ON THE STRUCTURE AND DYNAMICS OF ALUMINOPHOSPHATE GLASS-FORMING MELTS MEASURED BY DYNAMIC LIGHT SCATTERING Tri D. Tran A THESIS Submitted to the faculty of the Graduate School of the Creighton University in Partial Fulfillment of the Requirements for the degree of Master of Science in the Department of Physics. Omaha, NE May, 2014
Abstract I report the results of a series of dynamic light scattering studies of molten aluminophosphate liquids, [Al(PO 3 ) 3 ] x [NaPO 3 ] 100 x, obtained by photon correlation spectroscopy. The study covers a range of compositions designed to manipulate the network connectivity of the glass-forming liquid in order to investigate how the change in the chemical structure of covalent network affects its dynamics. The dynamic structure factor was measured to provide quantitative information about the liquid s dynamics through the non-ergodic level, stretching exponent, and fragility. The fragility was shown to decrease with increased network connectivity and the result is identical to both ultraphosphates and chalcogenides. I argue that this pattern fits with predictions of the rigidity theory and may be linked to a common dependence on the entropy of the liquid as a function of the network connectivity described by a two-state bond model. i
Acknowledgements First, I would like to specially thank my advisor, Dr. David Sidebottom, for his indispensable guidance and advice throughout my years at Creighton University. Without his continual support, my journey through the convoluted, yet fascinating, world of physics would have been rough. The knowledge imparted by Dr. Sidebottom is invaluable and will continue to be a source of my motivation. Also, would like the express my gratitude to the physics department facility, staff members, and fellow students. The friendly and helpful environment was like a second home to me. In addition, I would like to gratefully acknowledge the National Science Foundation (grant # DMR-0906640) for the financial support toward my research. Last, but not least, I would like to express deep gratitude to my parents and family who have supported me, physically and mentally, throughout my time at Creighton. ii
Contents 1 Background 1 1.1 Inspiration.................................. 1 1.2 Glass.................................... 3 1.3 Liquid Dynamics.............................. 5 1.4 Fragility................................... 7 1.5 Dynamic Structure Factor.......................... 9 1.6 Rigidity Theory............................... 11 1.7 Chalcogenides................................ 13 1.8 Oxide Liquids................................ 15 1.9 Sodium Ultraphosphates.......................... 17 1.10 Aluminophosphates............................. 23 2 Experimental Methods 25 2.1 Dynamic Light Scattering and Photon Correlation Spectroscopy..... 25 2.2 Sample Preparation............................. 27 2.3 Experimental Design............................ 29 2.4 Data Analysis................................ 30 3 Results 37 3.1 Glass Transition Temperature........................ 37 3.2 Fragility................................... 38 3.3 Bond Model................................. 41 3.4 Stretching Exponent: β........................... 44 3.5 Non-ergodic level: f q............................ 47 iii
CONTENTS iv 4 Conclusion 49 Appendices A Analysis 52 A.1 [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100........................... 53 A.2 [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99............................ 57 A.3 [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5.......................... 61 A.4 [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96............................ 66 A.5 [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95............................ 70 A.6 [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93............................ 74 A.7 [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90........................... 79 A.8 [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88........................... 84 A.9 [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85........................... 89 A.10 [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80........................... 94 A.11 [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70........................... 98 A.12 [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60........................... 104 A.13 [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50........................... 110 B Derivation 116 B.1 Derivation of Eq. 3.8............................ 117
List of Figures 1.1 Fragility vs. n or r for Sodium Ultraphosphates and Chalcogenides.. 2 1.2 Phase Transition Illustration........................ 4 1.3 The Atomic Structure of Glass and a Crystal Illustration.......... 5 1.4 Cage Effect................................. 6 1.5 Difference Between α-relaxation and β-relaxation............. 7 1.6 Angell s Plot of Various Glass-forming Liquids.............. 8 1.7 Dynamic Structure Factor, S( q, t)...................... 10 1.8 Overconstrained vs. Underconstrained System............... 12 1.9 Deviation for the Total Number of Angular Constraints.......... 13 1.10 Fragility vs. r for Chalcogenides..................... 14 1.11 n vs. r.................................. 16 1.12 NMR Spectrum for Sodium Ultraphosphates................ 17 1.13 Different Structural Units of Sodium Ultraphosphates........... 18 1.14 log 10 τ avg (s) vs. T g /T for Sodium Ultraphosphates............. 19 1.15 Recreation of Figure 1.1.......................... 20 1.16 β vs. log 10 τ avg (s) for Sodium Ultraphosphates............... 21 1.17 f q (T g ) vs. 1000/m for Sodium Ultraphosphates.............. 22 2.1 Scattering diagram............................. 26 2.2 Experimental Setup............................. 29 2.3 [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 Spectra...................... 30 2.4 [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 Autocorrelation Function............ 32 2.5 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5.......... 33 v
LIST OF FIGURES vi 2.6 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5........... 34 2.7 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5............... 35 2.8 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5.............. 36 3.1 T g Comparison Between Experimental and Brow s Results........ 37 3.2 Aluminophosphate Compositions Plot of log 10 τ avg (s) vs. T g /T(K).... 38 3.3 Fragility Comparison between Sodium Ultraphosphates, Chalcogenides, and Aluminophosphates........................... 40 3.4 Fragility and the Bond Model........................ 44 3.5 Aluminophosphate Compositions Plot of β vs. log 10 τ avg (s)........ 45 3.6 Aluminophosphate Compositions Plot of the Slope β vs. n....... 46 3.7 Aluminophosphate Compositions Plot of f q (T g ) vs. 100/m........ 47 A.1 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100.......... 53 A.2 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100.......... 54 A.3 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100........... 55 A.4 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100............ 55 A.5 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100................ 56 A.6 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100............... 56 A.7 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99........... 57 A.8 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99........... 58 A.9 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99........... 59 A.10 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99............ 59 A.11 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99................ 60 A.12 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99................ 60 A.13 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5......... 61 A.14 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5......... 62 A.15 (3) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5......... 63 A.16 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5.......... 64 A.17 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5........... 64
LIST OF FIGURES vii A.18 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5............... 65 A.19 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5.............. 65 A.20 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96........... 66 A.21 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96........... 67 A.22 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96........... 68 A.23 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96............ 68 A.24 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96................ 69 A.25 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96................ 69 A.26 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95........... 70 A.27 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95........... 71 A.28 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95........... 72 A.29 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95............ 72 A.30 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95................ 73 A.31 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95................ 73 A.32 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93........... 74 A.33 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93........... 75 A.34 (3) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93........... 76 A.35 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93........... 77 A.36 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93............ 77 A.37 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93................ 78 A.38 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93................ 78 A.39 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90.......... 79 A.40 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90.......... 80 A.41 (3) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90.......... 81 A.42 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90........... 82 A.43 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90............ 82 A.44 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90................ 83 A.45 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90............... 83 A.46 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88.......... 84
LIST OF FIGURES viii A.47 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88.......... 85 A.48 (3) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88.......... 86 A.49 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88........... 87 A.50 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88............ 87 A.51 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88................ 88 A.52 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88............... 88 A.53 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85.......... 89 A.54 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85.......... 90 A.55 (3) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85.......... 91 A.56 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85........... 92 A.57 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85............ 92 A.58 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85................ 93 A.59 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85............... 93 A.60 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80.......... 94 A.61 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80.......... 95 A.62 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80........... 96 A.63 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80............ 96 A.64 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80................ 97 A.65 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80............... 97 A.66 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70.......... 98 A.67 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70.......... 99 A.68 (3) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70.......... 100 A.69 (4) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70.......... 101 A.70 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70........... 102 A.71 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70............ 102 A.72 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70................ 103 A.73 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70............... 103 A.74 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60.......... 104 A.75 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60.......... 105
LIST OF FIGURES ix A.76 (3) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60.......... 106 A.77 (4) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60.......... 107 A.78 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60........... 108 A.79 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60............ 108 A.80 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60................ 109 A.81 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60............... 109 A.82 (1) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50.......... 110 A.83 (2) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50.......... 111 A.84 (3) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50.......... 112 A.85 (4) Autocorrelation Spectra for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50.......... 113 A.86 log 10 τ avg (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50........... 114 A.87 log 10 τ avg (s) vs. T g /T(K) for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50............ 114 A.88 β vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50................ 115 A.89 f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50............... 115
List of Tables 1.1 Table of δ 6, δ 5, and δ 4 for Different mol% of Al 2 O 3 Using Data from Brow [18].................................. 23 3.1 Summary of T g, Fragility, and n for [Al(PO 3 ) 3 ] x [NaPO 3 ] 100 x where x is the mole percent........................... 39 A.1 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.808 ± 0.0004, τ, and β for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100 Between Temperatures of 602.8K and 587.2K.................................... 53 A.2 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.808 ± 0.0004, τ, and β for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100 Between Temperatures of 585.4K and 572.0K.................................... 54 A.3 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.835 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99 Between Temperatures of 611.5K and 599.0K.................................... 57 A.4 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.835 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99 Between Temperatures of 597.2K and 582.9K.................................... 58 A.5 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.828 ± 0.0009, τ, and β for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 Between Temperatures of 624.3K and 589.8K.................................. 61 A.6 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.828 ± 0.0009, τ, and β for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 Between Temperatures of 598.8K and 587.9K.................................. 62 x
LIST OF TABLES xi A.7 (3) The Value of the Amplitude, f q Evaluated when A coh = 0.828 ± 0.0009, τ, and β for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 Between Temperatures of 585.9K and 573.2K.................................. 63 A.8 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.835 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96 Between Temperatures of 648.5K and 626.7K.................................... 66 A.9 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.835 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96 Between Temperatures of 623.7K and 605.0K.................................... 67 A.10 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.770 ± 0.0006, τ, and β for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95 Between Temperatures of 649.8K and 633.2K.................................... 70 A.11 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.770 ± 0.0006, τ, and β for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95 Between Temperatures of 628.7K and 612.0K.................................... 71 A.12 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.803 ± 0.0002, τ, and β for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93 Between Temperatures of 695.4K and 668.8K.................................... 74 A.13 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.803 ± 0.0002, τ, and β for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93 Between Temperatures of 666.3K and 652.1K.................................... 75 A.14 (3) The Value of the Amplitude, f q Evaluated when A coh = 0.803 ± 0.0002, τ, and β for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93 Between Temperatures of 648.6K and 626.8K.................................... 76 A.15 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.840 ± 0.0002, τ, and β for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90 Between Temperatures of 731.6K and 704.4K.................................... 79
LIST OF TABLES xii A.16 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.840 ± 0.0002, τ, and β for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90 Between Temperatures of 701.1K and 672.5K.................................... 80 A.17 (3) The Value of the Amplitude, f q Evaluated when A coh = 0.840 ± 0.0002, τ, and β for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90 Between Temperatures of 668.2K and 645.5K.................................... 81 A.18 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.846 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88 Between Temperatures of 766.6K and 734.8K.................................... 84 A.19 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.846 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88 Between Temperatures of 731.3K and 710.3K.................................... 85 A.20 (3) The Value of the Amplitude, f q Evaluated when A coh = 0.846 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88 Between Temperatures of 707.3K and 681.3K.................................... 86 A.21 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.838 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85 Between Temperatures of 782.0K and 751.6K.................................... 89 A.22 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.838 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85 Between Temperatures of 748.1K and 725.5K.................................... 90 A.23 (3) The Value of the Amplitude, f q Evaluated when A coh = 0.838 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85 Between Temperatures of 723.0K and 696.8K.................................... 91 A.24 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.839 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80 Between Temperatures of 824.5K and 769.3K.................................... 94
LIST OF TABLES xiii A.25 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.839 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80 Between Temperatures of 765.2K and 726.0K.................................... 95 A.26 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.718 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 Between Temperatures of 884.1K and 854.3K.................................... 98 A.27 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.718 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 Between Temperatures of 854.3K and 831.8K.................................... 99 A.28 (3) The Value of the Amplitude, f q Evaluated when A coh = 0.718 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 Between Temperatures of 831.8K and 805.5K.................................... 100 A.29 (4) The Value of the Amplitude, f q Evaluated when A coh = 0.718 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 Between Temperatures of 805.5K and 766.2K.................................... 101 A.30 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.797 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 Between Temperatures of 941.1K and 914.9K.................................... 104 A.31 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.797 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 Between Temperatures of 914.9K and 890.5K.................................... 105 A.32 (3) The Value of the Amplitude, f q Evaluated when A coh = 0.797 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 Between Temperatures of 882.5K and 859.0K.................................... 106 A.33 (4) The Value of the Amplitude, f q Evaluated when A coh = 0.797 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 Between Temperatures of 859.0K and 813.5K.................................... 107
LIST OF TABLES xiv A.34 (1) The Value of the Amplitude, f q Evaluated when A coh = 0.762 ± 0.0003, τ, and β for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 Between Temperatures of 997.5K and 945.9K.................................... 110 A.35 (2) The Value of the Amplitude, f q Evaluated when A coh = 0.762 ± 0.0003, τ, and β for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 Between Temperatures of 914.9K and 917.2K.................................... 111 A.36 (3) The Value of the Amplitude, f q Evaluated when A coh = 0.762 ± 0.0003, τ, and β for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 Between Temperatures of 917.2K and 890.8K.................................... 112 A.37 (4) The Value of the Amplitude, f q Evaluated when A coh = 0.762 ± 0.0003, τ, and β for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 Between Temperatures of 883.6K and 862.9K.................................... 113
Chapter 1 Background 1.1 Inspiration Glass is unique, and it is often misrepresented due to its inherent complexity of mixed matter properties. Much like a solid, the structure of glass consists of a network of bonds, ionic or covalent, but it also retains random properties typically associated with liquid making it difficult to accurately understand glass. Nevertheless, glass plays an important role in the advancement of civilization through its virtually unlimited and versatile applications (e.g., optical fibers to provide fast communication and shockproof screens for electrical devices). Furthermore, there is a natural artistic attraction towards glass as it is used to create many architectural structures like the glass stairway stretching about 30 ft in New York City. These novelties, however, are only possible with continuous research on improving and discovering different refinement techniques such as tempering, ion-exchange used in gorilla glass [1], and chemical alterations. For example, the dynamic properties of phosphate glass featured in several applications (e.g., fiber optics, solid-state lasers, borosilicate in durable cookwares, and nuclear waste encasement) depend on the response of the systematic network of covalent bonds to chemical alterations. If sodium ions are introduced into phosphate glass, the covalent bonds within the network are chemically broken resulting in a decrease of the network connectivity. (See 1.8 Oxide Liquids). This alteration causes the glass transition temperature, T g, and solubility to decrease. If aluminum ions are added instead of sodium ions, the solubility still decreases, but T g rises. 1
CHAPTER 1. BACKGROUND 2 While the findings are intriguing, the physics of the chemical manipulation of phosphate is poorly understood, especially the reaction of the structure and the dynamics to chemical additives. Thus, it becomes the focus of our research group to provide further insight upon the matter. Figure 1.1: The figure displays the fragility comparison for sodium ultraphosphates and chalcogenides with their respective network connectivity parameters. Initial research began with Fabian [2] using a dynamic light scattering technique called photon correlation spectroscopy to measure the fragility of several samples of phosphate pentoxide, P 2 0 5, liquid with varying concentrations of sodium oxide, Na 2 O. (See 2.7 Sodium Ultraphosphates and 1.4 Fragility). In general, he observed an increase in the fragility with increasing sodium additives as n (i.e., the network connectivity parameter based on the average number of bridging oxygen atoms) decreased from 3 to 2. (See 2.6 Oxide Liquids). Interestingly, a remarkable coincidence in the fragility behavior was found from the results of chalocogenide liquids except with r (i.e., the network connectivity
CHAPTER 1. BACKGROUND 3 parameter based on the average number covalent bonds) as seen in Figure 1.1. (See 2.5 Chalcogenides). Moreover, the fragility appeared to reach a minimum around n = 2.4 suggesting a tie to the rigidity theory where, at that value, the number of constraints and the degrees of freedom are equivalent. (See 2.4 Rigidity Theory). As a continuation of the investigation, my research project analyzes the effect on the dynamics of metaphosphate, NaPO 3, when the systematic network is altered by chemically replacing NaPO 3 structures with aluminum oxide or alumina structures. I hypothesize that the fragility will decrease with alumina content because the network will gradually become an ion-free network and it will behave identically to ultraphosphates and chalcogenides in Figure 1.1, except n increases from 2 to 6 [3] for my compositions. If this holds true, the next objective is to understand the cause for the universal fragility behavior as a method to relate the dynamics to the systematical changes of the liquid s structural network. 1.2 Glass The unique mix of crystal and liquid properties present in glass originate from the difference between the transition of the glass state and that of the common known states (i.e., gas, liquid, and solid as in a crystal). The different phase transitions are represented through the change in volume, proportional to entropy, at specific temperatures in Figure 1.2. Starting from the gas state, as the temperature decreases, the gas cools and condenses to a liquid upon reaching the boiling temperature, T b. During this process the temperature is static but a volume discontinuity occurs. This is a characteristic of a first order transition which describes that the first derivative of the Gibbs potentials suffers a discontinuity across the phase boundary [4]. If the substance is allowed to further cool towards solidification, the path diverges into two depending on the cooling rate. If the cooling rate is slow, the liquid will encounter another first-order transition at the freezing temperature, T f, where the liquid solidifies into a crystal. During this process, the liquid undergoes nucleation and crystallizes because there is sufficient time for the molecules to rearrange into a favorable ordered array with minimized potential energy. If the cooling rate is fast, the substance bypasses T f, and enters an intermediate state called the viscous
CHAPTER 1. BACKGROUND 4 liquid or supercooled liquid. Figure 1.2: Liquids possess two different means of solidification. Path 1 describes the process of crystallization at T f which is characterized by the first order transition. Path 2 depicts the process of the glass transition at T д identified by a continuous volume transition [5]. As a supercooled liquid continues to cool, the volume slowly decreases until the temperature is at T g where a different transition then occurs as it does not suffer a discontinuity in volume. During this phase transition, the dramatic increase in viscosity does not allow sufficient time for molecules to obtain a favorable orientation, and thus they solidify in random orientation similar to a liquid [5]. This process is known as vitrification and the result is an amorphous, non-crystalline solid, or glass. The glass state is considered a type of solid. However, there are distinctions between the glass and its crystal state counterpart. The glass state is not an actual thermodynamic state because the glass and liquid states are not directly connected [4]. Instead, an intermediate state, the supercooled liquid, serves as a bridge between the two states. As a consequence, the glass state is also a non-equilibrium state as the molecules are still energetic and have not reached a structure that favors the least potential energy, but yet the
CHAPTER 1. BACKGROUND 5 long-range motion seems to cease like a crystal. Thus, the glass state is a non-equilibrium thermodynamic state and the differences between the crystal state and glass state are noticeable through their atomic structure. A crystal possesses a high degree of short-range order because the nearest neighbor separation and the bond lengths are exactly the same for each individual atom within the structure. It also has a high degree of long-range order because the location of distinct atoms can be easily determined. However, glass lacks any presence of long-range order and possesses a similar degree of short-range order where the amount of nearest atoms is the same and has nearly the same nearest neighbor separation and bond lengths. (See Figure 1.3 for an illustration of this concept). The lack of long-range order implies randomness at a larger separation but has semi-order structure at atomic scale. Since there is a sense of randomness associated with glass, it can be concluded that the entropy of glass is larger than its crystal counterpart. Figure 1.3: The left figure shows the atomic structure of a crystal, while the right figure represents the atomic structure of a glass. The glass state resembles the crystal state except the bond length, bond angle, and the position of each atom is random. 1.3 Liquid Dynamics The slowing dynamics of a liquid are convey by either the structural relaxation time, τ, which is a characteristic time at which the slowest measurable process relaxes to
CHAPTER 1. BACKGROUND 6 equilibrium [4] or equivalently by the viscosity, η. These two quantities are related in a Maxwell fluid where the material possesses both elastic and viscous properties by the mechanical equation described as η G д τ, (1.1) where G g represents the shear modulus. Using G g with a with typical value of 10 10 Pa [6] and viscosity of 10 12 Pa s at T g, the structural relaxation at T g equates to 100 seconds. Figure 1.4: The illustration shows an example how the cage effect (dotted pattern area) restricts the motion of particles to rattle within an enclosed volume due to the neighboring particles. Each particle experiences the cage effect and not only the ones indicated in the figure. However, several relaxation processes occurring at different time scales are associated with the progression of a liquid to a glass. Thus, it is essential to isolate the relaxation time that represents the rearrangement of the macroscopic structure within a liquid. Picture a warm liquid undergoing a glass transformation by decreasing the temperature. As the temperature decreases, the motion of the particles slows down and becomes selfinhibited through the development of a cage effect. At this point, the relaxation time starts bifurcating into two processes, α-relaxation and β-relaxation, and becomes more distinguishable as the temperature decreases further as seen in Figure 1.5. The β-relaxation occurs on a shorter time scale and is caused by the microscopic structural rearrangement as the particles rattle within an enclosed volume resulting from the cage effect (i.e., a cage-like effect where individual particles are restricted to move within a localized volume
CHAPTER 1. BACKGROUND 7 due to their surrounding particles). This is illustrated in Figure 1.4. The α-relaxation occurs on a longer time scale and is characterized by the global rearrangement of the system leading to a viscous flow. The α-relaxation corresponds to the structural relaxation time and becomes important when determining the fragility. Figure 1.5: The diagram of relaxation time vs. inverse temperature indicates a distinct separation of the relaxation process into two, the α-relaxation(red) and the β-relaxation(blue) [4]. 1.4 Fragility First discussed by Angell [7], the fragility quantifies the behavior of the viscosity as the temperature approaches T g and how this behavior is sensitive to the chemical makeup of the liquid. He plotted the viscosity as a function of T g /T for several liquids and concluded that liquids are separated into two categories depending on the comparison between the measured viscosity and the viscosity (i.e., relaxation time) given from the empirical formula called the Vogel-Tamman-Flucher (VTF) described as τ avд = τ o exp B, (1.2) T T o
CHAPTER 1. BACKGROUND 8 where B is a constant associated with the activation energy of the specific liquid [4]. Liquids following the VTF are considered fragile liquids and display a very non- Arrhenius curvature (curve plots in Figure 1.6) whose viscosity sharply increases as it approaches T g. These liquids are held together by weak interacting bonds, like Van der Waals or ionic bonds, and possess isotropic forces. Strong liquids, however, exhibit an Arrhenius behavior (linear plots in Figure 1.6) with T o = 0 in Eq. 1.2 and their viscosity increases more gradually as it reaches T g. These liquids are illustrated as linear plots in Figure 1.6. Strong liquids tend to form a network consisting of covalent bonds and possess directional intermolecular forces. As a precautionary, the two terms, fragile and strong, do not imply anything about the degree of brittleness of a glass-forming liquid. Instead, the fragility quantifies the easiness of a system to change from one random glassy state (i.e., configuration) to another random glassy state with the same energy [4]. In other words, it describes the ability to self-rearrange the intermolecular networks within the liquid. Figure 1.6: The Arrhenius diagram from Angell shows viscosity behavior of various glass-forming liquids of two categories: strong and fragile. (The erratic behavior of water is explained in the article by Angell [7]). The figure is reprinted with permission from Elsevier.
CHAPTER 1. BACKGROUND 9 As a direct correlation to the study conducted by Angell, the fragility index provides a conventional way to determine the fragility of a glass-forming liquid. The fragility index measures the rate in which the relaxation time increases near T g and is defined as m = d loд 10τ avд d(t д /T ). (1.3) T T д As evident from Figure 1.6, the more non-arrhenius (i.e., fragile) the liquid is, the greater the fragility index. If the fragility index is greater than 70 then the liquid is considered to be extremely fragile. In contrast, strong liquids usually have a fragility index less than 30. 1.5 Dynamic Structure Factor One method to determine the fragility index is by measuring the dynamic structure factor, S( q, t). S( q, t) theoretically describes the dynamics of the liquid through the relative density fluctuation and shows the two relaxation times (i.e., α-relaxation and β-relaxation) as separated decays in Figure 1.7. For liquids possessing viscoelastic properties, the α-relaxation is characterized by a non-exponential decay in a form of a stretch exponent equation depicted as [ ] S( q,t) = f q exp (t/τ) β, (1.4) where the parameters β, f q, and τ are respectively called the stretching exponent, the non-ergodic level, and the structural relaxation time. The β parameter characterizes the non-exponential behavior of S( q, t) and has been interpreted to either (1) reflect the degree of cooperativity of a homogenous relaxation process or (2) describe a heterogeneous relaxation process in which different regions are relax exponential but with a distribution of relaxation rates [8]. For the sake of convenience, the later interpretation for β will be used to differentiate its behavior in fragile and strong liquids. From literature accounts, fragile liquids exhibit β values much smaller than one [9] and reflect a broad distribution of relaxation rates [10]. This is due to the diverse environment within the liquid because of the weakly interacting bonds. However, strong liquids have β values closer to one and thus, a narrow distribution of
CHAPTER 1. BACKGROUND 10 relaxation rates. Therefore, the smallness of β describes the broadness of the distribution of relaxation rates. Figure 1.7: β-relaxation and α-relaxation are characterized by different decay processes and are separated by non-ergodic level, f q. Different β values affect S( q, t) by changing the shape of the curve. Separating the two relaxation processes is a plateau known f q illustrated in Figure 1.7. It describes the fraction of S( q, t) that decays by the slow relaxation. In other words, f q describes the transition from the β-relaxation to the α-relaxation and is related to the amount of structural decay at short time scale. At f q, the initial decay of the β-relaxation is temporarily arrested by the cage effect before allowing the particles to diffuse throughout the liquid and undergo the decay of the α-relaxation. In a fragile liquid, f q has a low value since the Van der Waals bonds permit a substantial amount of localized motion. While in strong liquids, the localized motion is hindered by the covalent structure since it possesses a high level of cohesion resulting in a high f q. Thus, f q should exhibit an inverse relationship to the fragility index.
CHAPTER 1. BACKGROUND 11 Determining the fragility through the measurement of S( q, t) provides a means to understand the dynamics of a liquid and Angell has categorized the bonds featured in both fragile and strong liquids. However, a proper quantitative description regarding the fragility and the systematical changes in the structure of the liquid is still unclear. One attempt to address this problem lies in the rigidity theory and its representation of the network within these liquids. 1.6 Rigidity Theory The rigidity theory (RT) [11] analyzes the relationship between network connectivity and its effects on a system of particles by exploring the consequences of added constraints. The theory classifies systems as either floppy or rigid, and this classification is synonymous to the notion of underconstrained and overconstrained glass postulated by Phillips [12]. The idea of underconstrained or a floppy system is analogous to a freely moving rope in three dimensions. The deformation of the rope is achievable because the rope possesses a vast amount of configurations. However, the flexibility of the rope is reduced if constraints, like fixing a point on the rope, are continuously added and will eventually become rigid at a certain degree. The addition of constraints beyond this threshold results in an overconstrained system. This idea of underconstrained and overconstrained could be observed in a covalent bonded network found in glasses where the collective atoms are analogous to the freely moving rope and the bonds are constraints of the network. Overall, the rigidity theory determines the sturdiness of the system by comparing the number of degrees of freedom (DOF) to the number of constraints as a function of the average bond number, r. Consider a system composed of two different regions, rigid and floppy. The floppy regions are local volumes where the number of DOF exceeds the number of constraints. While in the rigid regions, the number of constraints exceeds the number of DOF. At a low r, the system is described as underconstrained because the floppy regions dominate throughout with isolated rigid regions. As the r increases, the rigid regions increase in volume until the system reaches a critical average coordination number, r c. At this critical value, the rigid regions percolate into existence as the number
CHAPTER 1. BACKGROUND 12 of DOF equals the number of constraints. Any additional increase of r results in a highly constrained system with the number of constraints exceeding the number of DOF. Figure 1.8 illustrates the progression from an underconstrained to an overconstrained system. Figure 1.8: The left figure represents an underconstrained system where the rigid regions (red) are isolated. However in an overconstrained system, illustrated on right figure, the floppy regions (blue) are isolated and the rigid regions percolated throughout the system [11]. A method to calculate r c is by equating the number of DOF with the number of linearly constraints, N c, for network of N atoms containing n r atoms with r bonds such that N = n r, (1.5) The atoms are limited to their translational degrees of freedom giving a total of 3N for the network, so that at r c 3N = N c. (1.6) The number of linear constraints is composed of two types: bond length and bond angle. In the first constraint, each bond fixes a separation length between two atoms and so constitute an r/2 bond constraint per atom where the 1/2 is used to correct for overcounting. The second type of constraints are called angular constraints which arise from the angular direction of the bonds and only exists for r > 2. (Note that r = 1 is excluded because it does not affect the connectivity of the network). The number of angular constraints represents the number of angles necessary to describe the position of each individual bond similar to identifying the number of Lagrangian constraints in the system. For example, imagine an r
CHAPTER 1. BACKGROUND 13 atom connected to two other atoms by r = 2 bonds as illustrated in Figure 1.9. Here, the number of angle constraints is one. If more bonds are added, each additional bond results in two more angular constraints to accurately define the position of the added bond giving a total of 2r - 3 linearly independent angular constraints. The details of the derivation are also shown in Figure 1.9. Figure 1.9: The illustration show how each additional bond results in 2 more angular constraints, starting from r = 2, to adequately described the position of the added bond. So, the total linear independent constraints associated with the network is Substituting N c into Eq. 1.6 gives r N c = n r 2 + (2r 3). (1.7) r r 3N = n r 2 + (2r 3). (1.8) By separating terms and values, r c is then determined to be when the transition occurs. r rrn r N = r = 2.4, (1.9) 1.7 Chalcogenides Past attempts to study the RT prediction have looked at systems where r could be controlled across the transition. For example, Böhmer and Angell [13] studied chalcogenide
CHAPTER 1. BACKGROUND 14 glasses where the network is characterized as a covalently bonded network consisting of three elements: Germanium, Arsenic, and Selenium. This, in turn, allows r to range from 2 to 4 based on each individual elements specific valence electrons of four, three, and two respectively and mixing the appropriate molar fraction in Ge x As y Se 1 x y, r = 2(1 x y) + 3y + 4x. (1.10) Figure 1.10: The illustration shows the fragility dependence on r for chalcogenides. Böhmer and Angell plotted the fragility against r and noticed several aspects presented in Figure 1.10. For r < 2.4, the fragility exhibits a rapid decrease as r approaches 2.4 followed by a local minimum appearing at that value. For r > 2.4, the fragility appears to remain stabilized at 30. These observations are interesting since they seem to coincide with the transition occurring at r c predicted by RT. The low fragility comprised of a network of ionic bonds correlates to a floppy system with a low number of constraints.
CHAPTER 1. BACKGROUND 15 While the high fragility with the network of covalent bonds corresponds to the rigid system of a constrained network. 1.8 Oxide Liquids As an interest in the results from by Böhmer and Angell, our research group began to investigate oxide liquids. A classic example of a strong oxide liquid is phosphorous pentoxide, P 2 O 5, in which the directional forces create a recurring short-range order in the form of a tetrahedral structure by surrounding a phosphorus atom with four oxygen atoms. Three of these oxygen atoms are bridging in the sense that they are shared by neighboring tetrahedron. While, the last oxygen atom is double bonded to the center phosphate and does not form a bridge. The network within these liquids is then comprised of tetrahedral structures connected by bridging oxygen atoms. However, the addition of alkali (e.g., sodium oxide) alters the network structure by breaking bridging oxygen links as seen in Fabian s research [2]. Thus, the network connectivity parameter of these oxide liquids becomes dependent on their average bridging oxygens per structural unit, n, instead of the average covalent bonds per atom like in chalcogenides, r, and differs with different concentration of alkali [14]. Although n and r are different, our research purposed that they are the equivalent in the sense of network topology (i.e., network connectivity parameter of the respective liquid) [14]. In chalcogenides, each atom contains specific valence electrons. The atoms behave as nodes, while the bonds contribute to the network connectivity as they bridge one atom to another. However, oxide networks contain discrete structural units, Q n, formed by oxygen atoms surrounding a center non-oxygen atom which are well defined by Nuclear Magnetic Resonance (NMR) studies [15]. These structural units are rather rigid in the sense that their internal bonds do not vary substantially with regards to length and angle and so the entire structural unit behaves like a node. (See Figure 1.11). The oxygen atom is shared between two different structural units and acts as a bridging component for network connectivity in oxide liquids similar to the bonds that are responsible for the network connectivity in chalcogenides.
CHAPTER 1. BACKGROUND 16 Figure 1.11: The illustration on the left is an oxide network composed of phosphorus (red) and oxygen (blue) atoms. Here, the bridging oxygen atoms define the network connectivity of an oxide network (i.e., n ). While in chalcogenides the covalent bonds between Germanium (green), Arsenic (yellow), and Selenium (light blue) atoms describe the network connectivity for chalcogenides (i.e., r ). Despite their different definitions, they define the network connectivity of their respective liquids in terms of the bridging component. (For convenience, the non-bridging oxygen for P 2 O 5 is not shown in 2-D figure.) Further details on NMR and the technique to determine n are beyond the scope of this thesis but NMR simply identifies specific structural units, Q n, where n represents the number of bridging oxygens and quantitatively measures the number of different Q n units that the oxide network possesses. Briefly, the technique involves the immersion of an amorphous solid in a directional constant magnetic field. This immersion triggers a polarization of the nuclear magnetic moments, µ m, associated with the nuclear spin state with the least potential energy. Afterward, a signal of proper frequency is applied to induce a transition from a lower to a higher energy spin state defined as a spin flip. Once the signal is turned off, the nuclear magnetic moments relax to their original energy state by emitting measurable signals at the resonance frequency associated with the spin flip. The results of the technique is a spectrum of resonance shifts like that shown in Figure 1.12 from which the integrated area can be used to determine the specific Q n units and
CHAPTER 1. BACKGROUND 17 their fractional populations. This, in turn, characterized n as n = f n n, (1.11) where f n is the fraction of structural units with Q n. n Figure 1.12: An example of a NMR spectrum from Brow and Kirkpatrick [15]. Different concentrations of Na 2 O (mol %) affect the number of Q n species in phosphate glass. The figure is reprinted with permission from Elsevier. 1.9 Sodium Ultraphosphates In a previous study, Fabian [2] investigated sodium ultraphosphate liquids characterized as [Na 2 O] x [P 2 O 5 ] 100 x, (1.12) where the molar percent, x, ranged from 0% to 50% and was chosen because the network connectivity varies with the concentration of Na 2 O. As previously mentioned, P 2 0 5
CHAPTER 1. BACKGROUND 18 network consists of tetrahedral structural units, PO 4. In this circumstance where the tetrahedral has just three bridging oxygens, the structural unit is classified as Q 3 unit. The addition of an Na 2 O induces a depolymerization process where one bridging oxygen atom is replaced by two non-bridging oxygens that are linked by a weak ionic bond arising from the sodium ion. This decreases the connectivity and converts some of the Q 3 to Q 2 units depending on the molar percent of Na 2 O. When x = 50%, all the structural units are converted into Q 2 units, NaPO 3. Thus statistically, the network connectivity is based on the fraction of Q 2 and Q 3 units in the ultraphosphate regime and are described from a NMR study by Brow [16] as f (Q 3 ) = 100 2x 100 x, (1.13) and f (Q 2 ) = x 100 x. (1.14) Now using the definition for n in Eq.1.11, the overall network connectivity for sodium ultraphosphates is ( 100 2x ) ( n = 3 + 2 100 x x 100 x ). (1.15) Figure 1.13: Originally, P 2 O 5 consists of three bridging oxygen atoms. The bridging oxygen is reduced to two when Na 2 O is introduced.
CHAPTER 1. BACKGROUND 19 Figure 1.14: The figure shows that the fragility of ultraphosphates typically decreases with Na 2 O additions. Data from Fabian [2]. In the Fabian study, he investigated the effect of alkali addition on the fragility by plotting the log 10 τ avg (s) versus T g /T with their respective Na 2 O concentrations shown in Figure 1.14 where τ avg is defined by Moyihan [17] as τ avд = τ Γ(1/β). (1.16) β Fabian generally observed a reduction in fragility (i.e, slope) as the Na 2 O concentration increases which is typical given the depolymerization of the network. In terms of the network connectivity, the fragility of measured ultraphosphates was plotted as a function of n in Figure 1.15 together with data from chalcogenides. As mentioned in the introduction, the results from chalcogenides and ultraphosphates are remarkably similar. The near equivalent fragility between the two liquids suggests that fragility is solely dependent on the network connectivity of the liquids despite their chemistry.
CHAPTER 1. BACKGROUND 20 Figure 1.15: For convenience, Figure 1.1 is recreated here. The fragility of sodium ultraphosphates and chalcogenides are similar with their respective network connectivity. Fabian also considered the stretching exponent by plotting β as a function of log 10 τ avg (s) shown in Figure 1.16. In his observation, far from T g, the increasing concentration of Na 2 O results in a lowering of β across the compositions. This effect indicates a broadening of the distribution of relaxation rates present throughout the liquid and are usually associated with liquids with high fragility (i.e., m > 40) as anticipated. Furthermore, Fabian observed two different β behaviors. For composition up to 30 mol% of Na 2 O, β decreases when T g is approached (i.e., as τ avg approaches 100 seconds) and agrees with many literature accounts [2]. For composition above 40 mol% of Na 2 O, β increases as T g is approached and this behavior is rarely seen. One hint for this unusual behavior is the influence of Na 2 O on the liquid dynamics because as T g is approached, the effect from the Na 2 O seems to diminish as β increases and reaches a common value of 0.5 for all compositions above 40 mol% of Na 2 O. Fabian has speculated the culprit for this
CHAPTER 1. BACKGROUND 21 diminishing effect lies with the coupling of the motion of the ions and the viscoelastic relaxation of the network. Figure 1.16: Experiments have shown that β decreases as the temperature approaches T д. However, Fabian s experiment also shows an increasing β behavior as T д is reached. It is speculated that the interaction between the motion of the ions and the viscoelastic relaxation is the cause for this [2]. When two systems are coupled, they interact with each other in an indistinguishable manner. When decoupled, the systems are independent from each other. In our study, the two systems are the motion of the ions and the viscoelastic relaxation of the network. Far from T g, the motion of the ions and the viscoelastic relaxation tend to converge as they both have similar rapid timescales. Thus, they are coupled causing the viscoelastic relaxation to be sensitive to the ions motion resulting in a broader distribution of relaxation rates and thus a small β when many ions are present. As temperature decreases toward T g, the viscoelastic relaxation becomes slower than the ions motion and becomes insensitive to the motion of the ions. The two motions are decoupled [2]. In fact at T g, the viscoelastic
CHAPTER 1. BACKGROUND 22 relaxation is arrested while the ions motion continues to persevere. Figure 1.17: The introduction of alkali elements causes f q to decrease, but slightly deviated from the predicted line. At the higher alkali content, the ions become more efficient at cross-linking the network causing the higher f q than expected [2]. Finally, Fabian reported on the effect of alkali content on the non-ergodic level, f q. He saw that the f q for ultraphosphates exhibits an inversely linear relationship to fragility showing that f q decreases with Na 2 O content. However, there is a deviation where f q is slightly higher than the predicted line (i.e., dashed line) based on previous results for compositions above 20 mol% of Na 2 O [2]. It is suggested that beyond 20 mol% of Na 2 O the excessive sodium ions become more effective at cross-linking the depolymerized network due to the limited terminal oxygen atoms. This generates a greater cohesion in the liquid to reduce the fast localized relaxations and thus caused f q to rise more than expected in Figure 1.17.
CHAPTER 1. BACKGROUND 23 1.10 Aluminophosphates To further investigate oxide liquids and follow up with the results of sodium ultraphosphates study, the effects of the addition of alumina, Al 2 O 3 (specifically Series I classification) in a phosphate network was analyzed. Series I classification of aluminophosphate glass is chemically described as [Al (PO 3 ) 3 ] x [NaPO 3 ] 100 x, (1.17) where x is the mole percent of aluminum phosphate, Al(PO 3 ) 3. Contrary to sodium ultraphosphates where the mixing of alkali element decreases n, an NMR study by Brow [18] shows the addition of alumina increases n through a process of polymerization. His results confirm that the Q 2 structural units of NaPO 3 are predominantly replaced by octahedral alumina structural units with six bridging oxygens, Q 6, or by much smaller fraction of Q 5 and Q 4 at higher alumina concentrations. Table 1.1: Table of δ 6, δ 5, and δ 4 for Different mol% of Al 2 O 3 Using Data from Brow [18] mol% of Al 2 O 3 mol% of Al(PO 3 ) 3 δ 6 δ 5 δ 4 2.5 5.3 1 0 0 5.0 11.1 1 0 0 7.5 17.6 0.96 0.02 0.02 10.0 25.0 0.94 0.04 0.02 12.5 33.3 0.92 0.06 0.02 15.0 42.9 0.89 0.09 0.02 17.5 53.8 0.86 0.13 0.01 20.0 66.7 0.71 0.21 0.08 25.0 100.0 0.50 0.33 0.17 Mol% of Al 2 O 3 was also converted to mol% of Al(PO 3 ) 3 and shown on table. Assuming n is only affected by the amount of phosphorus atoms with Q 2 units and aluminum atoms with Q 6, Q 5, or Q 4 units, the fraction of P and Al are f p = 100 + 2x 100 + 3x, (1.18)
CHAPTER 1. BACKGROUND 24 and f Al = x 100 + 3x. (1.19) Therefore by using Eq. 1.11, n is determined as n = 2f p + (6δ 6 + 5δ 5 + 4δ 4 )f Al, (1.20) where δ 6, δ 5, and δ 4 are fractions of each Al coordinated state given by Brow [18] in Table 1.1.
Chapter 2 Experimental Methods 2.1 Dynamic Light Scattering and Photon Correlation Spectroscopy Light scattering techniques are useful in identifying the atomic structure and dynamics of matter. Unlike NMR and Raman spectroscopy that measure quantitative information for amorphous solids, our research applies a form of Rayleigh scattering called dynamic light scattering to understand the attributes of amorphous liquids near T g. Rayleigh scattering is a phenomenon in which a particle, much smaller than the wavelength of light, re-radiates the incident electromagnetic, EM, wave. When an EM wave is incident on a collection of particles, the electron clouds are perturbed with the same periodic frequency as the incident electric field of the EM wave. This oscillation of the electron clouds separates the two opposing charges to create oscillating induced dipole moments which then become the source of the scattering radiation. In the case of Figure 2.1, the light scattering is elastic which refers to having the identical scattering and incident wavelength. Photon correlation spectroscopy (PCS) utilizes the principle of dynamic light scattering to monitor the slow dynamics (i.e., τ avg > 1 µs) and characterizes the viscoelastic (VE) relaxation of a liquid near T g. This technique relies heavily on the ability of the substance to cause a variation in the scattered intensity or scatter electric field to produce an autocorrelation function. If the particles in a material are stationary, the scattered intensity does not suffer from fluctuation and no autocorrelation function is produced. 25
CHAPTER 2. EXPERIMENTAL METHODS 26 Figure 2.1: As a EM wave is incident on particles, it creates oscillating dipole within the material and re-radiate the same EM wave for the case of elastic scattering. However, if the particles are constantly in motion, the intensity will fluctuate due to the relative motion of the particles. This behavior is usually observed in liquids exhibiting Brownian motions. For oxide liquids, they do not possess characteristics of a Brownian particle. Instead, the intensity fluctuation originates from the inherent density fluctuations generated by the continuous rearrangement of atoms or structural units driven by thermal severing and reconnecting of bonds within the network. The PCS technique employs the Weiner-Khintchine theorem to show that the power spectrum of the scattering light, I q (w), is the Fourier transform of the autocorrelation of the scattered electric field, Eq(0)E q (t), when applied to light scattering [19]. I q (ω) = e iωt E q(0)e q (t) dx. (2.1) E q (t) is the net electric field detected far from the scattering volume with N scatterers described as E q (t) = E o N f i ( q)e i q r i (t). (2.2) i=1 E o is the incident electric field, f i ( q) is the form factor of the scattering source, and q is the scattering wave vector defined as ( ) q = k s k i = 4πn θ sin, (2.3) λ source 2
CHAPTER 2. EXPERIMENTAL METHODS 27 where θ is the scattering angle and n is the index of refraction of the medium. From Eq. 2.2, the autocorrelation of the scattering electric field yields E q (0)E q (t) = E o 2 N f i ( q)e i q r i (0) N f j ( q)e i q r j (t). (2.4) i j Assuming that all particles are identical with identical form factors, Eq. 2.4 becomes proportional to the time-dependent dynamic structure factor, S( q, t), and is given as E q (0)E q (t) = E o 2 f (q) 2 N N 1 e i q r i (t) r j (0) = I q {S( q, t)}. (2.5) N i,j Measuring the progress of the scattering electric field for an extended duration is impracticable because it is difficult to detect the phase change in the electric field due to its sensitivity to any imperfection from the optical equipment. However, through the Siegart relation [20], this problem becomes irrelevant as it gives an alternative way of detection. The Siegart relation, I q (0)I q (t) = I q 2 + A coh E q (0)E q (t) 2, (2.6) relates the electric field correlation function to the intensity-intensity correlation function whenever the total detectable electric field results from a statistical collection of independent scatterers. Substituting Eq. 2.5 into Eq. 2.6, the autocorrelation function, C(t), then provides a means to directly measure S( q, t) by measuring intensity (i.e., counting photons) and correlating against itself, C(t) = I q(0)i q (t) I q 2 = 1 + A coh S( q, t) 2. (2.7) 2.2 Sample Preparation Samples of different compositions of aluminophosphate, [Al(PO 3 ) 3 ] x [NaPO 3 ] 100 x, of 10 to 20 grams were prepared by mixing the appropriate amount of aluminum metaphosphate, Al(PO 3 ) 3 (99.95%; Chemsavers, Inc.), with sodium metaphosphate, NaPO 3 made in the
CHAPTER 2. EXPERIMENTAL METHODS 28 laboratory. The NaPO 3 was prepared in batches of 40 to 50 grams by initially mixing amounts of ammonium dihydrogen phosphate, NH 4 H 2 PO 4 (99.99%; Sigma-Aldrich), with sodium carbonate, Na 2 CO 3 (99.95%; Acros Organics), in a container. A small amount of the mixture was poured in a Pyrex beaker to thinly fill the bottom and was placed in a furnace at 350 C to 400 C to trigger the reaction of 2[Na 2 CO 3 ] 50 + 4[NH 4 H 2 PO 4 ] 50 CO 2 + 3H 2 O + 2NH 3 + 2[Na 2 O] 50 [P 2 O 5 ] 50 (2.8) to yield NaPO 3. More mixture was gradually added to the reaction vessel to avoid overflow due to foam produced from the release of CO 2 and NH 4 from the reaction. This process continued until all the mixture was completely reacted resulting in a dry-white substance. The substance was scraped from the beaker and ground into a fine powder using a mortar and pestle and was later combined with Al(PO 3 ) 3 for a specific composition of aluminophosphate. Each powdered sample was placed into a clean quartz ampoule, 6 mm ID x 8 mm OD x 200 mm in length, made from standard quartz tubing. These ampoules were initially cleaned using glassware soap (Micro90) and filtered water (Millipore). Afterward, the internal wall of the ampule was cleaned by filling it with 10% hydrofluoric acid and allowing it to set for five to ten minutes in a fume hood. Then, it was vigorously rinsed with filtered water (Millipore) and flame dried. After the sample was added, the external wall of the ampoule was cleaned using lens sheet (Thor lab) and acetone prior to placing it upright in a furnace at 450 C. The furnace temperature was then slowly increased to 950 C-1200 C, depending on the aluminum content. The sample slowly melted for at least one hour after reaching the high temperature. A vacuum, of approximately 100 mtorr, was then applied to draw out the unwanted gas bubbles. The final sample was a clear, transparent melt. The furnace was then lowered to 500 C before the sample was transferred to an optical furnace where PCS experiments were preformed.
CHAPTER 2. EXPERIMENTAL METHODS 29 2.3 Experimental Design The purpose of the experimental construction, illustrated in Figure 2.2, is to collect and digitize scattered light from various samples. The components of the setup consist of a 532 nm continuous solid state laser, an optical furnace, and several optical elements. The laser beam is focused by a lens through the center of the ampoule housed by an optical furnace. The optical furnace is constructed using two stainless steel cylinders of 13 cm and 4 cm separated by a 3 mm spacing for the beam to penetrate through the center undisturbed. In addition, an 8 mm center hole is drilled extending vertically through the entire first cylinder and roughly 0.5 cm through the second cylinder. This holds the ampoule vertically in the beam. For heating purpose, nichrome wire, enclosed in an electrically insulating sheath, is coiled around the surface of furnace and the temperature is controlled by regulating the current using a commercial controller (Omega Engineering). In addition, the components of the two cylinders are encased with insulated material to aid in maintaining a control temperature. Figure 2.2: A representation of our optical experiment setup. The intensity of the scattered light is collected to produce a autocorrelation function. An example of autocorrelation function of polystyrene sphere is shown. From this, appropriate parameters are measured.
CHAPTER 2. EXPERIMENTAL METHODS 30 The scattered light from the sample first passes through optical filters, a laser line filter (about 10 nm linewidth) to block unwanted light generated by the furnace and a Glan Thompson polarizer to collect the vertical-vertical (VV) scattering component. Then the scattered light is focused onto a 50 µm pinhole stationed about 50 cm in front of the photonmultiplier tube, PMT (Thorne EMI), which is placed 90 from the incident beam. The collected light is amplified, discriminated, and digitized before it is delivered to a digital correlator (Correlation.com Flex 01-12) to produce a computerized intensityintensity autocorrelation function which relates to the S( q, t) as in Eq. 2.7. 2.4 Data Analysis Figure 2.3 shows an example of autocorrelation functions collected with varying temperature for one of the aluminophosphate compositions in this study. The profile of the spectra Figure 2.3: The spectra of [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 shows that the relaxation time moves with temperature and sometimes exhibit a noticeable second decay.
CHAPTER 2. EXPERIMENTAL METHODS 31 progressively shifts to the right as the temperature decreases and normally exhibits one relaxation decay. For these cases, the correlation function was fitted according to { [ ]} 2, C(t) = 1 + A exp (t/τ) β (2.9) where A = A coh f q 2, (2.10) which was derived by substituting the stretch exponential function, Eq. 1.4, into the autocorrelation function, Eq. 2.9. The coherence factor, A coh, depends on instrumental factors such as the size of the pinhole and the separation distance between the detector and the pinhole. It serves as a calibrated measurement to determine f q for each correlation function and is calibrated using a standard aqueous suspension of polystyrene spheres for a value of 0.81 ± 0.04. However, on frequent occasions within our experiment, some of the autocorrelation functions also exhibit a weaker second relaxation as seen in Figure 2.3. The cause of this effect was not precisely identified, but we suspect it is possibly due to residual bubbles or particulates undergoing a very slow Brownian motion. Nevertheless, the second relaxation time is well separated from the viscoelastic relaxation and can be easily accommodated with the addition of a floating baseline in a form of the constant, C o, together with truncation of the autocorrelation prior to the second decay. This transforms Eq. 2.9 to { [ ]} 2. C(t) = 1 + C o + A exp (t/τ) β (2.11)
CHAPTER 2. EXPERIMENTAL METHODS 32 Depending on the situation, each autocorrelation function is fitted according to Eq. 2.9 or Eq. 2.11 using a commercial data analysis software (KalediaGraph) to determine the values of A, τ, β, and C o (if necessary). They are represented by m1, m2, m3, and m4 respectively in Figure 2.4. Figure 2.4: An example of fitting the autocorrelation function to Eq. 2.11 to measure several parameters.
CHAPTER 2. EXPERIMENTAL METHODS 33 Once the fitting parameter are determined, T g can be estimated by extrapolating the point where τ avg equals 100 s on a graph of log 10 τ avg (s) vs. 1000K/T g (K). Recall that τ avg is defined by Eq. 1.16. For [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5, T g is 571.2K. Figure 2.5: The figure illustrates how T д is determined from the plot of log 10 τ avд (s) vs. 1000/T(K) when τ avд = 100 s. For the case of [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5, T д was determined as 571.2K.
CHAPTER 2. EXPERIMENTAL METHODS 34 Once T g is known, the fragility index can be determined from the slope of a plot of log 10 τ avg (s) vs. T g /T(K) according to the definition of fragility index in Eq. 1.3. For [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5, the fragility index is 54.2 shown in Figure 2.6. Figure 2.6: The fragility index is determined by the slope of a linear fit in the figure. In this case the slope has a value of 54.2.
CHAPTER 2. EXPERIMENTAL METHODS 35 To consider the other parameters, plot of β vs. log 10 τ avg (s) is generated for each composition as was done by Fabian [2]. In the case of [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5, β increases as the temperature approaches T g in Figure 2.7. Figure 2.7: In this figure, β increases as T д is approached. This indicates a narrowing in the distribution of relaxation rates.
CHAPTER 2. EXPERIMENTAL METHODS 36 The non-ergodic level, f q, is determined using Eq. 2.10 together with values of the A coh obtained by calibration. The value of f q near T g was then assessed by constructing a graph of f q vs. log 10 τ avg (s) like that shown for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 in Figure 2.8. Figure 2.8: The figure is an example on how f q near T g is measured. The flat slope shows that f q (T) does not vary much with temperature. (The data analysis is similar for other compositions of aluminophosphate and is provided in the appendix.)
Chapter 3 Results 3.1 Glass Transition Temperature As a prerequisite to determine the fragility, T g of each studied aluminophosphate composition was obtained using the method provided in the Data Analysis section. Previously Figure 3.1: With increasing concentration of Al 2 O 3, Brow [3] and our experiment, show a linear increase in T g. The results differ by 4% due to different method used to determine T g. 37
CHAPTER 3. RESULTS 38 mentioned and supported by Brow [3], the relationship between T g and the alumina content is linear due to additional energy needed to disrupt the network bonds accompanied by the increasing alumina content. Our results reproduce this linear relationship, but when comparing our T g with values reported by Brow [3] for the identical series of aluminophosphate, the two measurements differ systematically by 4% as seen in Figure 3.1. This is likely due to different methods used in determining T g. In our experiment, T g is determined by τ avg = 100 s. But in Brow, T g is determined from the transition in specific heat and is sensitive to the cooling rate used. Nevertheless, both results exhibit a similar linear dependence on alumina content which attest to the chemical integrity of my samples. 3.2 Fragility Figure 3.2: The plot of log 10 τ avg (s) vs. T д /T(K) for all aluminophosphate compositions ranging from 0 to 50 mole % of Al(PO 3 ) 3 indicates a general decrease in the slope.
CHAPTER 3. RESULTS 39 The results of the fragility of all studied aluminophosphate compositions are summarized on the plot of log 10 τ avg (s) vs. T g /T(K) as illustrated in Figure 3.2. A general decrease in the slope (i.e., fragility) with increasing alumina content is observed which corresponds to a less fragile liquid. As a consequence, the aluminophosphate network should increase in the amount of covalent bonds following Angell s [7] conclusion for strong liquids. This also coincides with Brow s NMR [3] study except in terms of n. The increase in alumina content triggers a polymerization process by replacing the Q 2 with Q 6 units (i.e., n increases) causing a more constrained network (i.e., rigid) and thus as anticipated, reflects a decrease in the fragility similar to previous studied liquids. Table 3.1: Summary of T g, Fragility, and n for [Al(PO 3 ) 3 ] x [NaPO 3 ] 100 x where x is the mole percent x(%) T g (K) fragility (m) δ 6 δ 5 δ 4 n 0.0 568.6 76.4 ± 6.3 1 0 0 2 1.0 576.1 71.4 ± 4.4 1 0 0 2.039 2.5 571.2 54.2 ± 5.6 1 0 0 2.093 4.0 598.1 53.0 ± 2.8 1 0 0 2.143 5.0 604.8 44.2 ± 2.9 1 0 0 2.174 7.0 617.9 42.5 ± 2.1 1 0 0 2.231 10 627.5 35.1 ± 2.1 1 0 0 2.308 12 642.2 34.3 ± 1.7 1 0 0 2.352 15 663.7 33.9 ± 1.6 1 0 0 2.410 20 688.7 36.1 ± 1.5 0.96 0.02 0.02 2.491 30 745.9 36.4 ± 1.5 0.93 0.05 0.02 2.618 40 786.5 34.0 ± 1.7 0.90 0.08 0.02 2.704 50 826.3 34.7 ± 2.6 0.88 0.13 0.01 2.770 The table shows that increasing the concentration of alumina results in a decrease in n and fragility, but a increase in T д. To gain further insight upon the relationship between the fragility and the structural network of aluminophosphates, the fragility of the aluminophosphates is plotted as a function of n in Figure 3.3. ( n was calculated for the aluminophosphate compositions using Eq. 1.20 and is summarized in Table 3.1 which also includes the values of T g and the fragility for the corresponding compositions). Together with the data from chalcogenides and ultraphosphates, the collective result demonstrates a universal evolution of the fragility as a function of the liquid s respective network connectivity. This further
CHAPTER 3. RESULTS 40 supports the hypothesis that the fragility solely depends on the network connectivity despite differences in chemistry. Furthermore, Figure 3.3 supports different behavior in the fragility depending on the specific range of n that was previously seen with chalcogenides and ultraphosphates. From n = 2 to 2.4, the fragility rapidly decreases with increasing n. A weak minima appears at n = 2.4, followed by a weak local maxima near n = 2.6 might be present. Otherwise, the fragility is reasonbly constant (m 35) from a range of 2.4 < n < 2.7 and it approaches a lower limit of fragility (m = 17) as n nears a value of 3. Figure 3.3: The figure shows the fragility of ultraphosphates, chalcogenides, and aluminophosphates as a function of their network connectivity. The fragility behavior is similar across these liquids indicating a dependence only on the network connectivity and shows a possible connection to the RT. The unique behaviors in the fragility across different ranges of n, especially the local minima, seem to mirror the transformation of a continuous random network from one that is underconstrained to one that is overconstrained in RT. However, the RT does not
CHAPTER 3. RESULTS 41 offer any proper connection to the dynamics of a liquid. To make such connection, it is necessary to relate the fragility to some sort of thermodynamic model which relates the dynamics to the status of the network of bonds within the liquid. 3.3 Bond Model The bond model proposed by Angell and Rao [21] is a two-state model where particles are connected by weak linkages to form a bond lattice. Each linkage can accommodate one of the two states: broken or connected with a respective difference in free energy. If these linkages are assumed to be independent events and posses identical breaking enthalpy, H, the equilibrium fraction of broken bonds, X b, is determined by modeling the bond lattice state as a thermodynamic reaction following the article by Angell, Richard, and Velikov [22]. Similar to a chemical reaction process, G is produced when the initial bond lattice state (A) with all intact linkages change to a state (B) with broken linkages, A B. (3.1) G is the free energy of standard-state [23] described as G = H T S = ktlnk eq, (3.2) where K eq, the equilibrium constant, is K eq = X b X c = Substituting Eq. 3.3 into Eq. 3.2 and solve for X b yields X b 1 X b, (3.3) X b = [ 1 + e ( H T S)/kT ] 1. (3.4) According to Angell and Rao [21], the entropy change is caused by two factors, configuration entropy, S conf, and vibrational entropy, S vib. S conf is associated with the statistical distribution of the broken and intact linkages and it increases as more linkages are broken. In particular, it measures the number of additional alternative configurations
CHAPTER 3. RESULTS 42 of the network when a linkage is broken. The S vib contribution, as they suggested, arises whenever there is a decrease in the average vibrational frequency caused by breaks in the linkages. The bond model is a statistical model and does not explicitly predict the dynamic properties of a liquid (i.e., viscosity or relaxation time). To overcome this limitation, Angell and Rao [21] proposed that the probability for a flow event, P(T), is inversely proportional to τ avg and is expressed as an exponential function of the fraction of broken linkages, { P (T ) = exp f }, (3.5) X b (T ) where f is a critical local broken constraint fraction [14]. τ avg is then replaced with P(T) 1 in Eq. 1.3 for the definition of the fragility index which then equates to m = ( { }) f d loд 10 exp X b (T ) d(t д /T ). (3.6) T T д Further simplifying Eq. 3.6 by using logarithmic rules and factoring the constant out yields ( ) 1 d m = 0.434f X b (T ) ). (3.7) d(t д /T ) T T д The next step is evaluate the derivative of Eq. 3.7, but it is problematic because it requires evaluating the derivative with respect to inverse temperature. However using Eq. 3.8 (See Appendix B Derivation), it becomes easier to solve. d 2 d(t д /T ) = T d dt. (3.8) By substituting Eq. 3.8 into Eq. 3.7 definition for the fragility index gives T д ( m = 0.434f T 2 d T д 1 X b (T ) dt ) (3.9) T T д which becomes m = 0.434f T 2 dx b (T ) T д X 2 b (T ) dt, (3.10) T T д
CHAPTER 3. RESULTS 43 once the derivative of X 1 (T) with is done. Rewriting Eq. 3.10 in its limit form yields b 0.434f T 2 dx b (T ) m = lim T T д T д X 2 b (T ), (3.11) dt which simplifies to m = 0.434f T д X 2 b (T g) X b T, (3.12) when the limit is evaluated. Numerous numerical simulations were conducted by Angell and Rao [21] to investigate the relationship between the fraction of broken bonds to both the temperature and the value of H and S. They concluded that X b (T) is roughly linear and that the slope of X b / T is proportional to S. Thus according to Eq. 3.12, a direct proportionality between fragility and entropy exists and may provide a rational explanation for the common fragility dependence of the network connectivity. Returning to the rope analogy introduced in section 1.6 Rigidity Theory, we find a natural connection to S conf as a function of n. At n = 2, the network consists of polymeric chains which, like a rope, may twist and turn in various directions with each chain possessing a large number of configurations, or microstates. However, as n slowly increases, cross-links between these polymeric chains are formed which restrict their motion, similar to the idea of pining down a rope. Each chain suffering from the constraint severely reduces the number of available configurations for any given chain. As n further increases, the chains continuously suffer from more of the pin down constraints in order to further limit the available configurations of the chains. This action should cause a rapid decrease in S conf with the number of cross-links which we observed in our results for regions between n = 2 and 2.4. This also implies an equivalent decrease in fragility according to the relationship described in Eq. 3.12. At some level of cross-linking, the number of configurations will reduce to one particular configuration resulting in S conf 0. Presumably, this will occur near n c = 2.4 where the number of constraints equals the number of degrees of freedom as depicted by the RT. Beyond n c, the addition of excessive cross-links does not affect S conf because the network is locked in one configuration. However, the fragility does not reach its lower
CHAPTER 3. RESULTS 44 limit at n c, but does at n = 3 as shown in Figure 3.4. S conf alone is unable to explain for this and both the local minima at n = 2.4 and the local maxima at n = 2.6. The small possibility of vibrational frequency (i.e., S vib ) may be present after the diminution of S conf and we can only speculate that it may account for the unique fragility behavior below n = 2.4 in the Figure 3.4. Figure 3.4: The bond model indicates a rapid decrease in the fragility as n goes from 2 to 2.4 due to S conf. Beyond n = 2.4, S vib begins to emerge as the network becomes rigid. 3.4 Stretching Exponent: β In Figure 3.5, the β values of various compositions of [Al(PO 3 ) 3 ] x [NaPO 3 ] (100 x) are plotted as a function of log 10 τ avg (s). In all cases, with the exceptions of [Al(PO 3 ) 3 ] 50 [NaPO 3 ] 50, β increases as T g is approached (i.e., τ avg = 100 s). This coincides with our hypothesis that decreasing temperature causes a narrowing in the distribution of relaxation rates as
CHAPTER 3. RESULTS 45 Figure 3.5: The figure is a summary plot of β vs. log 10 τ avд (s) for aluminophosphate compositions. All compositions, except 50 mol% of Al(PO 3 ) 3, show a positive slope indicating a broadening in the distribution of relaxation rates as T д is approached. Not all compositions are represented in Figure 3.5. (See Appendix A). the ions motion begins to decouple from the VE relaxation. Furthermore, the increase in alumina content was previously shown to lower the fragility which should cause an increase in β far from T g since a higher β reflects a stronger liquid (i.e., smaller fragility). Unfortunately, it is hard to distinguish this and the difference between β behavior across compositions with just Figure 3.5 because alumina content does not differ much between aluminophosphate compositions. To better evaluate the effect of alumina on β across measured aluminophosphate compositions and with the ultraphosphates, the slope of β with respect to the log 10 τ avg (s) vs. n is plotted in Figure 3.6. Since the smallness in β indicates the broadening in the relaxation rates, the slope of β defines how rapid the narrowing (i.e., positive slope) or broadening (i.e., negative slope) of the distribution of relaxation rates as T g is approached.
CHAPTER 3. RESULTS 46 Figure 3.6: The figure is a summary plot of slope of β vs. n for aluminophosphates and ultraphosphates. The slope of β for aluminophosphates stays positive longer than ultraphosphates. Results for aluminophosphates and sodium ultraphosphates shown in Figure 3.6 agree with each other except in the range of 2.2 < n < 2.5 where the slope of β for aluminophosphates remains positive but eventually drops into the negative zone. The likely culprit for this deviation is the influence of ions. Far from T g, the ionic motion is coupled with the VE relaxation and most likely influence the shape of S( q, t) and in turn, alters the β parameter. As previously mentioned, far from T g, we anticipate an increase in β as the ion concentration decreases towards a ion-free Al(PO 3 ) 3 composition and should result in a negative β slope. However, as the ion population decreases, the diversity of ion environments increases as Q 2 sites are replaced by Q 6 sites. This diversity promotes a broadening in the distribution of relaxation times leading to a reduction in β far from T g. This may explain why the slope of β remains more positive than that of ultraphosphates before decreasing.
CHAPTER 3. RESULTS 47 3.5 Non-ergodic level: f q It is difficult to accurately measure f q because is it not only dependent on the calibration of the coherence factor, but also on the presence of any unwanted, stray elastic scattering. In an ideal situation, the PMT should only observe light scattered from the interior of the sample when collecting data for f q. However, light scattered by the internal and external surface of the ampoule, residual bubbles, and particulates within the sample could be inadvertently mixed with the scattering light. If there is a sufficient amount of this unwanted scattering entering the PMT, a condition known as heterodyne detection results that reduced the signal-to-noise ratio which ultimately will lower the overall amplitude of the correlation function. Thus, the f q measurements may be lower than their true values despite efforts to avoid heterodyne detection. Figure 3.7: The figure is a summary plot of f q (T д ) vs. 100/m for aluminophosphates. The linear inverse relationship between fragility and f q (T д ) holds true for aluminophosphates.
CHAPTER 3. RESULTS 48 Nevertheless, the limiting value of f q near T g is plotted in Figure 3.7 as a function of inverse fragility together with the results from other previous study [2]. The results are consistent with a wide variety of melts to emphasize the inverse relationship between fragility and f q as anticipated. However, the aluminophosphates do not suffer from the rise in f q previously seen in sodium ultraphosphates which was speculated to be a consequence of ion cross-linking. For aluminophosphates, the increasing concentration of alumina causes a reduction in alkali ions accompanied by a strengthening of the covalent network where Q 2 sites are replaced by Q 6 sites. This reduced ion cross-linking and may explain why the f q values for aluminophosphates reside below those of ultraphosphates.
Chapter 4 Conclusion The project began as an interest in the physics of altered glasses whose properties change with the addition of different chemicals, and evolved into the discovery of an intrarelationship between dynamics and structural properties of glass. Fabian initiated this evolution by investigating sodium ultraphosphates where the addition of Na 2 O causes depolymerization. This results in a lowering of n together with a general rise in fragility. More importantly, his results are reproducible with data from chalcogenides and from my experiment with aluminophosphates. However, the addition of alumina to NaPO 3 causes n to increase as NaPO 3 units are replaced by Al 2 O 3 instead of a decrease in n in the case of ultraphosphates. Nevertheless, the similarity between the liquids suggests that fragility solely depends on their respective liquid network connectivity. With the introduction of the bond model, the connection between the fragility and the structure of the glass becomes apparent as the bond model concludes that fragility is proportional to the combination of configurational and vibrational entropy. Configurational entropy closely ties to the alternative definition of the rigidity theory which describes overconstrained and underconstrained systems based on the number of available configurations a system possesses. In fact, the results do show a dramatic decrease in fragility as n increases from 2 and agrees with the rapid decrease in the number of configurations for an underconstrained system when constraints were added. However when the transition from an underconstrained to an overconstrained system occurs at n = 2.4, supposedly the number of configurations reduces to one causing the fragility to reach 49
CHAPTER 4. CONCLUSION 50 a lower limit. Our results do not reflect this. Instead, the fragility rises after n = 2.4 and decreases shortly afterward toward a lower limit. We could only suspect that contributing factor for this behavior is the emergence of vibrational entropy that appears after n = 2.4. When the β parameter is investigated, all compositions except for the 50 mol% show an increase in β as the temperature approaches T g. The result suggests a narrowing in the distribution of relaxation rates and is expected as the effect of the ion coupling is diminished near T g similar to the β behavior in ultraphosphate compositions above 30 mol%. The difference in the β behavior between aluminophosphates and ultraphosphates appears in the slope of β where it typically remains positive for aluminophosphates before decreasing like ultraphosphates. Finally, the f q level for the aluminophosphate compositions generally remains constant with varying temperatures indicating little or no dependence on temperature. Furthermore, f q near T g has been shown to increase with alumina which we suspect was due to a greater cohesion throughout the liquid promoted by additional bridging oxygens. Also, f q near T g displays an inverse relationship with fragility and the result resonates with other literature accounts. The next possible step in this project is to continue the study on aluminophosphates. My research was halted at a composition of 50 mol% because of inadequate equipment to bring the temperature high enough to completely melt the composition to its liquid phase. However, with the new furnace, this high temperature is achievable. Furthermore, there are several different series of aluminophosphates that could be studied; each with different characteristics. For example, series IV aluminophosphates of the form, [NaAlO 2 ] x [NaPO 3 ] 100 x, (4.1) where x is the mole percent of NaAlO 2 [18] could be investigated. I anticipate that series IV aluminophosphates will exhibit the same fragility trend, but instead of replacing the majority of NaPO 3 units of Q 2 with alumina units of Q 6, they are mostly dominated by Q 5 and Q 4 [3]. I suspect the noticeable difference between series I and series IV aluminophosphates will be the behavior in β and f q since they are influenced by ion
CHAPTER 4. CONCLUSION 51 content which, unlike series I, is constant in Series IV aluminophosphates. Another possible direction for this research involves investigating other oxide liquids. A good candidate is a mixed glass of geranium oxide, GeO 2, and NaPO 3 described as [GeO 2 ] y [NaPO 3 ] 100 y, (4.2) where y is the mole percent of GeO 2. The importance of germanium oxide lies in its applications due to its high refractive index and other properties. When combined with phosphate glass, the quality of the mixed glass can increase and be fine-tuned for appropriate purposes. Similar to aluminophosphates, the structure within this glass has been thoroughly examined by Ren and Eckert [24] using NMR. They conclude that this glass is a network consisting of Q 2 and Q 3 phosphate units linked together by Q 2 and Q 3 germanium units. Hopefully, the result from the germanium phosphates study will reproduce similar fragility results from previous liquids: ultraphosphates, chalcogenides, and aluminophosphates.
Appendix A Analysis 52
APPENDIX A. ANALYSIS 53 A.1 [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100 Figure A.1: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100 between temperatures of 602.8K and 587.2K. Table A.1: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.808 ± 0.0004, τ, and β for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100 Between Temperatures of 602.8K and 587.2K. Temperature (K) Amplitude (A) f q τ(s) β 602.8 0.381 ± 0.020 0.687 ± 0.018 0.00085 ± 0.00007 (0.276 ± 0.012) 602.7 0.368 ± 0.033 0.675 ± 0.030 0.00056 ± 0.00009 (0.284 ± 0.021) 598.7 0.264 ± 0.008 0.572 ± 0.009 0.00389 ± 0.00009 0.363 ± 0.015 598.2 0.286 ± 0.005 0.595 ± 0.005 0.00288 ± 0.00007 0.364 ± 0.008 596.2 0.278 ± 0.004 0.587 ± 0.004 0.0050 ± 0.0002 0.383 ± 0.009 593.7 0.309 ± 0.003 0.618 ± 0.004 0.0141 ± 0.0004 0.379 ± 0.007 591.7 0.291 ± 0.002 0.600 ± 0.003 0.0239 ± 0.0005 0.413 ± 0.006 589.9 0.256 ± 0.002 0.563 ± 0.003 0.0510 ± 0.0018 0.417 ± 0.009 589.4 0.293 ± 0.002 0.602 ± 0.002 0.0473 ± 0.0011 0.442 ± 0.006 587.2 0.337 ± 0.002 0.646 ± 0.003 0.1940 ± 0.0058 0.426 ± 0.006
APPENDIX A. ANALYSIS 54 Figure A.2: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100 between temperatures of 585.4K and 572.0K. Table A.2: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.808 ± 0.0004, τ, and β for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100 Between Temperatures of 585.4K and 572.0K. Temperature (K) Amplitude (A) f q τ(s) β 585.4 0.276 ± 0.002 0.584 ± 0.003 0.2390 ± 0.0097 0.448 ± 0.009 584.9 0.350 ± 0.001 0.658 ± 0.002 0.3640 ± 0.0088 0.429 ± 0.005 582.8 0.299 ± 0.001 0.608 ± 0.002 0.3900 ± 0.0116 0.451 ± 0.007 581.0 0.275 ± 0.001 0.583 ± 0.002 0.9520 ± 0.0392 0.470 ± 0.010 578.4 0.350 ± 0.001 0.658 ± 0.002 0.9960 ± 0.0236 0.524 ± 0.007 576.6 0.252 ± 0.002 0.558 ± 0.002 3.5600 ± 0.1950 0.463 ± 0.012 576.6 0.298 ± 0.002 0.607 ± 0.003 5.6300 ± 0.4190 (0.395 ± 0.013) 576.0 0.259 ± 0.002 0.566 ± 0.002 3.3000 ± 0.1830 0.479 ± 0.013 572.0 0.335 ± 0.005 0.644 ± 0.005 15.500 ± 1.5300 (0.446 ± 0.019) 572.0 0.335 ± 0.005 0.644 ± 0.005 37.100 ± 7.1500 (0.449 ± 0.036)
APPENDIX A. ANALYSIS 55 Figure A.3: T д is determined to be 568.6K from the graph of log 10 τ avд (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100. Figure A.4: The fragility index of 76.4 for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100 is determined from the slope on the graph of log 10 τ avд (s) vs. T д /T(K).
APPENDIX A. ANALYSIS 56 Figure A.5: The positive β slope of 0.0383 on the graph of β vs. log 10 τ avд (s) indicates a narrowing of the distribution of relaxation rates for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100. Figure A.6: f q near T д is estimated at 0.607 from the graph of f q vs. log 10 τ avg (s) for [Al(PO 3 ) 3 ] 0 [NaP0 3 ] 100.
APPENDIX A. ANALYSIS 57 A.2 [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99 Figure A.7: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99 between temperatures of 611.5K and 599.0K. Table A.3: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.835 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99 Between Temperatures of 611.5K and 599.0K. Temperature (K) Amplitude (A) f q τ(s) β 611.5 0.306 ± 0.025 0.606 ± 0.026 0.0013 ± 0.0002 0.276 ± 0.025 608.9 0.316 ± 0.012 0.615 ± 0.012 0.0034 ± 0.0002 0.382 ± 0.022 608.1 0.266 ± 0.001 0.565 ± 0.002 0.0050 ± 0.0004 0.395 ± 0.028 606.2 0.335 ± 0.008 0.633 ± 0.008 0.0089 ± 0.0005 (0.342 ± 0.014) 606.2 0.350 ± 0.009 0.647 ± 0.008 0.0089 ± 0.0006 0.362 ± 0.016 601.9 0.325 ± 0.004 0.624 ± 0.004 0.0276 ± 0.0012 0.414 ± 0.010 600.8 0.340 ± 0.005 0.638 ± 0.005 0.0203 ± 0.0010 0.368 ± 0.012 600.8 0.312 ± 0.007 0.612 ± 0.007 0.0223 ± 0.0017 0.418 ± 0.020 600.8 0.320 ± 0.008 0.619 ± 0.008 0.0207 ± 0.0016 0.407 ± 0.020 599.0 0.345 ± 0.004 0.643 ± 0.004 0.0770 ± 0.0037 0.385 ± 0.011
APPENDIX A. ANALYSIS 58 Figure A.8: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99 between temperatures of 597.2K and 582.9K. Table A.4: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.835 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99 Between Temperatures of 597.2K and 582.9K. Temperature (K) Amplitude (A) f q τ(s) β 597.2 0.316 ± 0.004 0.615 ± 0.004 0.0450 ± 0.0051 0.452 ± 0.021 597.2 0.285 ± 0.005 0.584 ± 0.005 0.0601 ± 0.0024 0.453 ± 0.010 596.3 0.310 ± 0.003 0.609 ± 0.003 0.1389 ± 0.0074 0.431 ± 0.012 593.7 0.282 ± 0.003 0.581 ± 0.004 0.1823 ± 0.0127 0.454 ± 0.016 593.7 0.339 ± 0.004 0.637 ± 0.004 0.2033 ± 0.0125 0.403 ± 0.012 593.7 0.351 ± 0.003 0.648 ± 0.003 0.3140 ± 0.0176 0.403 ± 0.010 590.3 0.329 ± 0.002 0.628 ± 0.003 0.7010 ± 0.0376 0.428 ± 0.010 586.6 0.316 ± 0.001 0.616 ± 0.002 2.4250 ± 0.0700 (0.404 ± 0.006) 582.9 0.338 ± 0.001 0.636 ± 0.002 10.350 ± 0.4864 (0.407 ± 0.008)
APPENDIX A. ANALYSIS 59 Figure A.9: T д is determined to be 576.1K from the graph of log 10 τ avд (s) vs. for[al(po 3 ) 3 ] 1 [NaP0 3 ] 99. 1000/T(K) Figure A.10: The fragility index of 71.4 for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99 is determined from the slope on the graph of log 10 τ avд (s) vs. T д /T(K).
APPENDIX A. ANALYSIS 60 Figure A.11: The positive β slope of 0.0346 on the graph of β vs. log 10 τ avд (s) indicates a narrowing of the distribution of relaxation rates for [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99. Figure A.12: f q near T д is estimated at 0.640 from the graph of f q vs. [Al(PO 3 ) 3 ] 1 [NaP0 3 ] 99. log 10 τ avg (s) for
APPENDIX A. ANALYSIS 61 A.3 [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 Figure A.13: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 between temperatures of 624.3K and 589.8K. Table A.5: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.828 ± 0.0009, τ, and β for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 Between Temperatures of 624.3K and 589.8K. Temperature (K) Amplitude (A) f q τ(s) β 624.3 0.404 ± 0.027 0.699 ± 0.025 0.00031 ± 0.00006 (0.251 ± 0.019) 620.4 0.492 ± 0.026 0.771 ± 0.020 0.00064 ± 0.00006 (0.268 ± 0.014) 615.9 0.395 ± 0.012 0.691 ± 0.011 0.00167 ± 0.00009 (0.308 ± 0.013) 611.7 0.282 ± 0.014 0.584 ± 0.015 0.0053 ± 0.0005 0.361 ± 0.027 611.6 0.398 ± 0.004 0.694 ± 0.004 0.0041 ± 0.0002 (0.325 ± 0.013) 607.6 0.337 ± 0.004 0.638 ± 0.004 0.0082 ± 0.0002 0.409 ± 0.009 606.2 0.345 ± 0.003 0.646 ± 0.003 0.0217 ± 0.0008 0.431 ± 0.010 600.9 0.344 ± 0.002 0.645 ± 0.003 0.0309 ± 0.0010 0.419 ± 0.008 598.8 0.381 ± 0.003 0.679 ± 0.003 0.0954 ± 0.0036 0.408 ± 0.008
APPENDIX A. ANALYSIS 62 Figure A.14: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 between temperatures of 598.8K and 587.9K. Table A.6: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.828 ± 0.0009, τ, and β for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 Between Temperatures of 598.8K and 587.9K. Temperature (K) Amplitude (A) f q τ(s) β 598.8 0.382 ± 0.003 0.679 ± 0.003 0.0941 ± 0.0034 0.420 ± 0.008 598.8 0.379 ± 0.002 0.677 ± 0.002 0.0830 ± 0.0027 0.405 ± 0.007 596.2 0.348 ± 0.002 0.648 ± 0.002 0.0959 ± 0.0031 0.431 ± 0.007 592.8 0.349 ± 0.002 0.649 ± 0.003 0.2360 ± 0.0098 0.459 ± 0.009 590.4 0.377 ± 0.002 0.675 ± 0.002 0.5960 ± 0.0187 0.415 ± 0.007 590.2 0.425 ± 0.001 0.717 ± 0.001 0.9820 ± 0.0220 0.434 ± 0.004 587.9 0.364 ± 0.001 0.663 ± 0.001 1.3300 ± 0.0809 0.422 ± 0.010 587.9 0.390 ± 0.002 0.687 ± 0.002 1.0600 ± 0.0363 0.429 ± 0.006
APPENDIX A. ANALYSIS 63 Figure A.15: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 between temperatures of 585.9K and 573.2K. Table A.7: (3) The Value of the Amplitude, f q Evaluated when A coh = 0.828 ± 0.0009, τ, and β for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 Between Temperatures of 585.9K and 573.2K. Temperature (K) Amplitude (A) f q τ(s) β 585.9 0.380 ± 0.001 0.678 ± 0.001 1.6200 ± 0.0492 0.450 ± 0.006 584.0 0.410 ± 0.001 0.704 ± 0.001 4.8000 ± 0.1640 0.463 ± 0.007 584.0 0.413 ± 0.001 0.706 ± 0.002 3.3200 ± 0.0789 0.483 ± 0.006 582.0 0.407 ± 0.001 0.701 ± 0.001 4.7700 ± 0.0993 0.474 ± 0.005 579.6 0.416 ± 0.001 0.709 ± 0.002 7.6800 ± 0.1690 0.464 ± 0.005 579.5 0.456 ± 0.001 0.742 ± 0.001 12.200 ± 0.2900 0.464 ± 0.005 577.1 0.455 ± 0.002 0.742 ± 0.003 15.100 ± 0.4110 0.478 ± 0.006 573.2 0.434 ± 0.002 0.724 ± 0.003 32.000 ± 0.9090 (0.427 ± 0.006)
APPENDIX A. ANALYSIS 64 Figure A.16: T д is determined to be 571.2K from the graph of log 10 τ avд (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5. Figure A.17: The fragility index of 54.2 for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5 is determined from the slope on the graph of log 10 τ avд (s) vs. T д /T(K).
APPENDIX A. ANALYSIS 65 Figure A.18: The positive β slope of 0.0245 on the graph of β vs. log 10 τ avд (s) indicates a narrowing of the distribution of relaxation rates for [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5. Figure A.19: f q near T д is estimated at 0.716 from the graph of f q vs. [Al(PO 3 ) 3 ] 2.5 [NaP0 3 ] 97.5. log 10 τ avg (s) for
APPENDIX A. ANALYSIS 66 A.4 [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96 Figure A.20: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96 between temperatures of 648.5K and 626.7K. Table A.8: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.835 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96 Between Temperatures of 648.5K and 626.7K. Temperature (K) Amplitude (A) f q τ(s) β 648.5 0.222 ± 0.006 0.511 ± 0.007 0.0021 ± 0.0002 0.367 ± 0.025 645.3 0.302 ± 0.009 0.595 ± 0.008 0.0038 ± 0.0002 0.402 ± 0.020 642.5 0.344 ± 0.011 0.635 ± 0.010 0.0048 ± 0.0003 0.362 ± 0.018 639.6 0.284 ± 0.006 0.577 ± 0.007 0.0085 ± 0.0005 0.402 ± 0.016 635.4 0.325 ± 0.005 0.618 ± 0.005 0.0242 ± 0.0012 0.379 ± 0.011 632.8 0.361 ± 0.005 0.651 ± 0.004 0.0462 ± 0.0022 0.368 ± 0.010 630.4 0.310 ± 0.004 0.603 ± 0.003 0.0488 ± 0.0024 0.403 ± 0.012 628.2 0.305 ± 0.003 0.598 ± 0.004 0.0723 ± 0.0037 0.427 ± 0.012 626.7 0.329 ± 0.003 0.621 ± 0.003 0.1270 ± 0.0057 0.410 ± 0.009
APPENDIX A. ANALYSIS 67 Figure A.21: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96 between temperatures of 623.7K and 605.0K. Table A.9: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.835 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96 Between Temperatures of 623.7K and 605.0K. Temperature (K) Amplitude (A) f q τ(s) β 623.7 0.286 ± 0.002 0.579 ± 0.003 0.2020 ± 0.0088 0.433 ± 0.009 620.7 0.361 ± 0.002 0.651 ± 0.003 0.4610 ± 0.0200 0.405 ± 0.008 620.7 0.280 ± 0.003 0.573 ± 0.003 0.3990 ± 0.0268 0.410 ± 0.015 620.7 0.295 ± 0.002 0.588 ± 0.002 0.4140 ± 0.0171 0.445 ± 0.009 617.8 0.304 ± 0.002 0.597 ± 0.001 0.8880 ± 0.0384 0.428 ± 0.009 614.4 0.302 ± 0.002 0.595 ± 0.002 1.9300 ± 0.0713 0.453 ± 0.007 611.9 0.360 ± 0.002 0.650 ± 0.004 3.1100 ± 0.1070 0.480 ± 0.008 608.5 0.341 ± 0.001 0.633 ± 0.002 5.7300 ± 0.1410 0.458 ± 0.005 605.0 0.356 ± 0.001 0.646 ± 0.001 8.9400 ± 0.4160 0.479 ± 0.011
APPENDIX A. ANALYSIS 68 Figure A.22: T д is determined to be 598.1K from the graph of log 10 τ avд (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96. Figure A.23: The fragility index of 53.0 for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96 is determined from the slope on the graph of log 10 τ avд (s) vs. T д /T(K).
APPENDIX A. ANALYSIS 69 Figure A.24: The positive β slope of 0.0294 on the graph of β vs. log 10 τ avд (s) indicates a narrowing of the distribution of relaxation rates for [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96. Figure A.25: f q near T д is estimated at 0.650 from the graph of f q vs. [Al(PO 3 ) 3 ] 4 [NaP0 3 ] 96. log 10 τ avg (s) for
APPENDIX A. ANALYSIS 70 A.5 [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95 Figure A.26: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95 between temperatures of 649.8K and 633.2K. Table A.10: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.770 ± 0.0006, τ, and β for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95 Between Temperatures of 649.8K and 633.2K. Temperature (K) Amplitude (A) f q τ(s) β 649.8 0.282 ± 0.004 0.605 ± 0.005 0.01687 ± 0.00065 0.369 ± 0.009 645.4 0.283 ± 0.005 0.606 ± 0.05 0.0387 ± 0.0024 0.372 ± 0.013 645.4 0.296 ± 0.002 0.620 ± 0.003 0.0459 ± 0.0015 0.377 ± 0.007 637.6 0.333 ± 0.003 0.658 ± 0.003 0.1396 ± 0.0048 0.398 ± 0.007 633.7 0.333 ± 0.002 0.658 ± 0.002 0.3525 ± 0.0099 0.405 ± 0.005 633.7 0.337 ± 0.002 0.661 ± 0.002 0.5496 ± 0.0148 0.389 ± 0.004 633.2 0.310 ± 0.002 0.635 ± 0.003 0.2788 ± 0.0107 0.414 ± 0.007
APPENDIX A. ANALYSIS 71 Figure A.27: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95 between temperatures of 628.7K and 612.0K. Table A.11: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.770 ± 0.0006, τ, and β for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95 Between Temperatures of 628.7K and 612.0K. Temperature (K) Amplitude (A) f q τ(s) β 628.7 0.317 ± 0.001 0.642 ± 0.001 0.7060 ± 0.0234 0.423 ± 0.006 624.6 0.242 ± 0.001 0.561 ± 0.001 1.7200 ± 0.0341 0.427 ± 0.004 620.4 0.270 ± 0.001 0.592 ± 0.002 1.4450 ± 0.0525 0.425 ± 0.008 620.4 0.257 ± 0.001 0.578 ± 0.002 3.3380 ± 0.0793 0.455 ± 0.005 616.0 0.330 ± 0.001 0.655 ± 0.001 8.0140 ± 0.2015 0.459 ± 0.005 612.0 0.380 ± 0.001 0.702 ± 0.001 11.7500 ± 0.3214 0.435 ± 0.005
APPENDIX A. ANALYSIS 72 Figure A.28: T д is determined to be 604.8K from the graph of log 10 τ avд (s) vs. 1000/T(K) for[al(po 3 ) 3 ] 5 [NaP0 3 ] 95. Figure A.29: The fragility index of 44.2 for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95 is determined from the slope on the graph of log 10 τ avд (s) vs. T д /T(K).
APPENDIX A. ANALYSIS 73 Figure A.30: The positive β slope of 0.0334 on the graph of β vs. log 10 τ avд (s) indicates a narrowing of the distribution of relaxation rates for [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95. Figure A.31: f q near T д is estimated at 0.647 from the graph of f q vs. [Al(PO 3 ) 3 ] 5 [NaP0 3 ] 95. log 10 τ avg (s) for
APPENDIX A. ANALYSIS 74 A.6 [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93 Figure A.32: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93 between temperatures of 695.4K and 668.8K. Table A.12: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.803 ± 0.0002, τ, and β for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93 Between Temperatures of 695.4K and 668.8K. Temperature (K) Amplitude (A) f q τ(s) β 695.4 0.471 ± 0.066 0.766 ± 0.054 0.0003 ± 0.0001 (0.278 ± 0.035) 685.9 0.466 ± 0.042 0.762 ± 0.035 0.0013 ± 0.0001 (0.307 ± 0.022) 683.5 0.318 ± 0.011 0.629 ± 0.011 0.0017 ± 0.0001 0.370 ± 0.019 681.5 0.395 ± 0.016 0.702 ± 0.014 0.0017 ± 0.0001 0.352 ± 0.018 678.5 0.358 ± 0.003 0.668 ± 0.003 0.0026 ± 0.0001 0.376 ± 0.019 676.2 0.386 ± 0.011 0.694 ± 0.010 0.0040 ± 0.0003 0.380 ± 0.021 673.6 0.359 ± 0.009 0.669 ± 0.008 0.0063 ± 0.0004 0.403 ± 0.017 671.2 0.380 ± 0.008 0.688 ± 0.008 0.0104 ± 0.0006 0.396 ± 0.015 668.8 0.363 ± 0.005 0.672 ± 0.005 0.0147 ± 0.0006 0.408 ± 0.011
APPENDIX A. ANALYSIS 75 Figure A.33: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93 between temperatures of 666.3K and 652.1K. Table A.13: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.803 ± 0.0002, τ, and β for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93 Between Temperatures of 666.3K and 652.1K. Temperature (K) Amplitude (A) f q τ(s) β 666.3 0.345 ± 0.003 0.655 ± 0.003 0.0317 ± 0.0012 0.425 ± 0.009 663.8 0.402 ± 0.004 0.708 ± 0.003 0.0279 ± 0.0010 0.388 ± 0.009 663.8 0.390 ± 0.004 0.697 ± 0.004 0.0361 ± 0.0013 0.399 ± 0.008 660.4 0.403 ± 0.004 0.709 ± 0.003 0.0911 ± 0.0026 0.385 ± 0.006 660.4 0.350 ± 0.002 0.660 ± 0.002 0.0681 ± 0.0028 0.424 ± 0.009 658.0 0.378 ± 0.003 0.686 ± 0.003 0.0803 ± 0.0027 0.430 ± 0.008 654.6 0.419 ± 0.003 0.723 ± 0.002 0.1851 ± 0.0052 0.393 ± 0.005 652.1 0.409 ± 0.002 0.714 ± 0.002 0.2524 ± 0.0080 0.451 ± 0.007
APPENDIX A. ANALYSIS 76 Figure A.34: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93 between temperatures of 648.6K and 626.8K. Table A.14: (3) The Value of the Amplitude, f q Evaluated when A coh = 0.803 ± 0.0002, τ, and β for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93 Between Temperatures of 648.6K and 626.8K. Temperature (K) Amplitude (A) f q τ(s) β 648.6 0.388 ± 0.002 0.695 ± 0.002 0.3500 ± 0.0128 0.472 ± 0.008 646.2 0.385 ± 0.003 0.692 ± 0.003 0.4892 ± 0.0200 0.458 ± 0.009 642.7 0.389 ± 0.001 0.696 ± 0.001 0.7560 ± 0.0248 0.466 ± 0.007 639.4 0.398 ± 0.002 0.704 ± 0.002 1.7830 ± 0.0937 0.417 ± 0.010 639.3 0.420 ± 0.001 0.723 ± 0.001 1.9040 ± 0.0586 0.489 ± 0.008 635.7 0.366 ± 0.001 0.675 ± 0.001 1.9000 ± 0.0664 0.517 ± 0.009 631.4 0.408 ± 0.001 0.713 ± 0.001 11.050 ± 0.3106 0.497 ± 0.007 626.8 0.379 ± 0.001 0.687 ± 0.001 14.220 ± 0.5052 0.489 ± 0.001
APPENDIX A. ANALYSIS 77 Figure A.35: T д is determined to be 617.9K from the graph of log 10 τ avд (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93. Figure A.36: The fragility index of 42.5 for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93 is determined from the slope on the graph of log 10 τ avд (s) vs. T д /T(K).
APPENDIX A. ANALYSIS 78 Figure A.37: The positive β slope of 0.0363 on the graph of β vs. log 10 τ avд (s) indicates a narrowing of the distribution of relaxation rates for [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93. Figure A.38: f q near T д is estimated at 0.698 from the graph of f q vs. [Al(PO 3 ) 3 ] 7 [NaP0 3 ] 93. log 10 τ avg (s) for
APPENDIX A. ANALYSIS 79 A.7 [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90 Figure A.39: The autocorrelation spectra obtained for[al(po 3 ) 3 ] 10 [NaP0 3 ] 90 between temperatures of 731.6K and 704.4K. Table A.15: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.840 ± 0.0002, τ, and β for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90 Between Temperatures of 731.6K and 704.4K. Temperature (K) Amplitude (A) f q τ(s) β 731.6 0.447 ± 0.079 0.729 ± 0.064 0.00010 ± 0.00005 (0.267 ± 0.043) 727.7 0.482 ± 0.061 0.758 ± 0.048 0.0002 ± 0.0001 (0.298 ± 0.035) 724.7 0.564 ± 0.077 0.819 ± 0.056 0.0002 ± 0.0001 (0.278 ± 0.027) 717.9 0.482 ± 0.036 0.757 ± 0.028 0.0005 ± 0.0001 (0.317 ± 0.021) 717.9 0.489 ± 0.046 0.763 ± 0.036 0.0004 ± 0.0001 (0.301 ± 0.026) 714.3 0.401 ± 0.018 0.691 ± 0.016 0.0008 ± 0.0001 0.365 ± 0.019 711.0 0.436 ± 0.020 0.720 ± 0.017 0.0010 ± 0.0001 0.358 ± 0.018 707.7 0.424 ± 0.014 0.710 ± 0.011 0.0019 ± 0.0001 0.369 ± 0.016 704.4 0.426 ± 0.010 0.712 ± 0.008 0.0042 ± 0.0002 0.393 ± 0.014
APPENDIX A. ANALYSIS 80 Figure A.40: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90 between temperatures of 701.1K and 672.5K. Table A.16: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.840 ± 0.0002, τ, and β for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90 Between Temperatures of 701.1K and 672.5K. Temperature (K) Amplitude (A) f q τ(s) β 701.1 0.431 ± 0.008 0.716 ± 0.007 0.0068 ± 0.0003 (0.378 ± 0.011) 696.9 0.418 ± 0.005 0.705 ± 0.005 0.0102 ± 0.0003 (0.398 ± 0.010) 693.1 0.401 ± 0.005 0.691 ± 0.005 0.0119 ± 0.0004 0.403 ± 0.010 688.7 0.401 ± 0.003 0.691 ± 0.002 0.0367 ± 0.0010 (0.434 ± 0.007) 688.7 0.427 ± 0.004 0.713 ± 0.004 0.0302 ± 0.0012 0.411 ± 0.009 685.3 0.396 ± 0.003 0.686 ± 0.002 0.0315 ± 0.0009 0.451 ± 0.007 680.9 0.399 ± 0.003 0.689 ± 0.003 0.0633 ± 0.0023 0.459 ± 0.009 676.6 0.361 ± 0.002 0.655 ± 0.002 0.0813 ± 0.0029 0.465 ± 0.009 672.5 0.392 ± 0.002 0.683 ± 0.002 0.1221 ± 0.0037 0.479 ± 0.008
APPENDIX A. ANALYSIS 81 Figure A.41: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90 between temperatures of 668.2K and 645.5K. Table A.17: (3) The Value of the Amplitude, f q Evaluated when A coh = 0.840 ± 0.0002, τ, and β for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90 Between Temperatures of 668.2K and 645.5K. Temperature (K) Amplitude (A) f q τ(s) β 668.2 0.393 ± 0.003 0.684 ± 0.002 0.2573 ± 0.0113 0.457 ± 0.010 668.1 0.401 ± 0.002 0.691 ± 0.002 0.2958 ± 0.0111 0.475 ± 0.009 664.0 0.393 ± 0.002 0.684 ± 0.001 0.4820 ± 0.0134 0.478 ± 0.007 659.2 0.415 ± 0.002 0.703 ± 0.001 0.9058 ± 0.0333 0.495 ± 0.010 659.2 0.361 ± 0.002 0.656 ± 0.001 1.3060 ± 0.0432 0.479 ± 0.008 655.2 0.426 ± 0.001 0.712 ± 0.001 2.1640 ± 0.0615 0.513 ± 0.008 655.2 0.402 ± 0.001 0.692 ± 0.001 2.5890 ± 0.0716 0.487 ± 0.007 650.2 0.309 ± 0.001 0.606 ± 0.001 2.8070 ± 0.1026 0.466 ± 0.008 645.5 0.356 ± 0.001 0.651 ± 0.001 5.5060 ± 0.0228 0.459 ± 0.009
APPENDIX A. ANALYSIS 82 Figure A.42: T д is determined to be 627.5K from the graph of log 10 τ avд (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90. Figure A.43: The fragility index of 35.1 for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90 is determined from the slope on the graph of log 10 τ avд (s) vs. T д /T(K).
APPENDIX A. ANALYSIS 83 Figure A.44: The positive β slope of 0.0389 on the graph of β vs. log 10 τ avд (s) indicates a narrowing of the distribution of relaxation rates for [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90. Figure A.45: f q near T д is estimated at 0.630 from the graph of f q vs. [Al(PO 3 ) 3 ] 10 [NaP0 3 ] 90. log 10 τ avg (s) for
APPENDIX A. ANALYSIS 84 A.8 [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88 Figure A.46: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88 between temperatures of 766.6K and 734.8K. Table A.18: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.846 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88 Between Temperatures of 766.6K and 734.8K. Temperature (K) Amplitude (A) f q τ(s) β 766.6 0.437 ± 0.083 0.719 ± 0.068 0.00006 ± 0.00002 0.341 ± 0.045 762.1 0.519 ± 0.109 0.783 ± 0.082 0.00007 ± 0.00002 0.328 ± 0.049 759.1 0.433 ± 0.060 0.715 ± 0.050 0.00011 ± 0.00002 0.359 ± 0.039 756.6 0.399 ± 0.044 0.686 ± 0.038 0.00016 ± 0.00002 0.376 ± 0.037 753.0 0.378 ± 0.026 0.668 ± 0.023 0.00025 ± 0.00002 0.398 ± 0.028 749.7 0.386 ± 0.024 0.675 ± 0.021 0.00033 ± 0.00002 0.401 ± 0.028 747.3 0.444 ± 0.025 0.724 ± 0.020 0.00030 ± 0.00003 0.349 ± 0.022 744.2 0.349 ± 0.016 0.642 ± 0.014 0.00064 ± 0.00005 0.427 ± 0.029 741.2 0.352 ± 0.016 0.645 ± 0.015 0.00074 ± 0.00004 0.373 ± 0.021 737.7 0.412 ± 0.015 0.698 ± 0.012 0.00103 ± 0.00004 0.398 ± 0.016 734.8 0.401 ± 0.013 0.689 ± 0.011 0.00148 ± 0.0007 0.401 ± 0.017
APPENDIX A. ANALYSIS 85 Figure A.47: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88 between temperatures of 731.3K and 710.3K. Table A.19: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.846 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88 Between Temperatures of 731.3K and 710.3K. Temperature (K) Amplitude (A) f q τ(s) β 731.3 0.366 ± 0.007 0.658 ± 0.006 0.0022 ± 0.0001 0.459 ± 0.014 731.3 0.365 ± 0.009 0.656 ± 0.008 0.0021 ± 0.0001 0.457 ± 0.018 731.3 0.401 ± 0.009 0.689 ± 0.008 0.0019 ± 0.0001 0.410 ± 0.014 728.3 0.408 ± 0.008 0.694 ± 0.007 0.0029 ± 0.0001 0.419 ± 0.013 725.5 0.398 ± 0.007 0.686 ± 0.006 0.0036 ± 0.0001 0.452 ± 0.014 722.0 0.391 ± 0.006 0.680 ± 0.005 0.0058 ± 0.0003 0.473 ± 0.014 722.0 0.425 ± 0.012 0.709 ± 0.010 0.0058 ± 0.0004 0.412 ± 0.019 716.9 0.371 ± 0.003 0.662 ± 0.003 0.0098 ± 0.0004 0.470 ± 0.012 713.7 0.397 ± 0.005 0.685 ± 0.004 0.0145 ± 0.0012 0.418 ± 0.016 710.3 0.423 ± 0.007 0.707 ± 0.006 0.0233 ± 0.0024 0.461 ± 0.019
APPENDIX A. ANALYSIS 86 Figure A.48: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88 between temperatures of 707.3K and 681.3K. Table A.20: (3) The Value of the Amplitude, f q Evaluated when A coh = 0.846 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88 Between Temperatures of 707.3K and 681.3K. Temperature (K) Amplitude (A) f q τ(s) β 707.3 0.388 ± 0.004 0.677 ± 0.003 0.0368 ± 0.0015 0.470 ± 0.013 707.3 0.434 ± 0.004 0.716 ± 0.003 0.0339 ± 0.0016 0.447 ± 0.001 704.4 0.418 ± 0.003 0.703 ± 0.003 0.0364 ± 0.0012 0.477 ± 0.011 700.6 0.409 ± 0.004 0.695 ± 0.003 0.0596 ± 0.0036 0.483 ± 0.012 696.4 0.410 ± 0.004 0.696 ± 0.003 0.1222 ± 0.0065 0.498 ± 0.016 693.4 0.418 ± 0.002 0.703 ± 0.002 0.1708 ± 0.0078 0.486 ± 0.009 690.9 0.382 ± 0.002 0.672 ± 0.002 0.1627 ± 0.0064 0.465 ± 0.008 688.4 0.394 ± 0.002 0.682 ± 0.002 0.1911 ± 0.0063 0.497 ± 0.010 684.7 0.431 ± 0.003 0.713 ± 0.003 0.3954 ± 0.0184 0.503 ± 0.010 681.3 0.394 ± 0.002 0.682 ± 0.002 0.6151 ± 0.0222 0.536 ± 0.011
APPENDIX A. ANALYSIS 87 Figure A.49: T д is determined to be 642.2K from the graph of log 10 τ avд (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88. Figure A.50: The fragility index of 34.3 for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88 is determined from the slope on the graph of log 10 τ avд (s) vs. T д /T(K).
APPENDIX A. ANALYSIS 88 Figure A.51: The positive β slope of 0.0451 on the graph of β vs. log 10 τ avд (s) indicates a narrowing of the distribution of relaxation rates for [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88. Figure A.52: f q near T д is estimated at 0.688 from the graph of f q vs. [Al(PO 3 ) 3 ] 12 [NaP0 3 ] 88. log 10 τ avg (s) for
APPENDIX A. ANALYSIS 89 A.9 [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85 Figure A.53: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85 between temperatures of 782.0K and 751.6K. Table A.21: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.838 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85 Between Temperatures of 782.0K and 751.6K. Temperature (K) Amplitude (A) f q τ(s) β 782.0 0.395 ± 0.022 0.686 ± 0.019 0.00015 ± 0.00002 0.374 ± 0.031 777.5 0.345 ± 0.015 0.641 ± 0.014 0.00029 ± 0.00002 0.369 ± 0.025 773.2 0.433 ± 0.040 0.719 ± 0.033 0.00046 ± 0.00004 0.417 ± 0.035 769.3 0.405 ± 0.021 0.695 ± 0.018 0.00069 ± 0.00003 0.388 ± 0.019 765.4 0.416 ± 0.029 0.705 ± 0.025 0.00089 ± 0.00007 0.364 ± 0.025 761.9 0.308 ± 0.012 0.606 ± 0.012 0.0016 ± 0.0001 0.398 ± 0.025 758.5 0.388 ± 0.022 0.681 ± 0.019 0.0022 ± 0.0001 0.395 ± 0.019 758.4 0.386 ± 0.035 0.678 ± 0.031 0.0023 ± 0.0002 0.381 ± 0.034 755.1 0.330 ± 0.009 0.627 ± 0.009 0.0027 ± 0.0001 0.429 ± 0.021 751.6 0.391 ± 0.018 0.683 ± 0.016 0.0043 ± 0.0003 0.468 ± 0.025
APPENDIX A. ANALYSIS 90 Figure A.54: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85 between temperatures of 748.1K and 725.5K. Table A.22: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.838 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85 Between Temperatures of 748.1K and 725.5K. Temperature (K) Amplitude (A) f q τ(s) β 748.1 0.366 ± 0.008 0.661 ± 0.008 0.0040 ± 0.0002 0.447 ± 0.018 744.7 0.391 ± 0.009 0.683 ± 0.008 0.0070 ± 0.0005 0.497 ± 0.025 744.7 0.351 ± 0.001 0.647 ± 0.001 0.0077 ± 0.0004 0.425 ± 0.017 741.3 0.372 ± 0.005 0.667 ± 0.004 0.0091 ± 0.0004 0.471 ± 0.014 737.9 0.362 ± 0.006 0.657 ± 0.006 0.0137 ± 0.0007 0.482 ± 0.014 730.4 0.418 ± 0.005 0.706 ± 0.004 0.0207 ± 0.0011 0.472 ± 0.015 730.4 0.414 ± 0.003 0.703 ± 0.002 0.0324 ± 0.0009 0.457 ± 0.007 730.4 0.409 ± 0.005 0.699 ± 0.004 0.0371 ± 0.0017 0.433 ± 0.010 727.2 0.430 ± 0.003 0.716 ± 0.003 0.0338 ± 0.0010 0.443 ± 0.008 725.5 0.425 ± 0.004 0.712 ± 0.003 0.0449 ± 0.0018 0.476 ± 0.010
APPENDIX A. ANALYSIS 91 Figure A.55: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85 between temperatures of 723.0K and 696.8K. Table A.23: (3) The Value of the Amplitude, f q Evaluated when A coh = 0.838 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85 Between Temperatures of 723.0K and 696.8K. Temperature (K) Amplitude (A) f q τ(s) β 723.0 0.420 ± 0.003 0.708 ± 0.002 0.0666 ± 0.0019 0.445 ± 0.007 719.1 0.456 ± 0.003 0.738 ± 0.002 0.1120 ± 0.0033 0.441 ± 0.007 715.7 0.439 ± 0.018 0.724 ± 0.015 0.1515 ± 0.0035 0.455 ± 0.005 712.2 0.355 ± 0.002 0.651 ± 0.002 0.1724 ± 0.0054 0.464 ± 0.007 708.7 0.331 ± 0.002 0.628 ± 0.002 0.1833 ± 0.0055 0.454 ± 0.007 705.3 0.414 ± 0.002 0.703 ± 0.002 0.5562 ± 0.0108 0.487 ± 0.004 705.3 0.432 ± 0.001 0.718 ± 0.001 0.5684 ± 0.0086 0.474 ± 0.003 701.8 0.339 ± 0.001 0.636 ± 0.001 0.6260 ± 0.0154 0.468 ± 0.005 696.8 0.369 ± 0.002 0.664 ± 0.001 1.7250 ± 0.0214 0.495 ± 0.003
APPENDIX A. ANALYSIS 92 Figure A.56: T д is determined to be 663.7K from the graph of log 10 τ avд (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85. Figure A.57: The fragility index of 33.9 for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85 is determined from the slope on the graph of log 10 τ avд (s) vs. T д /T(K).
APPENDIX A. ANALYSIS 93 Figure A.58: The positive β slope of 0.0278 on the graph of β vs. log 10 τ avд (s) indicates a narrowing of the distribution of relaxation rates for [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85. Figure A.59: f q near T д is estimated at 0.694 from the graph of f q vs. [Al(PO 3 ) 3 ] 15 [NaP0 3 ] 85. log 10 τ avg (s) for
APPENDIX A. ANALYSIS 94 A.10 [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80 Figure A.60: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80 between temperatures of 824.5K and 769.3K. Table A.24: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.839 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80 Between Temperatures of 824.5K and 769.3K. Temperature (K) Amplitude (A) f q τ(s) β 824.5 0.474 ± 0.032 0.752 ± 0.025 0.00005 ± 0.00001 0.442 ± 0.034 808.3 0.456 ± 0.024 0.737 ± 0.019 0.00017 ± 0.00001 0.394 ± 0.019 803.2 0.436 ± 0.029 0.721 ± 0.024 0.00027 ± 0.00002 0.453 ± 0.033 798.8 0.445 ± 0.013 0.728 ± 0.011 0.00040 ± 0.00002 0.443 ± 0.021 795.0 0.464 ± 0.014 0.744 ± 0.011 0.00061 ± 0.00002 0.421 ± 0.017 791.0 0.462 ± 0.012 0.742 ± 0.010 0.00080 ± 0.00003 0.433 ± 0.016 786.5 0.451 ± 0.011 0.733 ± 0.009 0.00126 ± 0.00004 0.433 ± 0.015 782.0 0.436 ± 0.007 0.721 ± 0.006 0.00191 ± 0.00005 0.453 ± 0.012 777.7 0.460 ± 0.008 0.740 ± 0.007 0.00269 ± 0.00009 0.448 ± 0.013 773.5 0.470 ± 0.008 0.748 ± 0.006 0.00412 ± 0.00018 0.454 ± 0.015 769.3 0.444 ± 0.005 0.727 ± 0.004 0.00673 ± 0.00020 0.468 ± 0.010
APPENDIX A. ANALYSIS 95 Figure A.61: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80 between temperatures of 765.2K and 726.0K. Table A.25: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.839 ± 0.0005, τ, and β for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80 Between Temperatures of 765.2K and 726.0K. Temperature (K) Amplitude (A) f q τ(s) β 765.2 0.469 ± 0.008 0.748 ± 0.006 0.0078 ± 0.0005 0.434 ± 0.017 760.9 0.436 ± 0.003 0.721 ± 0.003 0.0135 ± 0.0003 0.467 ± 0.007 756.3 0.446 ± 0.003 0.729 ± 0.002 0.0254 ± 0.0006 0.479 ± 0.007 752.1 0.451 ± 0.003 0.733 ± 0.002 0.0396 ± 0.0012 0.467 ± 0.007 747.7 0.410 ± 0.002 0.699 ± 0.002 0.0407 ± 0.0013 0.478 ± 0.007 743.2 0.403 ± 0.001 0.693 ± 0.001 0.1490 ± 0.0038 0.489 ± 0.006 739.1 0.415 ± 0.002 0.703 ± 0.002 0.1070 ± 0.0039 0.482 ± 0.009 734.8 0.413 ± 0.001 0.702 ± 0.001 0.4140 ± 0.0091 0.483 ± 0.005 730.5 0.378 ± 0.001 0.671 ± 0.001 0.5020 ± 0.0117 0.481 ± 0.005 726.0 0.405 ± 0.003 0.695 ± 0.003 0.7610 ± 0.0557 0.509 ± 0.020
APPENDIX A. ANALYSIS 96 Figure A.62: T д is determined to be 688.7K from the graph of log 10 τ avд (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80. Figure A.63: The fragility index of 36.1 for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80 is determined from the slope on the graph of log 10 τ avд (s) vs. T д /T(K).
APPENDIX A. ANALYSIS 97 Figure A.64: The positive β slope of 0.0190 on the graph of β vs. log 10 τ avд (s) indicates a narrowing of the distribution of relaxation rates for [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80. Figure A.65: f q near T д is estimated at 0.620 from the graph of f q vs. [Al(PO 3 ) 3 ] 20 [NaP0 3 ] 80. log 10 τ avg (s) for
APPENDIX A. ANALYSIS 98 A.11 [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 Figure A.66: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 between temperatures of 884.1K and 854.3K. Table A.26: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.718 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 Between Temperatures of 884.1K and 854.3K. Temperature (K) Amplitude (A) f q τ(s) β 884.1 0.600 ± 0.046 0.914 ± 0.035 0.00009 ± 0.00002 (0.369 ± 0.051) 885.2 0.606 ± 0.060 0.918 ± 0.045 0.00009 ± 0.00001 (0.405 ± 0.032) 887.7 0.652 ± 0.129 0.953 ± 0.094 0.00008 ± 0.00001 (0.399 ± 0.026) 877.3 0.575 ± 0.021 0.895 ± 0.017 0.00015 ± 0.00001 0.473 ± 0.040 877.3 0.516 ± 0.048 0.847 ± 0.039 0.00014 ± 0.00001 0.410 ± 0.014 870.2 0.481 ± 0.009 0.818 ± 0.008 0.00031 ± 0.00002 0.488 ± 0.033 870.2 0.479 ± 0.024 0.817 ± 0.020 0.00029 ± 0.00001 0.488 ± 0.011 862.2 0.502 ± 0.009 0.836 ± 0.008 0.00051 ± 0.00001 0.435 ± 0.012 862.2 0.447 ± 0.009 0.789 ± 0.008 0.00047 ± 0.00002 0.464 ± 0.015 854.3 0.453 ± 0.015 0.794 ± 0.013 0.00078 ± 0.00003 0.509 ± 0.026
APPENDIX A. ANALYSIS 99 Figure A.67: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 between temperatures of 854.3K and 831.8K. Table A.27: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.718 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 Between Temperatures of 854.3K and 831.8K. Temperature (K) Amplitude (A) f q τ(s) β 854.3 0.431 ± 0.007 0.775 ± 0.007 0.00105 ± 0.00003 0.543 ± 0.016 854.3 0.469 ± 0.007 0.808 ± 0.006 0.00101 ± 0.00002 0.486 ± 0.012 846.4 0.500 ± 0.018 0.834 ± 0.015 0.00161 ± 0.00010 0.481 ± 0.028 846.4 0.488 ± 0.006 0.824 ± 0.005 0.00154 ± 0.00003 0.499 ± 0.010 846.4 0.490 ± 0.004 0.826 ± 0.003 0.00171 ± 0.00002 0.481 ± 0.006 838.6 0.491 ± 0.006 0.827 ± 0.005 0.0030 ± 0.0001 0.495 ± 0.012 838.6 0.484 ± 0.003 0.821 ± 0.002 0.0036 ± 0.0001 0.484 ± 0.005 838.6 0.456 ± 0.006 0.797 ± 0.005 0.0029 ± 0.0001 0.552 ± 0.017 831.8 0.463 ± 0.002 0.803 ± 0.002 0.0087 ± 0.0001 0.513 ± 0.005 831.8 0.479 ± 0.009 0.817 ± 0.007 0.0085 ± 0.0002 0.517 ± 0.006
APPENDIX A. ANALYSIS 100 Figure A.68: The autocorrelation Spectra obtained for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 between temperatures of 831.8K and 805.5K. Table A.28: (3) The Value of the Amplitude, f q Evaluated when A coh = 0.718 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 Between Temperatures of 831.8K and 805.5K. Temperature (K) Amplitude (A) f q τ(s) β 831.8 0.499 ± 0.006 0.833 ± 0.005 0.0117 ± 0.0005 0.492 ± 0.013 825.8 0.499 ± 0.005 0.833 ± 0.004 0.0118 ± 0.0004 0.494 ± 0.012 825.8 0.500 ± 0.004 0.834 ± 0.003 0.0131 ± 0.0004 0.494 ± 0.008 819.9 0.477 ± 0.002 0.815 ± 0.002 0.0388 ± 0.0008 0.476 ± 0.006 819.9 0.499 ± 0.002 0.833 ± 0.002 0.0325 ± 0.0006 0.481 ± 0.005 819.9 0.498 ± 0.002 0.832 ± 0.002 0.0497 ± 0.0010 0.463 ± 0.004 819.9 0.534 ± 0.003 0.862 ± 0.002 0.0251 ± 0.0006 0.492 ± 0.007 812.7 0.446 ± 0.001 0.788 ± 0.001 0.0626 ± 0.0007 0.494 ± 0.003 812.7 0.469 ± 0.002 0.808 ± 0.001 0.0416 ± 0.0006 0.508 ± 0.004 805.5 0.530 ± 0.002 0.859 ± 0.002 0.1126 ± 0.0020 0.506 ± 0.004
APPENDIX A. ANALYSIS 101 Figure A.69: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 between temperatures of 805.5K and 766.2K. Table A.29: (4) The Value of the Amplitude, f q Evaluated when A coh = 0.718 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 Between Temperatures of 805.5K and 766.2K. Temperature (K) Amplitude (A) f q τ(s) β 805.5 0.515 ± 0.001 0.847 ± 0.001 0.0869 ± 0.0011 0.501 ± 0.003 797.6 0.403 ± 0.001 0.749 ± 0.001 0.3054 ± 0.0036 0.506 ± 0.002 797.6 0.486 ± 0.001 0.822 ± 0.001 0.1714 ± 0.0020 0.515 ± 0.003 789.7 0.493 ± 0.002 0.828 ± 0.001 0.6298 ± 0.0073 0.538 ± 0.003 789.7 0.542 ± 0.001 0.868 ± 0.001 0.6093 ± 0.0066 0.505 ± 0.003 781.7 0.487 ± 0.001 0.823 ± 0.001 0.9053 ± 0.0305 0.511 ± 0.009 781.4 0.470 ± 0.001 0.809 ± 0.001 0.9590 ± 0.0115 0.548 ± 0.004 772.5 0.501 ± 0.001 0.835 ± 0.001 3.5000 ± 0.0498 (0.502 ± 0.004) 766.2 0.548 ± 0.001 0.873 ± 0.001 7.5610 ± 0.1990 (0.479 ± 0.006)
APPENDIX A. ANALYSIS 102 Figure A.70: T д is determined to be 745.9K from the graph of log 10 τ avд (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70. Figure A.71: The fragility index of 36.4 for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70 is determined from the slope on the graph of log 10 τ avд (s) vs. T д /T(K).
APPENDIX A. ANALYSIS 103 Figure A.72: The positive β slope of 0.0117 on the graph of β vs. log 10 τ avд (s) indicates a narrowing of the distribution of relaxation rates for [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70. Figure A.73: f q near T д is estimated at 0.812 from the graph of f q vs. [Al(PO 3 ) 3 ] 30 [NaP0 3 ] 70. log 10 τ avg (s) for
APPENDIX A. ANALYSIS 104 A.12 [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 Figure A.74: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 between temperatures of 941.1K and 914.9K. Table A.30: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.797 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 Between Temperatures of 941.1K and 914.9K. Temperature (K) Amplitude (A) f q τ(s) β 941.4 0.469 ± 0.049 0.767 ± 0.040 0.00021 ± 0.00001 0.493 ± 0.027 941.3 0.622 ± 0.054 0.884 ± 0.038 0.00019 ± 0.00002 0.548 ± 0.067 941.1 0.519 ± 0.028 0.807 ± 0.022 0.00014 ± 0.00001 0.434 ± 0.033 932.1 0.466 ± 0.018 0.765 ± 0.014 0.00013 ± 0.00004 (0.319 ± 0.039) 932.1 0.479 ± 0.035 0.776 ± 0.028 0.00025 ± 0.00002 0.515 ± 0.043 932.1 0.443 ± 0.009 0.745 ± 0.008 0.00028 ± 0.00001 0.530 ± 0.025 922.7 0.528 ± 0.014 0.814 ± 0.011 0.00046 ± 0.00001 0.500 ± 0.020 922.7 0.590 ± 0.073 0.861 ± 0.053 0.00039 ± 0.00005 0.431 ± 0.060 922.7 0.640 ± 0.040 0.896 ± 0.028 0.00039 ± 0.00003 0.414 ± 0.026 914.9 0.655 ± 0.090 0.907 ± 0.062 0.00051 ± 0.00008 0.398 ± 0.058 914.9 0.551 ± 0.015 0.831 ± 0.011 0.00082 ± 0.00004 0.455 ± 0.021
APPENDIX A. ANALYSIS 105 Figure A.75: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 between temperatures of 914.9K and 890.5K. Table A.31: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.797 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 Between Temperatures of 914.9K and 890.5K. Temperature (K) Amplitude (A) f q τ(s) β 914.9 0.497 ± 0.010 0.790 ± 0.008 0.00075 ± 0.00002 0.522 ± 0.017 902.7 0.533 ± 0.013 0.818 ± 0.010 0.0018 ± 0.0001 0.489 ± 0.021 903.9 0.554 ± 0.012 0.833 ± 0.009 0.0016 ± 0.0001 0.488 ± 0.019 904.2 0.533 ± 0.013 0.818 ± 0.010 0.0019 ± 0.0001 0.474 ± 0.019 904.0 0.513 ± 0.022 0.803 ± 0.018 0.0017 ± 0.0002 0.490 ± 0.038 898.3 0.547 ± 0.022 0.829 ± 0.017 0.0019 ± 0.0002 0.526 ± 0.040 898.3 0.496 ± 0.006 0.789 ± 0.005 0.0028 ± 0.0001 0.516 ± 0.012 898.3 0.503 ± 0.007 0.794 ± 0.005 0.0026 ± 0.0001 0.529 ± 0.016 890.6 0.519 ± 0.013 0.807 ± 0.010 0.0048 ± 0.0004 0.500 ± 0.027 890.6 0.480 ± 0.005 0.776 ± 0.004 0.0048 ± 0.0002 0.561 ± 0.014 890.5 0.471 ± 0.008 0.769 ± 0.007 0.0043 ± 0.0002 0.504 ± 0.018
APPENDIX A. ANALYSIS 106 Figure A.76: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 between temperatures of 882.5K and 859.0K. Table A.32: (3) The Value of the Amplitude, f q Evaluated when A coh = 0.797 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 Between Temperatures of 882.5K and 859.0K. Temperature (K) Amplitude (A) f q τ(s) β 882.5 0.560 ± 0.013 0.839 ± 0.010 0.0087 ± 0.0006 0.462 ± 0.021 882.5 0.489 ± 0.003 0.784 ± 0.003 0.0085 ± 0.0003 0.526 ± 0.011 882.5 0.496 ± 0.010 0.789 ± 0.008 0.0067 ± 0.0004 0.477 ± 0.019 874.6 0.462 ± 0.006 0.761 ± 0.005 0.0158 ± 0.0008 0.561 ± 0.016 874.6 0.506 ± 0.007 0.797 ± 0.006 0.0170 ± 0.0007 0.521 ± 0.032 874.6 0.438 ± 0.005 0.741 ± 0.004 0.0151 ± 0.0009 0.548 ± 0.013 866.8 0.402 ± 0.002 0.710 ± 0.002 0.0303 ± 0.0014 0.508 ± 0.013 866.8 0.412 ± 0.003 0.719 ± 0.002 0.0282 ± 0.0009 0.556 ± 0.012 866.8 0.520 ± 0.005 0.808 ± 0.004 0.0245 ± 0.0008 0.566 ± 0.012 859.0 0.490 ± 0.002 0.784 ± 0.002 0.0695 ± 0.0018 0.522 ± 0.009
APPENDIX A. ANALYSIS 107 Figure A.77: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 between temperatures of 859.0K and 813.5K. Table A.33: (4) The Value of the Amplitude, f q Evaluated when A coh = 0.797 ± 0.001, τ, and β for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 Between Temperatures of 859.0K and 813.5K. Temperature (K) Amplitude (A) f q τ(s) β 859.0 0.511 ± 0.003 0.800 ± 0.002 0.1053 ± 0.0032 0.510 ± 0.008 851.1 0.463 ± 0.002 0.763 ± 0.002 0.1672 ± 0.0044 0.478 ± 0.006 851.1 0.515 ± 0.002 0.804 ± 0.002 0.1827 ± 0.0062 0.549 ± 0.011 843.4 0.504 ± 0.002 0.795 ± 0.002 0.2840 ± 0.0071 0.518 ± 0.006 843.2 0.434 ± 0.001 0.738 ± 0.001 0.3191 ± 0.0143 0.533 ± 0.008 835.6 0.452 ± 0.002 0.753 ± 0.002 0.7853 ± 0.0318 0.562 ± 0.013 835.5 0.491 ± 0.001 0.785 ± 0.001 0.7475 ± 0.0331 0.568 ± 0.033 827.7 0.425 ± 0.001 0.730 ± 0.001 1.8870 ± 0.0500 0.553 ± 0.008 820.4 0.446 ± 0.001 0.748 ± 0.001 2.2170 ± 0.0419 0.545 ± 0.005 813.5 0.418 ± 0.001 0.725 ± 0.001 6.3350 ± 0.3376 0.542 ± 0.015
APPENDIX A. ANALYSIS 108 Figure A.78: T д is determined to be 786.5K from the graph of log 10 τ avд (s) vs. 1000/T(K) for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60. Figure A.79: The fragility index of 34.0 for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60 is determined from the slope on the graph of log 10 τ avд (s) vs. T д /T(K).
APPENDIX A. ANALYSIS 109 Figure A.80: The positive β slope of 0.0166 on the graph of β vs. log 10 τ avд (s) indicates a narrowing of the distribution of relaxation rates for [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60. Figure A.81: f q near T д is estimated at 0.720 from the graph of f q vs. [Al(PO 3 ) 3 ] 40 [NaP0 3 ] 60. log 10 τ avg (s) for
APPENDIX A. ANALYSIS 110 A.13 [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 Figure A.82: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 between temperatures of 997.5K and 945.9K. Table A.34: (1) The Value of the Amplitude, f q Evaluated when A coh = 0.762 ± 0.0003, τ, and β for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 Between Temperatures of 997.5K and 945.9K. Temperature (K) Amplitude (A) f q τ(s) β 997.5 0.803 ± 0.099 (1.026 ± 0.063) 0.00006 ± 0.00001 (0.357 ± 0.031) 997.2 0.789 ± 0.036 (1.018 ± 0.023) 0.00009 ± 0.00001 (0.421 ± 0.035) 988.8 0.741 ± 0.070 (0.986 ± 0.047) 0.00008 ± 0.00001 (0.375 ± 0.029) 972.6 0.626 ± 0.029 0.906 ± 0.021 0.00030 ± 0.00002 (0.411 ± 0.022) 972.5 0.490 ± 0.008 0.802 ± 0.007 0.00037 ± 0.00001 0.517 ± 0.013 959.4 0.512 ± 0.005 0.819 ± 0.004 0.00082 ± 0.00002 0.515 ± 0.010 959.4 0.486 ± 0.012 0.798 ± 0.010 0.00083 ± 0.00004 0.582 ± 0.029 959.4 0.485 ± 0.021 0.797 ± 0.017 0.00070 ± 0.00005 0.593 ± 0.049 945.9 0.448 ± 0.011 0.767 ± 0.010 0.00181 ± 0.00011 0.606 ± 0.032
APPENDIX A. ANALYSIS 111 Figure A.83: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 between temperatures of 945.9K and 917.2K. Table A.35: (2) The Value of the Amplitude, f q Evaluated when A coh = 0.762 ± 0.0003, τ, and β for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 Between Temperatures of 914.9K and 917.2K. Temperature (K) Amplitude (A) f q τ(s) β 945.9 0.507 ± 0.007 0.816 ± 0.006 0.0017 ± 0.0001 0.544 ± 0.014 933.2 0.490 ± 0.013 0.801 ± 0.011 0.0035 ± 0.0003 0.556 ± 0.034 933.2 0.505 ± 0.003 0.814 ± 0.003 0.0044 ± 0.0001 0.563 ± 0.008 933.2 0.554 ± 0.009 0.852 ± 0.007 0.0042 ± 0.0002 0.550 ± 0.019 925.2 0.508 ± 0.003 0.817 ± 0.002 0.0075 ± 0.0002 0.578 ± 0.012 925.2 0.501 ± 0.004 0.811 ± 0.003 0.0072 ± 0.0002 0.547 ± 0.012 925.2 0.511 ± 0.011 0.819 ± 0.008 0.0070 ± 0.0005 0.553 ± 0.022 925.2 0.541 ± 0.012 0.843 ± 0.009 0.0064 ± 0.0004 0.517 ± 0.025 917.2 0.474 ± 0.002 0.789 ± 0.002 0.0200 ± 0.0006 0.530 ± 0.008
APPENDIX A. ANALYSIS 112 Figure A.84: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 between temperatures of 917.2K and 890.8K. Table A.36: (3) The Value of the Amplitude, f q Evaluated when A coh = 0.762 ± 0.0003, τ, and β for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 Between Temperatures of 917.2K and 890.8K. Temperature (K) Amplitude (A) f q τ(s) β 917.2 0.450 ± 0.001 0.769 ± 0.001 0.0206 ± 0.0004 0.529 ± 0.005 917.2 0.478 ± 0.003 0.792 ± 0.002 0.0195 ± 0.0008 0.528 ± 0.011 908.0 0.497 ± 0.005 0.807 ± 0.004 0.0376 ± 0.0020 0.552 ± 0.017 908.0 0.485 ± 0.004 0.798 ± 0.003 0.0460 ± 0.0009 0.524 ± 0.005 908.0 0.459 ± 0.002 0.776 ± 0.001 0.0414 ± 0.0017 0.553 ± 0.013 899.0 0.485 ± 0.004 0.798 ± 0.003 0.1077 ± 0.0016 0.509 ± 0.004 899.0 0.557 ± 0.002 0.855 ± 0.001 0.0912 ± 0.0028 0.527 ± 0.008 890.8 0.501 ± 0.003 0.811 ± 0.002 0.2583 ± 0.0029 0.526 ± 0.003 890.8 0.483 ± 0.001 0.796 ± 0.001 0.2110 ± 0.0046 0.523 ± 0.006
APPENDIX A. ANALYSIS 113 Figure A.85: The autocorrelation spectra obtained for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 between temperatures of 883.6K and 862.9K. Table A.37: (4) The Value of the Amplitude, f q Evaluated when A coh = 0.762 ± 0.0003, τ, and β for [Al(PO 3 ) 3 ] 50 [NaP0 3 ] 50 Between Temperatures of 883.6K and 862.9K. Temperature (K) Amplitude (A) f q τ(s) β 883.6 0.475 ± 0.001 0.790 ± 0.001 0.3886 ± 0.0084 0.526 ± 0.006 883.6 0.449 ± 0.001 0.767 ± 0.001 0.2919 ± 0.0038 0.526 ± 0.003 876.3 0.523 ± 0.001 0.828 ± 0.001 0.7687 ± 0.0124 0.540 ± 0.005 876.2 0.487 ± 0.001 0.799 ± 0.001 0.5732 ± 0.0069 0.503 ± 0.029 868.7 0.515 ± 0.007 0.822 ± 0.006 1.1914 ± 0.0595 0.531 ± 0.009 868.7 0.573 ± 0.001 0.867 ± 0.001 1.3590 ± 0.0279 0.521 ± 0.006 862.9 0.580 ± 0.001 0.873 ± 0.001 3.1450 ± 0.1080 0.521 ± 0.009 862.9 0.441 ± 0.001 0.761 ± 0.001 2.5000 ± 0.0249 0.513 ± 0.003