational frequencies associated with the Boroxyl Ring, Tetra-, Di-, and Tri- borate mixed rings.

This data clearly shows the reduction of the 808 cm-1 mode as we increase to higher x. The 770

cm-1 mode shows initial growth, but reaches a maximum in Fractional Intensity at the x = 20

mol% composition. The 750 cm-1 mode shows a local maximum around x = 33 mol%, and the

mode centered around 705 cm-1 does not percolate until the x = 24 mol% composition.

Figures 5-18 through 5-21, show the changes in the center of the frequency band associated with the aforementioned modes.

Immiscibility

Range

S.R. I.P. Flex.

FIGURE 5-18: The 808 cm-1 mode demonstrates red-shifting after 15 mol% of BaO has been added to the base material. This suggests that the BR remains unmodified until we reach 15 mol% of BaO.

S.R. I.P. Flex.

FIGURE 5-19: The 770 cm-1 mode, associated with the Tetra-borate group, initially blue-shifts and then red-shifts after reaching the stress threshold 24 mol% of BaO, delineated by the intermediate phase.

FIGURE 5-20: The 740 cm-1 mode associated with the Di-borate shows a broad maximum in the intermediate phase 24 mol%≤ x ≤32 mol% where x indicates mol of BaO.

S.R. I.P. Flex.

FIGURE 5-21: The 705 cm-1 mode shows a maximum in the intermediate phase of compositional space.

FIGURE 5-22: The top panel is the global view of the normalized scattering strength of the Boson mode. The bottom panel indicates the monotonic reduction which may be due to the loss - of 2D character upon the creation of BO4 units, of the Boson mode scattering strength in the Stressed Rigid region of compositional space.

5.2.10 FTIR Results

In the FTIR experiments, flat pieces of the glass sample were mounted in a Seagull accessory of

a Thermo-Nicolet FTIR research grade 900 bench unit. The sample container was allowed to

purge with nitrogen gas (N2) for 30 minutes following sample loading. IR specular reflectance

measurements were recorded and absorption signals were extracted using Kramers-Kronig

transformation as demonstrated below:

1 = 1 + 2 2 cos − 𝑟𝑟 𝑛𝑛 2 𝑟𝑟 − 𝑟𝑟 𝜃𝜃 2 sin = 1 + 2 cos 𝑟𝑟 𝜃𝜃 𝑘𝑘 2 𝑟𝑟 − 𝑟𝑟 𝜃𝜃 r is the reflectivity measured directly from specular reflectance data and θ is the angle of incidence. The real part of the complex refractive index is indicated by n, which is also known as the index of refraction, and the imaginary part k, is known as the extinction coefficient. From the above equations, one can derive the real and imaginary part of the dielectric constant and then use this to calculate the frequency dependent absorption coefficient.

= ′ 2 2 𝜀𝜀 𝑛𝑛 − 𝑘𝑘 = 2 ′′ 𝜀𝜀 𝑛𝑛𝑛𝑛 2 = ′′ 𝜋𝜋𝜋𝜋 𝜀𝜀 𝛼𝛼 𝑛𝑛 Absorption Spectra is shown in figure 5-23.

FIGURE 5-23: Left - the present results from IR –reflectance measurements; Right - IR data -1 - from Yiannopoulous [25]. Note that the peak around 1600 cm is indicative of B R3 and B R2O triangular units not found in the work of Yiannopoulos [25]. ∅ ∅

The absorption coefficient spectra shown in figure 5-23, shows a band from 800-1200 cm-

1 which can be attributed to the B-Ø stretching vibration of BØ4 tetrahedra [25]. The band 1200

-1 -1 - cm -1600 cm can be attributed to the stretching of BØ3 and BØ2O triangular units [25]. This

data allows us to calculate the ratio of 4-coordinated Boron to 3-coordinated Boron (B4/B3),

which is shown in figure 5-24.

FIGURE 5-24: In red are the ratio of 4-coordinated Boron to 3-coordinated Boron (B4/B3) indicated by Yiannopoulos [25]. In blue is the B4/B3 ratio of the present work. Ar is the symbol suggested by Yiannopoulos [25] to indicate that these ratios were obtained by taking the “Area” -1 -1 under the spectral bands 800-1200 cm (B4) and 1200-1600 cm (B3).

Note that in figure 5-24, the current work shown in blue displays a lower B4/B3 ratio. The lower

ratio in the current work may be due to the fact that I have dry samples, whereas wet samples

would promote B-OH bonds which, in turn, would lower the B3 population and explain the

absence of the peak at 1550 cm-1 in the work of Yiannopoulos.

FIGURE 5-25: The number of four coordinated Boron (N4) derived from FTIR data of the present work (blue) and Yiannopoulos [25] (red). Notice that the present work follows the mean field prediction. The red may be lower due to the fact Yiannopoulos used two data points from NMR data to scale the FTIR data, where as in the present work I discovered three similar compositions and thus used three data points.

Chapter 6 Discussion

The present experimental research work conducted on Ba-Borate glasses using thermal, optical,

and mechanical probes, raises basic issues on these materials that are of broad interest in glass

science. We discuss some of these issues in this chapter, as follows:

6.1 Topological Phases of Ba-Borate glasses

6.2 Raman Scattering and aspects of glass structure

6.2.1 Raman Scattering as a probe of local, medium-range-structure, and extended-range-

structure

6.2.2 The nature of structural groupings contributing to the mixed-ring modes observed in Ba

Borates

6.3 IR reflectance is a quantitative probe of B4/B3 content of BaO modified B2O3 glasses

6.4 Microscopic origin of the Immiscibility range in the Equilibrium Phase diagram of the

BaO-B2O3 binary

6.5 Glass Network dimensionality considerations and the origin of the Boson mode in Borate

glasses

6.6 Boson modes and the Stress and Rigidity transition

6.1 Topological Phases of Ba-Borate Glasses

The nature of the glass transition continues to be a highly debated issue in the field of

fundamental glass science [55, 56]. It is generally believed that the glass transition is hysteretic in nature and that the overshoot in the glass transition endotherm, observed in a DSC measurement, is a measure of the enthalpy of the relaxation of glass. With waiting time, the

overshoot generally increases, i.e., the enthalpy of relaxation builds up as the glass

configurational entropy decreases. In 1998, Selvanathan et al. [9], identified for the first time that in binary Si-Se glasses, in a narrow but well defined glass composition range, 20 mol% < x

< 26 mol%, that not only is the enthalpy of relaxation minuscule, but the term actually does not age much with waiting time. These results suggested, for the first time, that the nature of the glass transition can become thermally reversing “in character” in select compositional ranges.

Subsequently, such “reversibility windows” were found in other chalcogenides [57, 58]; namely, modified oxides [18, 59], solid electrolytes [60], as well as in heavy metal oxide glasses [61]. In

2010, Micoulaut illustrated from modeling studies, that the vanishing of the enthalpy of relaxation constitutes evidence for the isostatic nature of a glass [15]. The term “isostatic” connotes that the count of bonding constraints (nc = 3) equals the degrees of freedom in a 3D

network. Over-constrained (nc > 3) and under-constrained (nc <3) glass networks will, in general, display large overshoots of the Tg endotherm, but isostatic networks will display a vanishing overshoot. These considerations suggest that the reversibility window observed in the present Ba-Borate glasses, constitutes evidence for the Intermediate Phase. The observation is qualitatively of a similar character to the ones observed recently in the alkali modified (Na- and

Li-) borate glasses. However, there are striking differences between the two cases; for example, the alkaline-earth additive is divalent, while the alkali additive is monovalent. These valence differences have a profound impact on glass structure, the phase diagrams of glass formation, and on the details of the topological phases.

Why is the Reversibility window Gaussian-like instead of square well in the present Ba-Borates?

We believe that the reason can be traced to the much heavier mass of the Ba cation (137 amu) that reduces the diffusivity of the ion at the reaction temperature of the melt (1700oC); thus inhibiting homogenization of the melt. In Na-Borates, the square well like reversibility window

constitutes a signature of homogeneous glasses, which is facilitated by the smaller mass of the

Na cation (23 amu) and rapidly diffusing in the base glass to establish the medium range structure. A similar circumstance prevails in Li-Borates. We recognize that the diffusivity of these cations in respective melts varies inversely as the square root of their masses [62]. In spite of our efforts to repeatedly melt the Ba-borates using an induction furnace followed by the use of a box furnace, we were not successful in getting a square well reversibility window.

I will comment on the narrower width of the IP in the present Ba-borate glasses, which is intimately tied to the medium–range-structure, after discussing their molecular structure in section 6.2.

On general grounds, one expects the glass compositions at x > 32 mol% of BaO to comprise the

Flexible phase of the present Ba-borates. The maximum of Tg near 33 mol% in these modified

borates stems from the dual role of the modifier cation, alkali or alkaline-earth. At low

concentrations, these cations serve to transform BO3 triangular units into BO4 tetrahedral ones,

thus, increasing the network connectivity as reflected in an increase of Tg (figure 6-1). At higher

concentrations (x > 33 mol%), these cations disrupt the connectivity of the glass network by

converting bridging O atoms into non-bridging oxygen (NBO). The disruption of the network connectivity will reduce the count of mechanical constraints and drive the glass network to be globally flexible. Furthermore, these NBO already begin to form at low x and their concentration increases rapidly at x > 15 mol% as Tg begins to saturate (figure 6-1).

Ba-borate glasses at x < 24 mol% are stressed-rigid. The identification is suggested by the very

strong excitation of the Boson mode in the base material, B2O3. I comment on the issue later in

section 6.6.

The dryness of the present Ba-borate glasses is supported by the high Tgs as observed in the

present samples compared to a previous report by N. Lower from Coe College, Iowa [63].

Additionally, the absence of bonded water in our glass samples is supported by our FTIR

absorption spectra that show little to no signal near 3200 cm-1, where a stretching mode [44] of

the bonded OH species is expected to be manifested. Figure 6-10 gives an example of FTIR

absorption response near 3200 cm-1 in the x =15 mol% and x= 20 mol% Ba-Borate glasses in the

present work.

FIGURE 6-1: Glass Transition temperature for the present experimental studies (blue) and the work of Lower et al. [63]. Notice that the present work shows higher Tg’s at lower molar concentrations of BaO.

In summary, the observation of the reversibility window in modulated DSC experiments fixes

the three topological phases in the present Ba-borates, with glasses at x < 24 mol% to be

stressed-rigid, those at x > 32 mol% to be in the flexible phase, and those in the intervening range 24 mol% < x < 32 mol% to be in the Intermediate Phase.

6.2 Raman Scattering and aspects of Glass-structure

6.2.1 Raman scattering as a probe of local, medium-range-structure and extended-range-

structure

Raman Scattering of the Ba-Borate glasses reveal rather rich lineshapes and decoding these have

provided valuable information on the local, medium range, and extended range structure. For instance, consider figure 6-2, which displays the observed Raman Scattering of a Ba-Borate glass at x = 25 mol% of BaO – several vibrational modes are observed here. The lowest frequency mode (< 100 cm-1) is the Boson mode. Subsequently, there are modes of characteristic local

- structures such as BØ4 tetrahedra, and BØ2O triangular unit with two bridging and one non-

bridging-oxygen. In addition, there are modes of characteristic rings, called mixed rings, that appear in the 650-850 cm-1 range. For example, Professor F. Galeener identified a mode

centered at 808 cm-1 in an unmodified borate base glass, which he attributed to a symmetric

stretch of Oxygen atoms in a planar three membered ring of BØ3 triangles (the Boroxyl-ring BR)

[53, 54]. The mode near 770 cm-1 in figure 6-2 also comes from a three membered ring, but is composed of two BO3 triangular units and one tetrahedral BO4 unit. The eigenvectors of the

mode in question are nearly the same as in the BR. A mixed ring such as this could be a

member of one or more of the possible structural groupings [64](SGs). Structural groupings

(SG) refer to extended range structures such as a Ba-Tetraborate or Na-Triborate. I will show that by tracking the fractional scattering strength of these mixed ring modes, one can independently establish the nature of SGs populated in the glass of interest rather precisely.

FIGURE 6-2: Raman Spectra of the x = 25 mol% sample displaying vibrational modes manifesting from local BO4 structures, medium range structure (symmetric stretch of mixed ring Tetra.-B), and extended range structure such as the Boson mode.

Utilizing Size Increasing Cluster combinatorics Approximations (SICA) and constraint counting arguments, on small rings, Dr. Micoulaut and Dr. Phillips demonstrated [52] that rigidity in networks is usually traced to characteristic rings. The case of the modified Borate is rather exceptional in one respect. Modified borates are the only glass system that have a combination of rings with the appropriate Na:B ratio or Ba:B ratio that resemble the SG of their crystalline phases [64]. These super-structural rings are usually described in the Borate literature [64] as

Structural Groupings. These SGs are therefore of great interest in specifying not only the local, and medium range structure (mixed rings), but also the extended range structure. In my experiments, I have paid special attention to the mixed-rings observed in the 650-850 cm-1 range.

(b) (Na2O)x(B2O3)100-x

* *

FIGURE 6-3: Vibrational regime of the symmetric stretch of the Boroxyl and mixed rings. This figure shows the growth/reduction, red-shifting of mixed rings as we progress up the vertical axis which corresponds to the increase of modifier content (mol% BaO or Na2O). The left panel(a) is the work on Sodium Borates Voinarooban [17] and the panel (b) on the right is the current work on Barium Borates.

Figure 6-3 shows Raman Scattering in the 700 to 900 cm-1 range, where vibrational modes of

mixed rings are populated. In the left panel, are the results on Na-Borates taken from the Thesis

[17] of Vignarooban. In the right panel, I display Raman spectra of Ba-borate glasses. It is important to note that in the sodium borates, we can see evidence of a mode near 770 cm-1

already at a low NaO molar content of 10 mol%, which is a feature not seen in corresponding 75

Ba-Borates. Indeed, I will show later that the absence of the modification of glass at low Ba- content is due to an immiscibility range, as liquid B2O3 and liquid BaO do not mix.

Furthermore, the BR mode systematically red-shifts as the base glass is modified with the growth

in scattering strength of the 770 cm-1 mode for the Na-Borates, however, the 808 cm-1 mode does not shift upon alloying BaO additive at low x. In both sets of spectra, I display colored arrows indicating the center frequency of a triad of modes populated at frequencies on the low side of the familiar Boroxyl ring mode at 808 cm-1. These triads of modes are identified with a characteristic of “mixed-rings” formed in the structure of respective glasses.

In the Raman experimental set up, the excitation laser light was polarized vertically. I performed

Raman polarization experiments by analyzing the scattered light in the vertical- and horizontal- polarizations. Results of these experiments, for a glass containing 15 mol % of BaO, is shown in figure 6-4. For instance, in the VV polarization (scattered light polarization along the vertical,

which is parallel to the incident light polarization), the scattering is pronounced. On the other

hand, in the VH polarization (i.e., the scattered light polarization along the horizontal which is

perpendicular to the incident light polarization), I observed the scattering to be considerably

weaker. These data unambiguously show that I not only have three distinct modes, but that each

of these modes possesses a strong depolarization-ratio = . Likewise, it conclusively 𝐼𝐼𝑉𝑉𝑉𝑉 � 𝜌𝜌 𝐼𝐼𝑉𝑉𝑉𝑉� proves that the eigenvectors of each of these modes represent a symmetric stretch of the intra-

ring bonds, which result in the movement of Oxygen atoms – as shown in the case of the

Boroxyl ring in figure 6-5.

FIGURE 6-4: Polarization measurements preformed on the x = 15 mol%. The black curve is the spectra of vertical polarization and the blue represents the spectra obtained capturing the “horizontally” polarized light. This is direct evidence of the existence of the triad of modes whose reduction of intensity indicates a symmetric stretch much like the Boroxyl Ring (BR) and demonstrated by Galeener and Thorpe [53, 54].

The red-shift of the mixed-ring modes in relation to the BR mode, can be understood by

recognizing that the bond length of an sp2 hybridized bond (1.37 Å) is shorter than a sp3

hybridized bond (1.43 Å), shown in figure 6-6. One then expects the effective spring constant of the symmetric-stretch ring mode to systematically red-shift as the fractional concentration of the sp3 bonds increases, as shown in figure 6-6.

FIGURE 6-5: Illustration of the Boroxyl Ring which constitutes the planar structure of the base glass. The black arrows indicate the vector displacement of the oxygen atoms during a symmetric Stretch excitation contributing to the 808 cm-1 Raman active mode.

The earlier work [17] on Sodium borates revealed several mixed ring modes. A plot of the frequency of these “ mixed- ring” modes as a function of the sp3 tetrahedral fraction of bonds is plotted in figure 6-6. On this plot, we have projected the data obtained on the current Ba-

Borates, shown in red. A clear correlation can be observed with a rather apparent meaning.

Presence of longer sp3 bonds in a three membered ring softens the symmetric stretch mode frequency; at first, it occurs linearly, but then super-linearly when the tetrahedral fraction exceeds 60 mol%.

Na Ba

FIGURE 6-6: Non-linear behavior (slope changes at 60 mol%) of the A1’ symmetric stretch vibration in various borate SGs. This figure was adapted from Vignarooban’s thesis [17]. The red data points are the SG present in the Ba-Borates (present effort) while the blue data points are on Na-Borates.

6.2.2 The nature of structural groupings (SGs) contributing to the mixed ring modes

observed in Ba Borates

Detailed fractional intensity data for the observed mixed-ring vibrational modes in the Sodium- and Barium- borates are respectively shown in figure 6-7 (a) and (b). In the Sodium- borates, one can see the 770 cm-1 mode scattering strength steadily increase as x > 0, to show a broad

maximum near, x = 25 mol% of Soda. The BR mode shows a complementary behavior in that

its scattering strength steadily decreases; this strongly suggests that microscopically, a BO3

triangle in a BR is being converted to a BO4 tetrahedral unit with a negative charge that is

compensated by a Na+ cation in the vicinity. The maximum in the fractional scattering strength of the 770 cm-1 mode near x = 25 mol% of soda is strongly suggestive that the “mixed-ring” is 79

part of a sodium triborate, i.e., 1 Na per 3 B atoms leading to a soda fraction of ¼ or 25 mol%.

Upon further increase of modifier content, we see another maximum near 33 mol% that can be

ascribed to Sodium Di-borate.

The fractional scattering strength in the Ba-Borates, by contrast, displays striking differences.

We do not observe growth of the 770 cm-1 mode until x > 15 mol% of BaO, and furthermore, the

Boroxyl ring mode stays at 808 cm-1. These are features that are most certainly suggestive of the immiscibility of the B2O3 liquid, and BaO liquid at elevated temperatures, as I will discuss in section 6.4. The 770 cm-1 mode is found to suddenly grow at x = 15 mol% and to show a

maximum near 20 mol% of BaO; this is an observation that strongly suggests that the “mixed-

ring” forms part of Ba-tetraborate, i.e., 1 Ba for 4 B atoms leading to a BaO molar content of 1/5

or 20 mol%. The second feature observed in the plot, is that of a mode near 750 cm-1, as illustrated in red, which shows a peak near a BaO mol% of 33 mol%. This leads to the finding that the mixed ring in question forms part of a Ba-diborate SG.

(Na2O)x(B2O3)100-x

(a)

(BaO)x(B2O3)100-x

(b)

FIGURE 6-7: Quantification of growth/reduction of fractional mode scattering strengths associated with the Boroxyl Ring, Tetra-, Di-, and Tri- borate SG in Na-modified (a) and Ba- modified glasses (b).

It is quite remarkable that the two SGs deduced from our detailed Raman Scattering experiments, viz, Ba-tetraborate and Ba-diborate, are actually the only two known crystalline compounds that

form in the well-established equilibrium phase diagram [39]. The results of figure 6-7 are strongly suggestive that characteristic “extended-range-structural-groupings”, based on the crystalline phases, define the structure of these modified oxides. These structural descriptions are far removed from the continuous-random-network description of modified borates advocated by several groups in the past [22, 65-67].

The structural groupings of Ba-tetraborate and Ba-diborate taken from [34] are shown in figure

- 6-8 and figure 6-9 respectively. As one would have expected, there are two [BO4] tetrahedra in

the vicinity of each Ba2+ cation that provide for charge compensation. The structure of Ba-

tetraborate is shown in figure 6-8.

Ba2+

B O Ba2+

FIGURE 6-8: Crystalline structure of Barium Tetra-borate[34]; the green dashed lines represent - direction of constraint between the Barium cation and the two [BO4] anions.

Ba2+

FIGURE 6-9: This SG forms the extended range structure of crystalline Ba-diborate [33]

EXAMPLE 6-1 Constraint Counting Arguments for the Barium Tetra-Borate

#Elements nc

2 ×Ba 2 𝑟𝑟 2

26 × O 2 2

12 B 3 9/2

4 B 4 7

r (constraints / r coordinated atom )n = nα + nβ = + (2r 3) 2 − c − [ + ( )] × + × + × = = = = . > 𝐫𝐫 𝐫𝐫 𝐫𝐫 + +𝟗𝟗 ∑ 𝐧𝐧 𝟐𝟐𝟐𝟐 − 𝟑𝟑 𝟐𝟐𝟐𝟐 𝟐𝟐 𝟏𝟏𝟏𝟏 𝟒𝟒 𝟕𝟕 𝟏𝟏𝟏𝟏𝟏𝟏 〈𝒏𝒏𝒄𝒄〉 𝟐𝟐 𝟐𝟐 𝟑𝟑 𝟏𝟏𝟏𝟏 𝟑𝟑 ∑𝐫𝐫 𝐧𝐧𝐫𝐫 𝟐𝟐𝟐𝟐 𝟏𝟏𝟏𝟏 𝟒𝟒 𝟒𝟒𝟒𝟒 83

EXAMPLE 6-2

# Elements n c

𝑟𝑟 2 ×Ba 2 2

10 × O 2 2

4 2 2 × 2 𝑂𝑂

4 B3 3 9/2

4 B4 4 7

( / ) = + = + (2 3) 2 𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑐𝑐 𝑛𝑛𝛼𝛼 𝑛𝑛𝛽𝛽 𝑟𝑟 − 9 [ + (2 3)] 14 × 2 + 4 × + 4 × 7 2 2 = = = 3.36 > 3 𝑟𝑟 14 + 4 + 4 ∑𝑟𝑟 𝑛𝑛𝑟𝑟 𝑟𝑟 − 〈𝑛𝑛𝑐𝑐〉 ∑𝑟𝑟 𝑛𝑛𝑟𝑟

Note, that in enumerating contraints, Example 6-1, one finds this species to be slightly stressed

2+ - rigid. Ba cation forms two ionic bonds, one with each [BO4] tetrahedral unit, and we thus estimate the count of bonding constraints taking its coordination number to be 2.0. On the other

+ - hand, Na cation will form one ionic bond with a [BO4] tetrahedral unit, and will possess a coordination number of 1. Using these coordination numbers, one can estimate the total count of bonding constraints for various SGs.

Edge-modified Barium di-borate SG

Ba2+

EXAMPLE 6-3

# Elements n c

3 ×Ba 2𝑟𝑟 2

10 × O 2 2

2 2 2 × 2 2 × ( ) 𝑂𝑂 4 B3 3 9/2 𝑂𝑂 𝑁𝑁𝑁𝑁𝑁𝑁

4 B4 4 7

( / ) = + = + (2 3) 2 𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑐𝑐 𝑛𝑛𝛼𝛼 𝑛𝑛𝛽𝛽 𝑟𝑟 − 9 [ + (2 3)] 16 × 2 + 4 × + 4 × 7 2 2 = = = 3.25 > 3 𝑟𝑟 16 + 4 + 4 ∑𝑟𝑟 𝑛𝑛𝑟𝑟 𝑟𝑟 − 〈𝑛𝑛𝑐𝑐〉 ∑𝑟𝑟 𝑛𝑛𝑟𝑟

Furthermore, upon enumerating bonding constraints for the Barium Di-borate SG, in Example

6-2, one finds this entity to be stressed-rigid. However, if one were to consider the creation of non-bridging oxygen (NBO), which undoubtely form, one can then certainly anticipate the formation of isostatically rigid Ba-tetrabotae and Ba-diborate SGs. These SGs would percolate in the narrow IP of the glasses which, inferred from calorimetric experiments (figure 6-7), occurs at the 24 mol% < x < 32 mol% range of BaO. Therefore, one expects the IP in Ba-borate glasses to be formed by a mix of the Ba-tetraborate, and Ba-diborate SGs dressed by NBO atoms on the surface to form an isostatically stree-free environment. Such dressing of NBO on

SG edges or surfaces is expected to be a peculiar feature of glasses which would not be present

- in corresponding crystals. Lowering the count of constraints the BØ2O species function as a

+ - “stress-releaser”. In sharp contrast, a Na cation modifier compensating a [BO4] cation can be shown to be isostatic, with the important consequence that the Na-Triborate and Na Diborate and

Na-Tri-pentaborate are isostatically rigid rings. Additionally, the IP in the Sodium-Borate glasses is not much wider (20 mol% < x < 40 mol%), but onsets when the Na-triborate species percolate in the network and comes to an end when the Na-Tri-pentaborate species cease to percolate in the glass backbone as discussed elsewhere [17, 18].

6.3 IR reflectance a quantitative probe of B4/B3 content of BaO modified

B2O3

The thermo-Nicolet FTIR system was used to record specular IR reflectance from the bulk

glasses. The reflectance data was then Kramer-Kronig transformed to deduce the optical absorption. These data, as a function of frequency, were systematically examined as a function

3+ of glass composition. It is widely recognized that the B species, as in the base B2O3 glass,

gives rise to features in the IR absorption in the 1200-1800 cm-1 range. On the other hand, as

4+ - BaO modifies the base glass and produces B species (as in the BO4 tetrahedral), the IR

response shifts to 800-1200 cm-1 range. These considerations have been useful in quantitatively separating the contribution to the absorption of the 4-fold B species from the 3-fold B ones.

The absorption data from the present experimental effort is shown on the left panel of figure

6-10. Professor Kamitsos’ group have also examined IR responses of Ba-Borate glasses [25],

and their data is summarized in the right panel of figure 6-10. The global pattern of the IR

response of glasses in the two panels of figure 6-10, reveals some striking similarities. One can

-1 clearly see the IR response due to B4 species (in the 800-1200 cm rage) to steadily increase

with BaO molar content, and this is most comforting. By integrating the IR absorption response

-1 -1 in the 800-1200 cm and 1200-1600 cm regions, I have obtained the contributions of B4 and

B3 species, respectively. Close examination of these data reveal prominent differences. For example, (i) at x = 37 mol% or 40 mol%, the B4 response in my samples is significantly smaller

than the one observed by Yannopoulos et al. (ii) A mode near 1600 cm-1 which is clearly

observed in my samples is not observed in the glasses of Yannapoulos et al. A close comparison

of the observed absorption of the x = 33 mol% BaO clearly shows a characteristic in my samples

at 1600 cm-1, but it is absent in the spectrum of the other group. This pattern is broadly observed

at other compositions as well. This feature of the data, namely the clear absence of the high

frequency response of B3 species, is suggestive of an intrinsic difference between our samples

and those of Yannopoulos et al. Unfortunately, the Tg reported by Yannopoulos et al. in their

paper was taken from those reported by the Iowa group [63], and these thermal data are

therefore not representative of the wetness or dryness of the glasses synthesized by Yannopoulos

et al. did not explicitly report Tgs in their samples.

Unfortunately, Yannopoulos did not show the IR absorption in the 3200 cm-1 range where the

OH-stretch of bonded water in Borate glasses is manifested. For example, in wet B2O3 glass samples, Vignarooban observed significant responses near 3200 cm-1(see figure 6-12). These were glasses of very dry B2O3 glass exposed to the laboratory environment for several minutes.

BO3

BO4

FIGURE 6-10: Left is the present results from IR –reflectance measurements, Right-IR data from Yiannopoulous [25]. Note that the peak around 1600 cm-1 is indicative of triangular units of - B 3 and B 2O .

∅ ∅ 88

-1 The peak observed at 3200 cm was absent in the dry B2O3 glasses. In my Ba-borate samples, I

did not observe evidence of an IR response in the 3200 cm-1 range, suggesting that the Ba-Borate glasses are dry. In the present work, special care was taken to ensure dry and homogeneous samples. In figure 6-11, I plot the Boron modified fraction, i.e., B4/B3, deduced from the IR

response of the present Ba-borate glasses as a function of BaO content x. In figure 6-11, I have

also included the corresponding B4/B3 fraction deduced by Yannopoulos et al. in their Ba-borate

glasses. This data clearly shows that the other group observes much greater modification of their

glasses than I have. I could not make homogeneous bulk glasses at x > 40 mol%, but the other

group has reported glass forming tendency up to x = 47 mol%. The principal difference in

synthesis between the two sets of samples can be traced to the alloying conditions. Yannopoulos used a box furnace to alloy the starting materials in the 1100-1300 oC range for 30 minutes in Pt

crucibles followed by a splat quench of the melts on polished Cu plates. I, on the other hand,

used an induction furnace to briefly react the same starting materials in Pt crucibles, followed by

a roller quench of the melts. I found that melts reacted in a box furnace, alone, did not look

homogeneous as studied by Raman Scattering or visible inspection. The Tg of my glasses are

found to be substantially higher than those reported by Lower et al [63]. In sharp contrast to the

Raman Scattering results in my glasses, Yannopoulos et al. did not observe a strong Boson mode

in their glasses. It is widely acknowledged that a strong Boson mode is generally regarded as the

signature of a well alloyed homogeneous glass.

FIGURE 6-11: IR response deduced B modified fraction, B4/B3 plotted as a function of BaO molar content in the present glasses (blue circles) and in the glasses of Yannopoulos et al. [25] (red circles).

FIGURE 6-12: FTIR absorption data taken from Vignarooban [17]. The peak around 3200 cm- 1 grows in the wet sample.

FIGURE 6-13: Full range of absorption data for the x = 15 mol% and x =20 mol% showing absence of B-OH absorption peak.

6.4 Microscopic origin of the Immiscibility range in the Equilibrium

Phase diagram of the BaO-B2O3 binary

An important feature of the BaO-B2O3 equilibrium phase diagram is the existence of an

immiscibility range between 0 < x < 15 mol % of BaO. B2O3 melts do not react or alloy with

BaO melts until there is a sufficient amount of the modifier for a homogeneous melt to form and

for glasses to result upon rapid cooling. In practical terms, the phenomenon is remarkable as

sketched in the figure below:

FIGURE 6-14: Equilibrium phase diagram of the B2O3-BaO binary[39] showing that three distinct crystalline compounds form at x = 20 mol% (Ba-tetraborate) , x = 33 mol% (Ba- diborate) and x = 50 mol% (Ba-meta Borate). In addition, an immiscible range exists between x = 0 and x = 15 mol% of BaO where melts do not alloy.

X = 15%

FIGURE 6-15: The left figure shows an attempt to synthesize a glass at 4 mol% of BaO , while the right figure shows a similar atempt at x = 15 mol% of BaO. Note that a homogenoes glass forms x = 15 mol% of BaO, but a stained and heterogeneous glass forms at x = 4 mol%. The heterogenous product at x = 4 mol% is composed of BaO inclusions in a B2O3 glass.

It is hard to obtain a transparent bulk glass at x < 15 mol% of BaO, but a transparent bulk glass

readily forms at x ≥ 15 mol% BaO (figure 6-15). The equilibrium phase diagram of the BaO-

B2O3 pseudo-binary phase diagram shows the existence of only three crystalline phases: Barium

Tetra-borate ([BaO]4[B2O3], x =20 mol%), Barium Di-borate ([BaO]∙2[B2O3], x=33 mol%),

and the Barium Meta-borate (BaO∙B2O3, x=50 mol%). In the immiscibility region, glasses remain macroscopically heterogeneous and appear to maintain BaO-rich regions suspended in a transparent matrix (B2O3-rich regions); the x = 4 mol% is displayed in figure 6-15 and is

characteristic of glasses in the immiscibility region. Above 15 mol%, glasses appear

homogeneous as illustrated by the 15 mol% glass, which is characteristic of glasses up to 40

mol% BaO. Upon closer examination of figure 6-7, we see can that the Barium Tetra-borate

group doesn’t nucleate until we reach 15 mol% BaO.

The Raman Scattering results of figure 6-3 at x = 4 mol% and 8 mol% of Ba are completely consistent with the notion of the immiscibility of BaO and B2O3 which melts at x < 15 mol% Ba.

In these spectra, we could not detect the 770 cm-1 mode, and furthermore, the 808 cm-1 BR mode

did not soften as it would have, had alloying taken place in the melt. Remarkably, as x increases

to 15 mol%, glass formation becomes facile as transparent bulk glasses are easily obtained by

melt quenching. The result can be understood on a molecular scale as follows: If BaO modifier

were to nucleate the Ba-pentaborate species, the immiscibility window would not exist. In the

pentaborate and triborate SGs, there is an odd number of BO4 tetrahedra formed per formula

2+ unit. With a divalent Ba modifier, on the other hand, one would need two BO4 units to form

close together for charge compensation. The BO4 units cannot be far apart in two separate

formula units otherwise the Coulomb binding energy becomes minuscule and the unit is

energetically unfavorable. Indeed, my Raman Scattering results unambiguously demonstrate

that there has to be at least 15 mol% of BaO for the Ba-tetraborate species to form. The Ba-

2+ tetraborate species has a pair of BO4 tetrahedra per Ba cation for charge compensation to occur. Thus, we can understand the existence of an immiscible range in Ba-Borate glasses on a molecular scale from these Raman Scattering data. Such an immiscible range does not appear with either the Li- or the Na- borates because in both instances, the Raman Scattering results clearly show that alkali pentaborate and tetraborate readily form. In summary, my Raman

Scattering data clearly reveal that a local minimum in the configurational energy must exist to nucleate the Barium Tetra-borate structural group for homogeneous Ba-Borate glasses to form.

6.5 Glass Network dimensionality considerations and the origin of the

Boson mode in Borate glasses

A Boron atom belongs to the Column-III of the periodic table with a valence of 3 and with an

2 1 2 electronic structure of s p . BO3 triangles are an ideal realization of the sp hybridization as first

enunciated by the pioneering work of Nobel Laureates Linus Pauling and Herbert Brown. The B

in the BO3 triangular unit forms three sigma bonds with the nearest neighbor, oxygen, in a strictly planar (2D) geometry. The Boroxyl ring makes use of three BO3 triangular units to form

a three-membered ring in a strictly planar geometry, as was elucidated by the late Frank Galeener

[53, 54]. Given that nearly 75 to 80 mol% of the BO3 units in B2O3 glass exist in Boroxyl rings,

one would be hard pressed to view B2O3 glass as a 3D Continuous Random network of BO3

triangles, as advocated by several pioneers of glass science in the early 1950s and 1960s [68].

B2O3 glass is better viewed as being composed of a 2D sub-network of BR embedded in a 3D

network. The BO3 triangular units, as the constraint counting algorithm shows, has a B atom

with 3/2 bond-stretching and two bond bending constraints, and the three Oxygen atoms have

each one bond-stretching and one bond bending constraint, to give a total count of [1.5 + 2 + 3 x

0.5 x 2]/2.5 = 6.5/2.5 = 2.6 constraints/atom. In 2D space, the BO3 triangular unit would be regarded as Stressed-rigid because the number of constraints per atom is larger than the degrees

of freedom (nc > 2).

Using ideas of Mosseri and Dixmier [69] on sub-network dimensionality, it was recently shown

that in Na-borate glasses, the effective count of constraints of 2.6 /atom for B2O3 will increase to

3.0 at x = 20 mol% as the global network becomes 3D with increasing soda concentration. The details of the calculations are discussed in [18].

These ideas will also apply to the case of Ba-Borates, largely because the 2D to 3D

morphological change is driven by a change in dimensionality of the local structures; 2D BO3

units into 3D BO4 units. The increase of network dimensionality drives the stressed-rigid BO3

triangular unit in conjunction with BO4 tetrahedra to become isostatic (nc = 3), thus optimizing the glass forming tendency.

There are some broad consequences of the dimensional crossover discussed above, and one of these bears directly on the origin of the Boson mode in these glasses. The Boson mode in these glasses is by far the strongest mode observed in Raman spectra of these B2O3 rich glasses.

Experiments show that the scattering strength of this mode decreases linearly with a decreasing

fraction of Boroxyl rings in the glasses at a low modifier content. The correlation between the

Raman Scattering strength of the Boson mode with the BR mode was first noted in the Sodium

Borate glasses, work conducted by Vignarooban [17], shown in figure 6-17. That correlation also persists in the Ba-Borate glasses as shown in the figure 6-16 below.

These data are persuasive in suggesting that the origin of the Boson mode is intrinsically tied to

the 2D character of the B2O3 rich- glasses. Recently, Naumis [70] has suggested that a Boson mode (low frequency mode) may exist in stressed-rigid glass if flexible springs (weaker bonds) coexist in a network composed of stiffer springs (stronger bonds). The B2O3 glass is an ideal example of such a realization as I comment next.

The sigma bonds in BR are viewed as the strong bonds (120 Kcal/mole) or stiffer springs that determine the 808 cm-1 normal mode of the three membered rings. The van der Waals

interaction (5 -10 kcal mole) between the BRs and the rest of the network is mediated by

Oxygen lone-pair electrons. The latter interactions represent the weaker bonds or softer springs

that serve to stabilize the 2D character of the rings in these glasses. One interesting correlation is

that the ratio of the sigma bond strength to the van der Waals bond strength (~18) scales with the

ratio of the vibrational mode frequency of the BR-mode to the Boson-mode (at 808 cm-1/50cm-1

= 16). Given this analogy, one then expects the count of flexible springs to steadily decrease as

the BR count in the glasses decreases upon modification of the base glass by alkali- or alkaline-

earth modifiers. These considerations provide, in a natural fashion, the correlation of the

observed scattering strength of the BR to the Boson mode.

As a final comment, I would like to remind the reader that in many discussions of B2O3 glasses,

one has often assumed BO3 units to be pyramidal as in AsSe3 or PSe3 in a 3D geometry. In such

geometry, the count bonding constraints of BO3 pyramids reveal that they are isosatic. For an isostatic network, Maxwell’s constraint counting would predict the count of floppy modes to

vanish in the mean-field picture. Additionally, if the Boson mode is a manifestation of the

floppy modes, one expects the Boson mode to not be observed in a B2O3 glass. I capture these

two different approaches to estimating constraints of BO3 triangular units in table 6-1. In this

table, the left panel describes what I believe to be the realistic origin of the Boson mode in B2O3

rich glasses. The right panel shows the perspective used in enumerating constraints in previous

models of these glasses.

FIGURE 6-16: Top left (a) and right (c) show the global trend in scattering strength of the Boson mode for the present work on (BaO)x(B2O3)100-x and the work conducted by Baranov et al. [71] on (Na2O)x(B2O3)100-x, respectively. The bottom panels show the Boson mode in the stressed–rigid regime correlating among both systems. Moreover, the BR mode shows correlative reduction in the present work.

FIGURE 6-17: Correlation of the Boroxyl Mode and the Boson mode in Sodium Borates from the work of Vignarooban [17]. Notice the similarities in the slope to the present work and that of Baranov.

TABLE 6-1: Relationship of the Boson mode for a Planar and a Trigonal BO3 Unit.

2D 3D

two angles three angles needed needed

3 1 3 1 = ( ) + 2 + 3 × × 2 = ( ) + 3 + 3 × × 2 2 2 2 2

𝑛𝑛𝑐𝑐 𝑛𝑛𝛼𝛼 �𝑛𝑛𝛽𝛽 � 𝑛𝑛𝑐𝑐 𝑛𝑛𝛼𝛼 �𝑛𝑛𝛽𝛽 � = 6.5 = 7.5

6.5 7.5 = = = 2.60 = = = 3 2.5 2.5 𝑛𝑛𝑐𝑐 𝑛𝑛𝑐𝑐 𝑟𝑟̅ 𝑟𝑟̅ 𝑁𝑁 𝑁𝑁

Stressed-Rigid Isostatic

Predicted Boson Mode Scattering High* 0 Strength

Observation of High Boson Mode

6.6 Boson mode and the Stress and Rigidity transitions in Ba-borate

glasses

The origin of the low frequency vibration in Raman Scattering of network glasses has stimulated

much discussion in the field of glass science over the past few decades. By now, it has become

apparent that several mechanisms give rise to such excitations [72]. As shown in the previous

section, for the case of B2O3 rich glasses, I identified a new mechanism for the Boson mode associated with the weak off-planar van der Waals interactions that stabilize the Boroxyl rings.

My experimental results on the Boson mode (figure 6-4) also reveal that the scattering strength of this mode shows a step-like increase near the rigidity transition at x = 32 mol% of BaO and then continues to increase at higher x (> 32 mol%). The rigidity transition marks the phase boundary between the IP and the Flexible phase. The observation is reminiscent of floppy mode excitation in the flexible phase of the present Ba-Borate glasses. Viewed in this fashion, the

Boson mode becomes the floppy mode excitation of the elastically flexible glasses.

Parallel results were noted earlier in two glass systems where the Boson mode scattering strength increased monotonically in the flexile phase. Specifically, in binary AsxS100-x glasses [73], the

IP was found to reside in the 22.5 mol% < x < 29.5 mol% range. The Boson mode scattering strength increased steadily as the As-content of the glasses decreased below x = 20 mol% upon entry into the flexible phase. The second example is that of the solid electrolyte pseudo-binary glass system (AgPO3)100-x (AgI)x [60], wherein the IP was found to reside in the 9.5 mol% < x <

37.8 mol% of AgI, with glasses at composition x > 37.8 mol% becoming flexible. The Boson

mode scattering strength increased almost linearly at x > 37.8 mol% with a slope quite close to

100 the expected behavior of the growth of floppy mode excitations in the Flexible phase of these glasses based on rigidity theory.

In conclusion, the present work on Ba-borate glasses suggest two distinct mechanisms that contribute to the Boson mode observed in Raman Scattering experiments. At low BaO content, x < 25 mol% of Ba, I identify the Boson mode with the weak off-planar van der Waals interactions that stabilize the planar Boroxyl rings in the glasses. At higher BaO content, particularly x > 32 mol% of Ba, the compositional behavior of these excitations closely parallels the floppy modes in Flexible glasses as predicted by Rigidity Theory

101

Chapter 7 Conclusions and Suggestions for future work

7.1 Conclusions on (BaO)x(B2O3)100-x binary glass system

Titled glasses over a wide composition range, 0 mol% < x < 40 mol%, were synthesized by

induction melting precursors and handling the glasses in a dry ambient. These high quality

glasses were comprehensively examined in Modulated DSC, Raman Scattering, Infrared

Reflectance, X-ray Diffraction and Molar volume measurements. The following conclusions

can be drawn on this project.

A. Modulated DSC experiments reveal (i) the compositional variation of the glass transition

temperature, Tg(x), shows a global maximum in the 25 mol% < x < 45% range of BaO, (ii)

a reversibility window is observed in the 24 mol% < x < 32 mol% range. The observation

fixes the three Topological phases in these glasses; compositions in the range 0 mol% < x <

24 mol% are in the stressed-rigid range, those in the 24 mol% < x < 32 mol% range in the

isostatically rigid Intermediate Phase, while those at x > 32 mol% in the elastically flexible

range.

B. Raman scattering measurements reveal (i) the low frequency Boson mode to be the most

intense mode observed in the base B2O3 glass, and its scattering strength to steadily decrease

with BaO content, scaling with that of the Boroxyl ring mode near 808 cm-1. The correlation

underscores that the Boson mode at low x results due to weak off-planar van der Waals

interaction mediated by the Oxygen lone pair electrons on the planar BRs with the rest of the

network that stabilize the planar ring structure. (ii) The rich lineshapes observed as a

function of BaO content reveal mixed-ring modes that can be traced to the formation of Ba-

102

tetraborate and Ba-diborate structural groupings- these are the only known species in the

crystalline Ba-Borates.

C. Compositional trends in IR absorption directly yield the fraction, B4/B3, of modified B as

BaO is alloyed in the base glass. Trends in Tg(x), Raman mode scattering strength of the

-1 - 1500 cm mode (due to BØ2O triangular units), B4/B3 fraction, each observable displays a

maximum in the reversibility window, underscoring the role of stress-relieving Non-bridging

Oxygen atoms in formation of the Intermediate Phase.

D. A microscopic understanding of the Immiscibility range in the titled glasses in the 0 < x <

15 mol% range can be traced to the formation of Ba-tetraborate as the only most Ba deficient

SG. For liquid B2O3 and BaO to alloy, there has to be sufficient BaO to nucleate the Ba-

tetraborate SG for glass formation to be observed.

7.2 Conclusions on (BaO)x[32 (B2O3) 68 (SiO2)]100-x pseudo-ternary glasses

Titled glasses over a wide composition range, 25 mol% < x < 48 mol% , were synthesized

by induction melting precursors and handling the glasses in a dry ambient. These high quality

glasses were comprehensively examined in Thermo Gravimetric Analyzer /Differential

Scanning Calorimeter (TGA/DSC), Raman Scattering, and Molar volume measurements. The

following conclusions can be drawn on this project.

A. TGA/DSC experiments reveal (i) the compositional variation of the glass transition

temperature, Tg(x), shows a global maximum in the 29 mol% < x < 32 mol% range of

BaO, (ii) a reversibility window is observed in the 29 mol% < x < 32 mol% range. The

observation fixes the three Topological phases in these glasses; compositions in the range 24

mol%< x < 29 mol% are in the stressed-rigid range, those in the 24 mol% < x < 32 mol%

103

range in the isostatically rigid Intermediate Phase, while those at 32 mol% < x < 48 mol% in

the elastically flexible range.

B. Raman scattering measurements reveal rich line-shapes that evolve systematically with the

BaO content of the glasses. Also evident from Raman studies the local structures formed in

2 − 3 − these glasses include Q (Si−O stretching in Si R4 with 2-briding oxygen) and Q (Si−O

4 − stretching in Si R4 with 3-briding oxygen) but not∅ in Q - , tetrahedral BO4, and B R2O

4 units. Furthermore∅ from Raman experiments intermediate𝑆𝑆𝑆𝑆∅ range order of the glasses∅ are

dominated by 4-membered rings containing and BO4 tetrahedral units in the ratio of 3:1

4 as in Reedmergenerite, and in the ratio of 𝑆𝑆2:2,𝑆𝑆∅ as in Danburite. These local structures are

inferred from their characteristic vibrational modes. The observed variations of Raman mode

scattering strengths show that in the low x-regime the modifier serves largely to assist in

polymerizing the backbone as BO3 triangles are converted to BO4 tetrahedra, while in the

high x regime, the modifier serves largely to depolymerize the network by creating NBO

2 3 − atoms, which results in the growth of Q species at the expense of Q ones, and B R2O

4 species in orthoborates grow at the expens𝑆𝑆𝑆𝑆∅e of BO3 ones. ∅

C. Molar, Vm(x), show a global minimum, features that we associate with the isostatically

rigid elastic phase, also called the intermediate phase

7.3 Suggestions for future work

A. It would be of much interest to examine the (SrO)x(B2O3)100-x and carry forward the

experimental plan performed for the Ba alkali-earth additive. We have a two-fold interest in this system. It is likely the reversibility window would be more square well like than Gaussian with

the SrO modifier because of the anticipated higher mobility of Sr (with a smaller mass, 89 amu)

104

than Ba (133 amu). Furthermore, AC conductivity measurements on Sr-Borate would permit us to see if the ion mobilities track the three topological phases and allow us to display the three

regimes of behavior expected, and as noted in Sodium borate glasses.

B. Because of the most unusual non-linear optical properties of the Ba-metaborate, it will be possible to synthesize Ba-Borate ceramics with Ba-metaborate inclusions that could be of much practical interest in Non-linear optics. Such work will be a natural extension of the present work at BaO content exceeding 40 mol% of BaO.

C. 11B NMR experiments on the high quality Ba-Borate glasses synthesized in the present program would be useful to perform, and build upon the existing knowledge base of glass structure established from the present work.

105

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Chapter 8 Appendix The Published work on the pseudo-ternary system (BaO)x[(B2O3)32(SiO2)68]100-x

The following pages present a copy of the paper published on modified Barium Borosilicate

glasses published in the Journal of Chemical Physics in the year 2014. Below you will find

references for where this published work may be obtained.

Topology and glass structure evolution in (BaO) x ((B2O3)32(SiO2)68)100 x ternary—Evidence of rigid, intermediate, and flexible phases

C. Holbrook, Shibalik Chakraborty, S. Ravindren, P. Boolchand, Jonathan T. Goldstein, and C. E. Stutz Citation: The Journal of Chemical Physics 140, 144506 (2014); doi: 10.1063/1.4869348 View online: http://dx.doi.org/10.1063/1.4869348 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/14?ver=pdfcov Published by the AIP Publishing

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Topology and glass structure evolution in (BaO) x ((B2O3)32(SiO2)68)100 x ternary—Evidence of rigid, intermediate, and flexible phases C. Holbrook, Shibalik Chakraborty, S. Ravindren, P. Boolchand, Jonathan T. Goldstein, and C. E. Stutz

Citation: The Journal of Chemical Physics 140, 144506 (2014); doi: 10.1063/1.4869348 View online: http://dx.doi.org/10.1063/1.4869348 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/14?ver=pdfcov Published by the AIP Publishing

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This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 50.5.17.99 On: Mon, 14 Apr 2014 18:56:58 THE JOURNAL OF CHEMICAL PHYSICS 140, 144506 (2014)

Topology and glass structure evolution in (BaO)x ((B2O3)32(SiO2)68)100−x ternary—Evidence of rigid, intermediate, and ﬂexible phases C. Holbrook,1 Shibalik Chakraborty,2 S. Ravindren,2 P. Boolchand,2 Jonathan T. Goldstein,3 and C. E. Stutz3 1AFRL/RYDP, 2241 Avionics Circle, B620, Wright-Patterson AFB, Ohio 45433-7707, USA 2Department of Electrical and Computing Systems, College of Engineering and Applied Science, University of Cincinnati, Cincinnati, Ohio 45221-0030, USA 3AFRL/RXAN, 3005 Hobson Way, B651, Wright-Patterson AFB, Ohio 45433-7707, USA (Received 24 November 2013; accepted 12 March 2014; published online 14 April 2014)

We examine variations in the glass transition temperature (Tg(x)), molar volume (Vm(x)), and Raman scattering of titled glasses as a function of modifier (BaO) content in the 25% < x < 48% range. Three distinct regimes of behavior are observed; at low x, 24% < x < 29% range, the modifier largely polymerizes the backbone, Tg(x) increase, features that we identify with the stressed-rigid elastic phase. At high x, 32% < x < 48% range, the modifier depolymerizes the network by creating non-bridging oxygen (NBO) atoms; in this regime Tg(x) decreases, and networks are viewed to be in the flexible elastic phase. In the narrow intermediate x regime, 29% < x < 32% range, Tg(x)showsa broad global maximum almost independent of x, and Raman mode scattering strengths and mode frequencies become relatively x-independent, Vm(x) show a global minimum, features that we associate with the isostatically rigid elastic phase, also called the intermediate phase. In this phase, medium range structures adapt as revealed by the count of Lagrangian bonding constraints and Raman mode scattering strengths. © 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4869348]

Covalent networks based on the chalcogenides were I. INTRODUCTION the first test systems used to check the prediction of a Rigidity theory1–4 has proved to be a remarkable inter- Maxwell rigidity transition. Interestingly, experiments re- pretive tool utilizing topological constraints to describe glassy vealed the existence of the three generic elastic phases,in- networks. According to this theory, covalent networks can be cluding IPs.14–16 In recent years these phases have also been described in terms of their bond lengths and angles, which observed in modified oxides such as the alkali silicates,17 al- can be viewed physically as mechanical constraints. With an kali germanates,18 and alkaline-earthsilicates.8 A significant increase of network connectivity, the number of mechanical step forward in extending rigidity theory to networks with constraints increases linearly. Here connectivity refers to the intrinsically broken 8-N bonding rule, including terminal or mean-coordination number (average number of bonds/atom), dangling ends19(such as Na+ or Ba2+), was the recent molec- a measure of network cross-linking. In 3D networks, when ular dynamic simulations20 in modified oxides which have the count of constraints per atom, nc, increases to 3, networks distinguished intact from broken constraints. This was made become isostatically rigid. Networks with nc < 3 are an elasti- possible by examining the variance (second moment) in bond- cally flexible phase, while those with nc > 3inastressed-rigid lengths and bond-angles associated with different atoms in phase. Thus as the network connectivity is steadily increased simulated networks. 21 one expects an elastic phase transition to be manifested when The phase diagram of the BaO − SiO2 − B2O3 ternary nc increases to 3, also known as the Maxwell rigidity percola- is shown in Figure 1 and displays patent compositions for tion transition. In 1998, it emerged that the condition, nc = 3, optical glass A (28 mol. % BaO), B (29 mol. % BaO), in real glassy systems with appropriate local structures, can C (25 mol. % BaO), and Schott AF45 (29 mol. % BaO, actually be fulfilled not at a singular composition, but over a alkali-free borosilicate) commercial grade glass in weight range of compositions defining an Intermediate Phase (IP).5, 6 percent. In the present work we have examined ternary The phase boundary between the Flexible Phase and the Inter- (BaO)x ((B2O3)32(SiO2)68)100−x glasses along the indicated mediate Phase represents the rigidity transition, while that be- green pathway in Figure 1. tween the Intermediate Phase and the Stressed-rigid phase the Particular care was taken to synthesize homogeneous stress transition.7 The rigid but unstressed nature of networks melts/glasses. The present glasses may also be viewed as cor- formed in the IP endows them with unusual functionalities, responding to the SiO2/B2O3 fraction K kept fixed at 2.12, 5 8 such as thermally reversing glass transitions, space filling, and the BaO/B2O3 fraction R varied in the 1.04 < R < 2.88 and qualitatively suppressed aging.9, 10 IPs can extend over range. Several earlier investigations of borosilicate glasses a finite range of compositions, as networks adapt11 to expel in the same range of K and R have been reported, but us- 22, 23 the stress creating redundant bonds by reconnecting, i.e., self- ing Na2O instead of BaO as the network modifier. Our organizing. Due to their unstressed nature,12 features of the IP goal here is to elucidate the role of the BaO modifier in have attracted interest in applications of these materials.4, 13 bringing about changes in thermal, optical, and mechanical

FIG. 2. DSC scan showing the glass transition of a glass at x = 29% taken FIG. 1. The green tie line joining BaO to a fixed SiO2/B2O3 molar ratio of K using a Q600 TGA/DSC. Endothermic heat flow is up. The inflection point = 2.12 shows the targeted system. The rectangle transparent box represents of the heat flow endotherm is taken as the Tg. the region of successfully quenched bulk glasses. Points A, B, C are weight percents of barium-oxide, boron-oxide, and silica reported in patented optical glass.24 The point AF 45 represents the Schott alkali-free commercial grade in a stream of dry N2 gas, and the endothermic heat flow near 25 ◦ glass. Tg recorded at a 10 C/min scan rate. The Tg is defined as the inflexion point of the endotherm, as shown in Figure 2 for a properties of the modified oxide glasses. Our results provide glass composition at x = 29%. Compositional trends in Tg(x) evidence for the three generic elastic phases observed in Ba- of the present Ba-borosilicate glasses show an increase at low borosilicate glasses as we will show in the present work. x, 24% < x < 29%, a decrease at high x, 32% < x < 48%, In Sec. II we provide the experimental results. In Sec. III and a broad maximum in the intermediate range of x, we discuss these results. The conclusions are summarized in 29% < x < 32%, where Tgs become nearly independent of Sec. IV. x (Figure 3). Such a behavior of Tg(x) has been noted earlier in alkali-borates and alkaline-earth borates27 but not in modi- II. EXPERIMENTAL fied silicates. Topologically, maximum Tg constitutes a signature for a A. Synthesis state where the network possesses its highest connectivity.28 Thirty gram size batches were synthesized using reagent- Thus the maximum of Tg in modified borates generally separates two distinct compositional regions, a low-x region grade H3BO3 (Sigma-Aldrich, 99.8%), BaCO3 (Sigma- where the modifier acts as a charge compensator (such as the Aldrich, 99.999% trace metal basis), and SiO2 (Sigma- Aldrich, 99.995% trace metal basis). The carbonate and acid creation of fourfold coordinated boron) promoting polymer- based precursors were utilized to introduce targeted binary ization of the network, and a high-x region where the modi- compounds due to their lower decomposition temperature, fier serves to depolymerize the network by creating nonbridg- which facilitates a lower melt mixing temperature. Mechan- ing oxygen (NBO). Noteworthy in Figure 3 are results of the ically mixed powders were de-carbonated and melted in a Pt90Rh10 crucible for several hours utilizing an inductively heated furnace. Inductive heating promotes sample homogenization through direct and localized heat. Here heating occurs via eddy currents in materials, in contrast to diffusion limited mixing in a conventional furnace which uses either surface-contact or radiant heating.26 Following inductive heating, the samples were equilibrated in a box furnace for several hours at a temperature of 1600 ◦C and then quenched on a brass plate. Glass transition temperature measurements (Tgs) of strictly box furnace melts were utilized for a compar- ative study, and this data will be presented next.

B. Differential scanning calorimetry Glass transitions were examined using a model Q600 DSC/TGA (Differential Scanning Calorimeter/Thermo FIG. 3. Tg(x) of bulk glasses showing three regimes labeled I, II, and III. Gravimetric Analyzer) from TA Instruments Inc. A 15 mg Regime II, shaded gray, shows a broad maximum of Tg, and corresponds to quantity of the bulk glass, placed in a gold pan, was heated the Intermediate phase. See text.

Box-Furnace melted (BFM) samples which show a lower Tg than those of induction melted glasses, and when examined in Raman scattering revealed to be inhomogeneous. These BFM samples were completely excluded from the present study.

C. Raman scattering Raman scattering studies made use of a T64000 Dis- persive Raman system from Horiba Jobin Yvon Inc. using a model BX41 microscope attachment from Olympus. The system was operated in a triple subtractive conﬁguration using a CCD detector. With the microscope attachment, using a 100× objective, spectra were acquired over a 1 μm resolution. This permitted us to check samples for homogeneity. The scattering was excited using the 514 nm line from an Ar-ion laser. The typical power used to excite the scattering was 1.5 mW exiting the 100× objective. Figure 4 illustrates the observed Raman spectra as a function of BaO content of glasses, obtained at room temperature. The observed line-shapes were deconvoluted29 as the superposi- tion of Lorentzian-Gaussian mix. An example of line-shape analysis9, 30 is provided for a glass composition at x = 26% in Figure 5. The deconvolution enabled mode widths, centroids, and scattering strengths to be established. In the spectra several generic features become transparent. Note that the S1 mode frequency (Figure 5) falls close to the broad peak centered −1 near the 400 cm of the SiO2-glass spectrum (Figure 4). We identify it with the stretch of the corner-sharing SiO4 tetrahedra of the base glass. The mode labeled R (Reedmergner- FIG. 5. Example of Raman spectrum of a glass at x = 26%, deconvo- ite) and D (Danburite) are identiﬁed with 4-membered rings luted as a Gaussian-Lorentzian mix. See Table I for mode assignments. of mixed SiO4 and BO4 tetrahedra, which comprise the ele- Variations in mode scattering strengths and frequency were established ments of intermediate range order. We shall discuss the mode with x. assignments in Sec. III. Variations in scattering strengths of the R- and D- modes (Figure 6) illustrate that the sum of their scattering strengths is almost constant with x. In other words, it has been

FIG. 4. Raman spectra of (BaO)x ((B2O3)32(SiO2)68)100−x glasses with in- FIG. 6. Scattering strength of R- and D-modes plotted as a function of Ba- creasing BaO content x indicated on the right of the spectra. Modes are la- Borosilicate glass composition x. The sums of the R- and D-mode scatter- 2 3 beled S1,R,D,Q , Q , B0, B1, B2, B3,andB4 and their assignments are ing strengths remain largely independent of x, suggesting that these modes summarized in Table I. Spectrum of SiO2 glass is shown at the bottom, and are generically related. The three regimes of variation are labeled as I, II, the D1 and D2 modes are due to 4-, and 3-membered rings. and III.

The scattering strength of modes labeled Q2 and Q3 (throughout this paper we use Nuclear Magnetic Resonance (NMR) notation, Qn, where n represents the number of bridging oxygen (BO)) also displays a complementary behavior. We note that the sum of their scattering strengths is nearly independent of x (Figure 7), suggesting that they are generi- 2 3 cally related. Modes Q and Q will be associated with SiO4 tetrahedra having 2 BO and 3 BO, respectively. Table I summarizes the 10 Raman active vibrational modes observed in the present glasses. The series of modes labeled B0 through B4 are thought to be associated with Boron centered local and intermediate range structures. Of −1 some interest is the B1 mode near 725 cm that we associate with mixed 3 membered rings composed of one 3- fold B and two 4-fold B. It steadily grows in scattering FIG. 7. Scattering strength of Q3-andQ2-modes plotted as a function of Ba- strength by a factor of 1.6 across the range of x examined. Borosilicate glass composition x. Note that in regime II (shaded), the Q3-and 2 Such a mode is observed in alkali-borate and is identiﬁed Q -mode scattering strengths remain independent of x suggesting an under- 31 lying structural arrest. with alkali-diborates structural groupings. We also observe −1 a mode labeled B0 near 460 cm that increases in scattering strength by a factor of 2 across the range of x exam- experimentally determined that the D mode grows at the ex- ined. The variation of the mode scattering strength is rapid pense of the R mode and therefore is generically related. in the low x regime but plateaus at high x. The mode is signa- This result will assist in understanding the evolution of glass ture of 4-fold tetrahedral B formed from BO3 triangles with a molecular structure as discussed in Sec. III B. Ba2+ serving as a charge compensating cation. Assignments

TABLE I. This table identiﬁes Raman Mode assignments, Mode Trends, and their FWHM (cm−1) determined experimentally.

Mode label Mode freq. (cm−1) Mode assig. FWHM (cm−1) Mode trends/Ref.

S1 390 A1 mode of SiO4 tetrahedra (SiO2-glass) 80 Scattering strength decreases with increasing x > 24%, Ref. 31 B0 460 A1 mode of BO4 tetrahedra 90 Scattering strength increases with increasing x > 24%, Ref. 31 R 530–560 Oxygen symmetric stretch in 96 Scattering strength decreases 4-membered rings of Reedmergnerite with increasing x > 24% and behavior complements the D-mode, Ref. 29 D 626 Oxygen symmetric stretch in 90 Scattering strength increases 4-membered rings of Danburite with increasing x > 24% and behavior complements the R-mode, Ref. 29 B1 725 Symmetric stretch of 3-membered diborate 115 Scattering strength increases rings composed of 2 BO4 and1BO3 with increasing x > 24%, unit as in alkali-diborates Ref. 31 B2 840 Symmetric stretch of planar orthoborate 30 Observed at x > 37% and 3− BO3 scattering strength increases conjointly with B3 mode at 1140 cm−1,Ref.31 Q2 940 Si−O− stretching in Q2 units 40 Scattering strength increases with increasing x > 24% and behavior complements the Q3mode, Ref. 35 Q3 1055 Si−O− stretching in Q3 units 110 Scattering strength increases with increasing x > 24% and behavior complements the Q2 mode, Ref. 35 B3 1140 Asymmetric stretch of planar 70 Observed at 40%, Ref. 31 3− orthoborate BO3 − B4 1425 B−O attached to large borate groups 120 Scattering strength increases with increasing x > 24%, Refs. 31 and 37

glasses. The minimum of molar volume seen in region II is similar to the one observed in Ba-silicates reported by Bourgel et al.8 We comment on these issues in Sec. III C.

III. DISCUSSION In this section we begin by providing justification for the Raman mode assignments (Sec. III A), which will permit us to elucidate the role of BaO as a modifier of the present Ba- borosilicate glasses. This will set the stage to discuss glass network structure evolution as BaO is alloyed in the borosilicate base network (Sec. III B). This will be followed by the identification of the three generic elastic phases in the present ternary glasses (Sec. III C).

FIG. 8. Molar volumes of (BaO)x ((B2O3)32(SiO2)68)100−x glasses examined as a function of BaO content, provide evidence of a narrowly defined minimum in the range 29% < x < 31%. A. Raman mode assignments, local and intermediate range structures 1. The R- and D-ring mode assignments of modes labeled B1, B2, B3, and B4 will be discussed in Sec. III A. Raman spectrum of bulk SiO2 reveals two sharp modes 32 labeled D1 and D2 as shown in Figure 4. Galeener et al. identified these modes with the presence of planar rings. D. Volumetric analysis −1 −1 The D1 mode at 497 cm and D2 at 606 cm are as- Mass density of the bulk glasses was measured using cribed, respectively, to 4-membered and 3-membered rings. a Mettler Toledo digital microbalance, model 185 with a The mode in question is a symmetric stretch of oxygen atoms hooked quartz fiber attached to the pan. Weights of bulk with the four Si atoms remaining stationary in a 4 mem- glasses were taken in air and in alcohol and the density de- bered ring (D1) as illustrated in Figure 9(a). In the present duced from Archimedes principle. In the measurements we Ba-borosilicates we identify the modes labeled R (535 cm−1) used bulk glass samples weighing at least 150 mg to achieve and D (626 cm−1) with 4-member rings constituted, respec- the 1/4% accuracy in density. Molar volumes Vm(x) were then tively, by 3 SiO4 and 1 BO4 tetrahedra (Figure 9(b)), and calculated from knowing the molecular weight of the glass 2SiO4 and 2 BO4 tetrahedra (Figure 9(c)). We would like and the measured density. The density of alcohol was cali- to draw an analogy between the R-mode at 535 cm−1 in brated using a Si single crystal (ρ = 2.330 g/cc). We sepa- our glasses, and a mode near 586 cm−1 reported by Manara rately used a Ge single crystal (ρ = 5.323 g/cc) to indepen- et al.29 in the mineral Reedmergnerite, which is composed of dently ascertain the accuracy of the density measurements. such rings. Furthermore, the D-mode near 626 cm−1 observed We observe Vm(x) to show a steep increase at low x in the in the present glasses may be compared to a mode near 614 24% < x < 28% range (region I), followed by a global min- cm−1 reported in the mineral Danburite by Manara et al.29 imum in the narrow range, 29% < x < 32% (region II), and The systematic blue-shift starting from the D1 mode in SiO2, a steady increase in the 32% < x < 42% range (region III) as to the R-mode and then the D-mode in the present glasses shown in Figure 8. can be traced to the shorter31 B–O bond length (1.43 Å) Variations in molar volumes provide important clues to compared to the Si–O bond length (1.59 Å) in respective the nature of the isostatically rigid elastic phase in network tetrahedral units. The shorter bonds lead to a stiffening of

FIG. 9. 4-membered (a) rings of SiO4 units in SiO2, (b) mixed rings of one BO4 and 3 SiO4 units as in Reedmergnerite, (c) mixed rings of 2 BO4 and 2 SiO4 units as in Danburite.

fied as the F2 scissor mode of SiO4 tetrahedra having, respectively, 2 and 1 NBO.8, 34 A Q4 species associated with 4 BO is known to have Raman activity near 1120 cm−1. In our experiments, the network at x = 24%, is apparently already modified enough that little or no evidence of a Q4 species could be observed even at the lowest x% composition. In sodium borosilicates, at a soda content of R = 0.78 or 25 mol. %, Manara et al.29 could observe evidence of Q4 mode in their Raman spectra. The growth in scattering of the Q2 mode at the expense of the Q3 highlights the fact that the Ba2+ cation is an effective modifier in depolymerizing the SiO2 base network by creating NBO, and that nearly all Q4 species convert to Q3 ones before homogeneous bulk glass formation can be observed in the present modified oxides. The sequence of modes labeled B0 through B4 come from FIG. 10. Raman vibrational mode frequency of D1 mode in SiO2 glass corre- modes of B-centered local units and larger structural group- sponding to n = 0, in 4 membered (SiO4)4−n(BO4)n rings. The R-mode and −1 D-mode in the present ternary glasses correspond to n = 1 and 2, respectively. ings as we comment next. The B0 mode near 460 cm repre- The stiffening of the 4-membered ring is driven by the shorter length of the sents the symmetric stretch of a BO4 tetrahedral unit, and has B–O bond relative to Si–O bond. See text. The smooth line is a polynomial been observed in Na-Borate glasses.31 The scattering strength fit. of this mode in the present glasses is found to increase by a factor of 1.5 in the 24% < x < 30% range, and at higher x the increase reduces significantly. That feature is in harmony the underlying spring constant (k) of the B–O bond com- with the present glassy networks polymerizing qualitatively pared to the Si–O bond, resulting in an up-shift in the nor- in the low x region, but that process slows down at high x, as 1/2 −1 mal mode frequency (ν = (k/m) ) of the symmetric stretch we shall comment in Sec. III C.TheB1 mode near 725 cm of the oxygen atom vibration observed in Raman scatter- represents a symmetric stretch of oxygen atoms in 3 mem- ing. One then expects the R-mode to blue shift relative to bered rings composed of 2 BO4 and1BO3 units as found in 31 the D1 mode in silica, and likewise the D-mode to blue- alkali-diborates. The scattering strength of this mode in our shift with respect to the R-mode (see Figure 10). The smooth glasses is quite similar to that of the B0 mode. In fact the pres- curve passing through the 3 data points in Figure 10 is the ence of this structural grouping in the glasses would require polynomial, that we also observe modes of BO4 tetrahedra as was found in the study of alkali-borates. The modes B and B are ob- −1 = + + 2 + 3 2 3 νR(cm ) 490 36.05n 10.92n 3.02n . (1) served at x > 37% in the present glasses, as well as in highly modified alkali borates.35, 36 These represent normal modes of In Eq. (1), the observed Raman mode frequency, νR, is related a planar orthoborate unit that is composed of a BO3 triangle to the count, n,ofBO4 tetrahedral units in the 4-membered = with the three oxygen near-neighbors of B being NBO, i.e., rings. Interestingly, an extrapolation of the curve to n 3 3− − − aBO unit. The 840 cm 1 mode is a symmetric A stretch, yields a mode frequency of 798 cm 1 for a hypothetical 4- 3 1 while the 1140 cm−1 an asymmetric stretch mode of the BO3− membered ring composed of 3 BO4 and 1 SiO4 tetrahedra. 3 The R- and D- ring modes serve as probes of medium unit. In the spectra (Figure 2) the scattering strength of the B B range structure of the glasses. The systematic growth of the 2 mode always exceeds that of the 3 mode. Furthermore, A D- ring fraction at the expense of the R-ring (Figure 6)issug- the line width of the symmetric 1 mode is found to be narrower (Table I) than the asymmetric one, consistent with the gestive that BO3 triangles converted to BO4 tetrahedral units with Ba2+ modifier, systematically replace SiO units in 4- assignment. 4 B −1 member R-rings, converting them to D-ones. The idea is in Finally, we come to the 4 mode (1420 cm ) that is harmony with the scattering strength of the D-mode growing found to display an increasing scattering strength with in- at the expense of the R-mode (Figure 6). creasing x in the present glasses (Figure 11) but with a rather striking feature that the variation displays a plateau in regime II, 29% < x < 32% composition range. Such a mode has also been observed in alkali-borates and has been assigned 2. Q2,Q3,B ,B ,B ,B ,andB mode assignments 0 1 2 3 4 to B–O− bond attached to large borate groups.36 Appearance There are other vibrational modes observed in Raman of this structural feature is viewed as a stress relief mecha- 2 3 spectra of the glasses (Figure 2) labeled as Q , Q , B0, B1, nism in which a BO is converted to a NBO as the backbone is B2, B3, and B4 that were commented earlier; we now discuss cut. Note that in regimes I and III the mode scattering strength their assignments. Our use of the notation Q3 and Q2 is moti- increases with x but in regime II it remains unchanged. The 3 2 vatedbytheSiQ and Q NMR notation, i.e., a Si having 3- plateau in scattering strength of the B4 mode in region II bridging and 2-bridging oxygen near- neighbors, respectively. is strongly suggestive that glasses in this composition range For instance, NMR work of Zhao et al.33 has provided a qual- are stress-free, and we will identify this regime with an iso- itative discussion on the structural role of NBO. In the Raman statically rigid intermediate phase in the present glasses (see spectra the modes near 940 cm−1 and 1055 cm−1 are identi- Sec. III C).

x = 24%. At the lowest composition of BaO, our results suggest that the Ba2+ cation serves both as a charge compensator creating BO4 tetrahedra and depolymerizes the network to create NBO to form Q3 species from Q4 ones, and B − O− bonds linked to large borate groups. How does the glass network structure evolve with increasing BaO content? We address the issue next. Low-x regime, 24% < x < 29% : In the low x regime, one observes (a) the concentration of D-rings increases from 20% to 30% (ii) while those of R-rings are depleted by about the same factor (Figure 6). The supply of the BO4 tetrahedra comes from the presence of either isolated BO3 triangles compensated by Ba2+ cations, and/ or from a characteristic tri- and/or di-borate groupings (B1 mode). In the same range of concentration, Q2 units grow from 2% to 9%, as their par- ent Q3 species deplete by the same amount (Figure 7). Thus the modifier produces slightly more Ba2+ cations that serve as −1 FIG. 11. Scattering strength variation of the of B4 mode near 1420 cm as charge compensators (10%) than NBO formers (7%), and one a function of x displaying a general increase with x except for regime II in expects, from topological considerations, the backbone to be- which we observe a plateau. come more connected. The glass transition temperature serves as direct measure of network connectivity28 and, indeed, one observes Tg to increase with x in this compositional regime B. Topology, glass structure evolution, (Figure 3). and T variation g High-x regime, 32% < x < 48% : In this range of Melts of SiO2 with B2O3 are immiscible and there is BaO content, the role of the modifier undergoes a qualita- no evidence of either crystalline phases or glass formation tive reversal—more NBO atoms are created than BO3 trian- 37 to occur. The result may be a consequence of the qualita- gles converted to BO4 ones. Raman scattering results show 2 tively different topology of B2O3 and SiO2; the former largely the scattering strength of Q species increases from 8% to composed of 3 membered BO3 triangles forming Boroxyl 28%, while that of the D-ring species increases from 32% to rings38, 39 in a quasi-2D network, while latter existing as a 42%. In other words Ba2+ cations are more effective at de- continuous random network (CRN) of SiO4 tetrahedra present polymerizing the network by creating NBO atoms rather than in a 3D network. Upon alloying a modifier such as an alkali- polymerizing it by serving as a charge compensator. These 32 35, 36 oxide or an alkaline-earth oxide, BO3 triangles con- conditions will lead the global connectivity of the glass net- 4 vert to BO4 tetrahedra while SiO4 tetrahedra (Q in NMR work to decrease. Not surprisingly, one finds Tg of the glasses notation) are modified to Q3 and then into Q2 tetrahedra as (Figure 3) systematically decreases in this compositional BO convert to NBO. A modifier thus facilitates mixing of regime. the BO4 tetrahedra with SiO4 ones forming mixed rings, as Intermediate-x range 29% < x < 32%: The narrow a variety of crystalline compounds form. A perusal of the regime is perhaps the most interesting one, and in some sense phase diagram of Figure 123 illustrates the richness of stoi- perhaps the most challenging one to understand. Globally chiometric crystalline phases present around the tie line join- Tg(x) shows a broad maximum centered near x = 30% with ing the center of the phase diagram (equimolar composition- little or no change across the narrow intermediate-x regime. (BaO)(SiO2)(B2O3)) to the BaO end point. It is also in this Raman scattering data on mode scattering strengths show part of the phase diagram that bulk glass formation is pro- (Figures 6 and 7) that the pair of modes, D and R, Q2 and moted, we suspect, because networks become near optimally Q3 display a plateau, i.e., little or no change with x. The same coordinated. behavior was noted earlier for the B4 mode, which increases We have already noted that upon alloying 24 mol. % in regimes I and III but displays a plateau in regime II. of BaO in the base (SiO2)68(B2O3)32 composition, almost In regime II other peculiarities are noted. We observe 4 2 all of the Q species associated with SiO2 are apparently Q mode frequency steadily red-shifts with increasing x, but converted into Q3 ones (Q3 mode near 1055 cm−1). There- shows a plateau-like behavior in regime II as illustrated in fore the network structure at the lowest modifier concen- the top panel of Figure 12. Q3 mode frequency, in gen- tration studied has local structures of Q3 and even some eral, red-shifts with increasing x, but also shows a plateau- 2 Q species. Each BO4 tetrahedra with 3 SiO4 ones, form like behavior in regime II as illustrated in the right panel of a mixed 4-member ring structural motif, which is also Figure 12. These data unmistakably demonstrate that glass found in the mineral Reedmergnerite.29 A fraction of mixed structure evolution appears to be suddenly arrested in regime rings composed of 2 BO4 and 2 SiO4 tetrahedra exist in II. On the other hand, frequency variation of the R- and a motif as found in the mineral Danburite. Thus a dis- D-modes (Figure 13) illustrates new features that reveal tinct fraction of mixed rings based on the Reedmergner- more subtle aspects of glass structure evolution. For ex- ite (R-mode), and Danburite (D-mode), and some B − ample, we find that R-mode frequency blue-shifts with in- − O attached to large borate groups (B4 mode) form at creasing x, but actually does so in three distinct steps

FIG. 12. Mode frequency variation of Q2-andQ3 modes as a function of BaO content in the present ternary glasses. Note Q2 (Q3) modes blue (red) shift, but both modes show a plateau in regime II. FIG. 13. Mode frequency variation of R and D modes as a function of BaO content in the present ternary glasses. Note the strikingly different frequency variation of these pair of modes, the R-mode displays a step on the top panel, that track the three distinct compositional regimes. In the the D-mode is almost independent of BaO content. Raman spectra of the glasses one can visually see the peak of the R-mode (Figure 2) shifts to higher frequency in going ent from Ba in our glasses. We believe that the significantly from x = 29% to 32%. Note that in regime I, the frequency lower breathing mode frequency in glasses (530–560 cm−1) increases rapidly at first but then saturates near x = 28%, the compared to the mineral (586 cm−1) stems from an excess of end of the regime I. The pattern is replicated in regimes II and charge around the rings in the glasses. In Raman scattering III. In sharp contrast, D-mode frequency variation is almost we find no evidence of Q4 Si species in our glasses, suggest- independent of x even though its scattering strength contin- ing that in both the R- and the D rings Q3 and possibly Q2 ues to increase with x but displays a plateau-like behavior in Si species are involved but no Q4 ones as in the minerals. regime II as was noted earlier (Figure 6). Thus the SiO4 tetrahedra in both these rings are associated The 4-membered ring structure in Danburite has two BO4 with NBO. The excess charge on the NBO will elongate the and 2 SiO4 tetrahedra, and will on average be associated with Si–BO bond length in the rings, and will lead to a red-shift of one Ba2+ cation. The growth of D rings at the expense of the breathing mode in the glasses. R-rings, with increasing x, is exactly what one would have The increase of the R-mode frequency with x in steps expected, given that a D-ring has a higher B-content than a (Figure 13(a)), particularly in regime II is a curious result. It R-ring. appears that the modifier localized around the Reedmergner- The mineral sample of Danburite (D) examined by Ma- ite rings (producing NBO) steadily migrates away from these 30 −1 nara et al. reveals a breathing mode of 614 cm , quite rings to produce additional BO4 tetrahedra (B0 mode) in di- −1 close to the value of 626 cm (Figure 13(b)) recorded in our borate groups (B1 mode), since the latter two modes steadily glasses. Danburite mineral has Ca as a modifier, which is iso- increase in scattering strength. It is widely known that tetra- valent to Ba. On the other hand, the mineral sample of Reed- hedra pack well, and the driving force for the Ba2+ migration mergnerite (R) examined by Manara et al. reveals a breath- away from the rings may well be compaction of the network ing mode of 586 cm−1, considerably higher than the range as a whole to lower its molar volume (Figure 8). The migra- of frequency 530–560 cm−1(Figure 13(a)) recorded in our tion of the modifier represents a network adaption feature that glasses. In Reedmergnerite mineral, Na is the modifier, differ- contributes to the formation of the IP in the present glasses.

C. Elastic phases of (BaO)x ((B2O3)32(SiO2)68)100−x TABLE II. Count of constraints per atom (nc). The number of BO bonds ternary glasses which connect the rings to the rest of network is given by ρ.

The variation of Tg in alkali borates reveals a broad max- R-Ring D-Ring imum near about 33 mol. % of alkali oxide,40 a behavior n ρ Qn n ρ Qn generally identified with the borate anomaly. At low alkali- c c oxide fraction (<33.3%) the glass network connectivity in- 3.56 8 Q4 3.46 8 Q4 creases as 3-fold B is transformed into 4-fold B with the 3.5 6 Q3 3.3 6 Q3 modifier acting as a charge compensator. At high alkali-oxide 2.846 2 Q2 3.1 4 Q2 (> 33.3mol. %), glasses qualitatively depolymerize as BO are steadily converted into NBO by alkali modifier. Recently the enthalpy of relaxation of Tg was examined For the case of the Danburite rings, we have likewise esti- as a function of soda content and showed an almost square- mated the count of constraints and find a similar pattern; SiO4 31 4 well like global minimum in sodium borate glass. Such win- tetrahedra in a Q configuration yield, nc = 3.46, but SiO4 2 dows are identified with isostatic nature of glasses formed in tetrahedra in a Q configuration yield nc = 3.1, i.e., nearly IPs.18 At low x (< 20 mol. %), glasses are in the stressed-rigid isostatic. These results are summarized in Table II. phase, those at high x soda-content (>40%) are in the flexible To visualize the striking implications of these Lagrangian phase while those at intermediate x in the isostatically rigid constraints, we have plotted in Figure 14, nc as function of IP. In the IP, Tgs show a broad maximum, and characteristic ρ, the number of contacts a ring makes with the rest of the rings based on mixed boroxyl rings containing both 3-fold and network. There are 3 sets of points each, for the R- and the 4-fold B as in sodium tri-borate and sodium-tetraborate are D-rings. formed. The IP occurs in the broad Tg maximum range. One For the case when Si is in a Q4 configuration both the R- expects similar behavior to be observed in the present glasses. and D-rings have 8 BO bonds (ρ) that connect them to the The three regimes of variation of Tg in the present glasses rest of the network, and lead to their becoming stressed-rigid (Figure 3) correspond to the three elastic phases; regime I the (nc > 3) as revealed by the count of constraints (Figure 14, stressed-rigid phase, regime II the IP, and regime III the flex- Table II). However an R-ring becomes flexible for the case ible phase. when Si is in a Q2 configuration, since there are only 2 BO bonds that constrain the ring to the rest of the network. The data of Figure 14 also show that an isostatically rigid 1. Adaptive 4-member rings and the intermediate backbone composed of R- and D rings will emerge in the phase present glasses if the fraction of these rings is almost equal, provided Si is in a Q2 configuration in both rings. For such a At the outset we note that our Raman scattering data mix of rings, it follows that n = 2.97, quite close to the ideal place the fraction of Reedmergnerite- and Danburite-rings in c value of 3 as can be seen from the data of Table II. These cal- aglassatx = 30% in the 20% − 25% range. The scattering culations are striking, given that the Raman mode scattering strength fraction, in the simplest case, also provides a measure strengths of these rings reveal (Figure 6) the R-ring fraction of the fraction of rings present in the network. We note that to be just slightly greater than the D-ring fraction in regime II. this fraction exceeds the Scher-Zallen41 volume percolation Yet another possibility to attain an isostatic configuration of threshold of 16.6% for volume percolation in 3D. Reedmergnerite- and Danburite-rings are known from the crystalline mineral form, to be composed of SiO4 tetra- 4 hedra, as Q species with BO4 tetrahedra. Enumeration of mechanical constraints/atom, nc, due to bond-stretching and bond-bending forces for Reedmergnerite rings shows that

nc = 3.56, (2)

i.e., they are over-constrained. Our Raman scattering data provide no evidence of Q4 modes (at 1110 cm−1). Thus, one can safely state that the mixed rings populated in the glasses are not the ones present in the crystalline minerals, but they are actually modified or more precisely they adapt as glass composition approaches the IP. If we assume that each SiO4 species present in Reedmergnerite-like rings in glasses are of the Q2 type, i.e., have 2 NBO associated with a Ba2+ cation, then our estimate of mechanical constraints of the ring shows FIG. 14. Plot of count of constraints for R- rings and D-rings as a function it to be flexible (see the Appendix), i.e., of the number of outside contacts connecting rings to the rest of the network. Open circles (◦) are for D-rings, filled circle (•) for R-rings. The smooth line through the data points is a guide to the eye. The horizontal broken line = nc 2.846. (3) corresponds to nc = 3.

the backbone is to have on average, three R-rings in a Q2 con- man mode scattering strengths show that in the low x-regime figuration for each D-ring in a Q3 configuration. The data of the modifier serves largely to assist in polymerizing the back- Table II, unambiguously show that for such a mix, nc = 2.995 bone as BO3 triangles are converted to BO4 tetrahedra, while close to the ideal value of 3 for an IP to form. The data of in the high x regime, the modifier serves largely to depoly- Table II also underscore the fact that if the D-ring scattering merize the network by creating NBO atoms, which results in 2 3 strength were to exceed the R-ring one, there is no possibility the growth of SiO4 Q species at the expense of Q ones, − to achieve an isostatic configuration of the backbone. These and BO2O species in orthoborates grow at the expense of are precisely the circumstances observed experimentally in BO3 ones. The low-x regime and the high-x regime represent Raman scattering strengths of Figure 6. Note that in regime the stressed-rigid and flexible elastic phases, respectively. The II the R-scattering strength always exceeds that of the D-ring narrow intermediate-x regime, wherein molar volumes show scattering strength by a small amount, but once they equalize a minimum and Raman mode scattering strengths and mode one goes out of regime II. frequency become largely x-independent, is a signature of the In Raman scattering, the fraction of Q2 species steadily isostatically rigid elastic phase. The intermediate-x regime increases at low x in regime I, and then plateaus in regime II glasses can be expected to be the most stable for applications. near x = 30% (Figure 7) before increasing precipitously in regime III. Clearly, a large concentration of Q2 species will ACKNOWLEDGMENTS depolymerize and soften the network. We suspect that these Q2 species are localized on Si atoms in the R-and D-rings, The present work was supported by Subcontract No. RSC and their plateau-like variation in regime II strongly suggest 12025 from University of Dayton on Contract No. FA8650- that the network is adapting to remain isostatic in that compo- 11-D-5401/0008 from AFOSR. All Raman scattering work sitional window. These topological considerations provide an was performed on a facility acquired with support from NSF excellent physical basis to understanding the observed elastic (Grant No. DMR-94-24556). phases in the present glasses. The rather high (660 ◦C) glass transition temperatures T g APPENDIX: BONDING CONSTRAINTS IN 4-MEMBER of the present glasses, made it difficult to measure the non- MIXED RINGS OF REEDMERGNERITE AND reversing enthalpy of relaxation using Modulated DSC to ob- DANBURITE serve a reversibility window, as was possible for the case of 6 sodium borates.31 Nevertheless, in this work we have shown We use tools of rigidity theory to understand the flexi- that glasses in regime II, i.e., 29% < x < 32% possess all the bility and rigidity of the Reedmergnerite and Danburite rings. features that we associate with an IP. In this compositional The starting point is to enumerate the count of bond-stretching regime, we observe (i) a broad maximum of Tg, (ii) a min- (α) and bond-bending (β) constraints in assembling a Reed- imum of molar volumes, and (iii) the evolution of the glass mergnerite ring composed of 4 tetrahedral units; 3 of SiO4 molecular structure that provides evidence of 4-member rings and 1 of BO4. For an atom possessing a coordination number based on Reedmergnerite and Danburite adapting, to render r, the count of bond-stretching constraints is given as the network isostatically rigid. These ideas based on network nα = r/2, (A1) topology have proved to be extremely powerful in understanding glass functionality. and the count of bond-bending constraints is given as

nβ = 2r − 3(A2) for r = 2 or greater in 3D. For r = 1, we will only have 1/2 of IV. CONCLUSIONS a bond-stretching constraint.19 Thus, for r = 4, such as tetra- Optically transparent bulk glasses corresponding to the hedral Si or B, we have a total of nc = nα + nβ = 7 constraints fraction SiO2/B2O3, K fixed at 2.12, and the modifier fraction per atom, while for a BO and NBO, with r = 2, the count of 2+ BaO/B2O3 = R, in the 0.75 < R < 1.50 range (or BaO con- constraints gives nc = 2 constraints/atom. A Ba cation will tent 24% < x < 42% range) were synthesized. Glass transi- share a pair of electrons with two BO4 tetrahedra. Therefore 2+ 2+ tion temperatures Tg(x) show three regimes of variation: a low one associates a Ba cation with two rings, i.e., 1/2 of a Ba x regime, 24% < x < 29%, in which Tg(x) steadily increases, cation per ring. We consider three types of Reedmergnerite 4 an intermediate range, 29% < x < 32%, wherein Tg(x)shows rings: first with Case A: when all Si is in a Q configuration; a broad flat global maximum, and a high x regime in the 32% second with Case B: when all Si is in a Q2 configuration; and 3 < x < 42% range wherein Tg(x) systematically decreases. last with Case C: when all Si is in a Q configuration. Raman spectra show rich line-shapes that evolve systematically with the BaO content of the glasses. The local struc- 1. Reedmergnerite ring composed of 3 SiO (Q4) units tures formed in these glasses include Q2 and Q3 (but not Q4) 4 2 − with 1 BO4 unit SiO4 units, tetrahedral BO4, and BO O units. The intermediate range order of the glasses are dominated by 4-membered This is the configuration of the mineral Reedmergner- rings containing SiO4 and BO4 tetrahedral units in the ratio ite in its crystalline form as schematically illustrated in of 3:1 as in Reedmergenerite, and in the ratio of 2:2, as in Figure 15. The ring stoichiometry can be written as b b Danburite. The local structures are inferred from their char- Si3B1O4O8 Ba1/2 where b and b designate the O atoms acteristic vibrational modes. The observed variations of Ra- in the ring and outside the ring. Next we enumerate the

b b nb Si3B1O4O2 O6 Ba7/2 where b and b designate the O atoms in the ring and outside the ring and nb the non-bridging oxygen atoms. Next, we enumerate the constraints/atom for the ring as a whole,

nc(3 + 1 + 2 × 6 + 3.5) 6 1 = 3 × 7 + 1 × 7 + 4 × 2 + 2 × 2 + 6 × 2 + + , 2 2 (A5)

where the coefﬁcient of nc designates the count of atoms associated with the ring. The 6 NBO associated with the 3 Ba2+ cations, and for each ionic interaction we count 1 bonding constraint shared between NBO and Ba2+, thus contributing 6 × 1/2 constraints. Ba2+ cation, a charge compensator, contributes 1/2 constraint per ring. Thus, 4 FIG. 15. Reedmergnerite ring composed of 3 SiO4 (Q )and1BO4 tetrahe- = dra connected to the rest of the network by ρ 8 bonds (bold lines). nc = 55.5/19.5 = 2.84. (A6)

2 Thus, Reedmergnerite made up of 3 SiO4 tetrahedra in a Q constraints/atom for the ring as a whole and denote it as nc, conﬁguration is under-constrained (nc < 3), and is mechan-

nc(3 + 1 + 4 + 4 + 1/2) ically ﬂexible. This is not unexpected given that it is cross linked to the rest of the network by only 2 bonds (Figure 16), = × + × + × + × × + 3 7 1 7 4 2 8 (1/2) 2 1/2, (A3) in sharp contrast to the case considered above where the R-

where the coefﬁcient of nc on the left side of Eq. (A3) is the ring is coupled to the network by 8 bonds (Figure 15). count of atoms associated with the ring. The Ba2+ cation as a

charge compensator, shares 2 bond stretching constraints with 3 3. Reedmergnerite ring composed of 3 SiO4 (Q ) units 2BO4 tetrahedra, or one bond stretching constraint per ring, with 1 BO unit and thus contributes 1/2 of a constraint per ring, 4 One can also enumerate constraints for a Reedmergnerite nc = 44.5/12.5 = 3.56. (A4) 3 ring composed of 3 SiO4 in a Q conﬁguration. The counting The Reedmergnerite ring in the crystalline mineral is mechan- reveals ically over constrained (n > 3), has ρ = 8 BO bonds connect- c n = 56/16 = 3.50. (A7) ing to the rest of the network (Figure 15) and is stressed-rigid. c Such a ring is also stressed-rigid since nc > 3.0. The present ring is coupled to the rest of the network by ρ = 5 bonds, 2. Reedmergnerite ring composed of 3 SiO (Q2) units 4 somewhat less than 8 for the case of Figure 15. One thus ﬁnds with 1 BO4 unit that as the connectivity of the SiO4 tetrahedra is steadily low- Next we consider a Reedmergnerite ring in which all the ered from Q4 to Q3 to Q2, the count of constraints of the Reed- 2 2 3SiO4 units are Q species. For each of the Si Q species we mergnerite (R)-rings steadily decreases from 3.56, to 3.50 to + associate a Ba2 cation creating two NBO as shown in 2.846. Figure 16. The ring stoichiometry now becomes

4. Global count of constraints for Reedmergnerite and Danburite rings We have counted mechanical constraints for Danburite rings as done above for Reedmergnerite rings, and these data are summarized in Table II.

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