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THE VARIATION O F .THE STRESS OPTICAL COEFFICIENT WITH GLASS COMPOSITION by Tod Renard Nissle A Thesis Submitted to the Faculty of the DEPARTMENT OF METALLURGICAL ENGINEERING In Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE WITH A MAJOR IN MATERIALS ENGINEERING. In the Graduate College THE UNIVERSITY OF ARIZONA 1 9 7 .3 STATEMENT BY AUTHOR This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judg ment the proposed use of the material is in the interests of scholar ship. In all other instances, however, permission must be obtained from the author. SIGNED: APPROVAL BY THESIS DIRECTOR This thesis has been approved on the date shown below: WALTER W. WALKER Associate Professor of Metallurgical Engineering Date ACKNOWLEDGMENTS Since they are responsible for the completion of this thesis, the author must give the inadequate reward of appearing on this page to: Mr. Warren Turner for his unerring explanations, apparatus innovations and otherwise; Mr. Roger Johnston, the master glassmaker; Dr. Walter W. Walker, Associate Professor of Metal lurgical Engineering at The University of Arizona, for his patience and aide in the dull task of finalizing the thesis; and Dr. Clarence L. Babcock, Professor of Optical Sciences at The University of Arizona, who, whenever the author began to stray, quietly appeared on the scene to point the way. This investigation was sponsored under Project THEMIS, administered by the United States Air Force Office of Scientific Research under Contract No. F44620-69-C-0024. TABLE OF CONTENTS Page LIST OF TABLES ABSTRACT ........ . . . . . . . . . . . ..................................... ............ i INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . LITERATURE REVIEW ....................................... 2.1 2.2 . 2 Bartholinus Discovers Birefringence ............ The Index of Refraction as a Measure of Birefringence ................... .......... 2.3 Huygens’ Misconception of Light Waves ... 2.4 The Nature of L i g h t .......... 2.5 Polarized Light and Birefringent Material . . . . . . 2.6 The Photoelasticity of Glass 2.7 The Theory of P h o t o e l a s t i c i t y .......... 2.8 The Variation of Birefringence with Composition .. . 2;9 Application of Data on the Variation of the Stress Optical Coefficient with Composition , 2.10 In Review . . . . . . . . :■ 3 . . . . . . x xiii CHAPTER 1 vii . LIST OF ILLUSTRATIONS PREPARATION OF GLASS SAMPLES USED TO MEASURE BIREFRINGENCE . . . . . . . . . . . . . . . . . . . . . 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 1 . . . . . . . 5 5 7 11 17 24 28 31 33 39 40 Weighing of Glass Constituents ... . . . . . . .. . . 41 Mixing of Glass Constituents . . 41 Melting the Batch .............. 41 Quenching, Washing and Drying the Glass . . . . . . . 47 Remelting and Stirring the Glass . . . . . . ........ 48 Pouring the Glass . .......... .. . . . ......... 48 Annealing the Glass Disk . . . . . . :. . . .. . . . . . 48 Residual Strain and Compositional Uniformity- . . . . . . 49 Overall Chemical Composition of the Glass . . . . . . 50 3.9.1 Procedure Followed in Measuring the Refractive Index of a G l a s s ............ 51 3.9.2 Evaluation of Measured Refractive Indices . . . . 5 1 3.10 Cutting and Finishing of the Glass S a m p l e .... 54 3.11 The Finished Piece of Glass . . . . . . . . . . .. . 54 iv V TABLE OF CONTENTS— Continued Page 4 OPTICAL SYSTEM AND MEASUREMENT P R O C E D U R E .......... 57 4.1 57 60 60 60 62 62 62 64 64 67 67 67 67 68 69 4.2 5 DATA 5.1 5.2 5.3 6 The Optical System .................... 4.1.1 Light Source . . . . ............ 4.1.2 Mdnochrometer or First Lens . . . . . . . . . 4.1.3 Interference Filter . . . . . ... 4.1.4 Pinhole ....................... . 4.1.5 Second Lens . . . . . . . . . . . . . . . . . 4.1.6 First Polarizer . 4.1.7 Glass Specimen Under Uniaxial Stress . . . . . 4.1.8 Soleil-Babinet Compensator ....... 4.1.9 Second Polarizer or Analyzer . . . . . . . . . 4.1.10 Third L e n s ................. 4.1.11 Photomultiplier and Oscilloscope . . . . . . . Experimental Procedure. . . . . . . . . . . . . . . . 4.2.1 Alignment . . . . . . 4.2.2 Orientation of the Polarizers . . . . . . . . 4.2.3 Measurement of the Retardation of the Ordinary Behind the Extraordinary Ray . . . ANALYSIS . . .......... 6.1 6.2 80 .... Calculation of Stress Optical Coefficients Statistical Analysis . . . ............ 5.2.1 Standard Deviation . . . . . . . . . . . . . . 5.2.2 Student's "t" Test . Linear Regression Analysis . . . . . . . . . . ... RESULTS AND DISCUSSION . . . . . . . . . .. . . . 70 ... . Initial Investigation . . . . . . . .. . . . . . .. Preparation of Glass Samples .......... ... . . . 6.2.1 Melting . . . . . 6.2.2 Annealing .......... 6.2.3 Grinding and P o l i s h i n g ............ 6.3 Measurement of the Stress Optical Coefficient . . . . . 6.4 Linear Regression Analysis . . . . . . . . .. . . . 6.4.1 Soda-Titania-Silica Glasses . . . . . . . . . 6.4.2 Soda-Alumina-Silica Glasses ... . . . . . 80 83 84 84 88 99 100 104 104 105 105 106 107 107 109 TABLE OF CONTENTS— Continued Page 7 CONCLUSIONS . . . . . . . . . . . . . I . . CHEMICAL ANALYSIS . APPENDIX B: COMPUTER PROGRAM USED FOR LINEAR REGRESSION ANALYSIS OF STRESS OPTIC D A T A ........ .. 117 EQUIPMENT EMPLOYED IN THE PREPARATION SAMPLES .......... 124 APPENDIX D : ........ 112 APPENDIX A: APPENDIX C: . . ... . . . . . . OFGLASS . . . . EQUIPMENT EMPLOYED IN MEASURING THE STRESS OPTICAL COEFFICIENT OF A GLASS SAMPLE . . . . . SELECTED BIBLIOGRAPHY 114 128 134 LIST OF ILLUSTRATIONS Figure Page 1. The double refraction of light by calcite . ......... 6 2. The longitudinal vibration of sound waves . . . . . . . . . 8 3. The transverse vibration of light waves ............. 9 4. The two perpendicular components, E and E , of the electric vector, E . . . . . . . . . . . ... 10 5. The electric and magnetic Vectors of a light wave . . . . . 6. Two dimensional representation of the electric field lines of an electron, e, .after Ernsberger (1970) . . . . ... . . . 12 14 7. Field of an oscillating electron after Ernsberger 8. Reduction of the electric vectors of a light ray, L, to one equivalent vector, E^ . .......... 16 Division of an incident beam of polarized light, L^, into two wavefronts, L and L , upon entering birefringent material . . . .e . . . ? . ................... 18 10. Linearly of plane polarized l i g h t ............. 20 11. Circularly polarized light 21 12. Elliptically polarized light . . . . . . . 22 13. Linearly polarized light ray, L^, becoming ellipticaily polarized by traversing a birefringent crystal . . . . . . 23 Linearly polarized light ray, L^, passing through isotropic glass . . . . . . . . . . ........ . . . . . . . . 25 Homogeneous glass acquiring the properties of a uniaxial anisotropic crystal by being subjected to uniaxial . ............ . compression, P . . . . . 26 Deformation of a glass block under uniaxial compression, P . . . . . . . . . . . . . . . . 27 9. 14. 15. 16. \ . . . .. . . vil ' (1970). 1 5 . . . . . . . . . . ■ . . . . . - ;■ viii LIST OF ILLUSTRATIONS— Continued Page Figure 17. 18. 19. 20. 21. 22. 23. Mueller concepts of "lattice effect" and "atomic effect" for glass under compressive force, P, after Ernsberger (1970) ... . . . ........ .. ... . . . . . . 30 Comparison of Adams and Williamson's and Pockels' data of birefringence vs. composition for flint glasses after Adams and Williamson (1919) . . . . . . . . 32 Variation of stress optical coefficient, C, with glass composition after Waxier and Napolitano (1957) . . . . .. 34 Linear variation of refractive index in phase fields of the NagO-CaO-SiOg glass system after Babcock (1968) . . . 36 Linear variation of Knoop hardness in phase fields of the NagO-CaO^SiOg glass system after.Georoff (1972) . . 37 Equations used to calculate data plotted in the N2S and N3C6S phase fields in Figures 20 and 21 after Babcock (1968) and Georoff (1972) . .. .. . . . . . . . . . .. 38 Preparation of a glass sample and important factors affecting glass homogeneity ................. . .. 42 Dimensions of the finished sample and criteria used • in its preparation . . . . . . , . ... . . . . . . . .. 55 25. Procedure for preparing finished glass samples . . . . .. 56 26. Optical bench arrangement . . .. 58 27. Continued optical bench arrangement .. 59 28. Polarization of light traveling through the optical, system . . . . . .. . . . . . .. . . . . . . . . , . . .. 61 29. Glan-Thompson polarizer 63 30. The Soliel-Babinet compensator 24. . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . 66 Affect of the positioning of the Babinet compensator on extinction points (y values) obtained . . . . . . . . . . 72 31.. Setting a Babihet compensator as a quarter wave plate 32. 65 . ix LIST OF ILLUSTRATIONS— Continued Figure 33. Page Definition of the "b” and "d" dimensions of the glass specimens . . . , . . . . . . . . . . . . . ... . 82 Stress optical coefficient, C, versus mole fraction of aluminum oxide for the five commercial laboratory melted glasses in the nepheline and corundum phase fields of the soda-alumina-silica ternary system ... . 101 Index of refraction, N, versus mole fraction of aluminum oxide for the five commercial laboratory melted glasses in the nepheline and corundum phase fields of the sodaalumina-silica ternary system after Babcock (1968) . . . 102 Knoop hardness number, K.H.N., versus mole fraction of aluminum oxide for the five commercial laboratory melted glasses in the nepheline and corundum phase fields of the soda-alumina-silica ternary system after Georoff (1972) . . . . . . . . . . . .. . . . . . . . ... 103 Lines of equal stress optical coefficient in primary phase fields D. (unknown) and E (Na^O.TiOg.SiOg) of the soda-titania-silica ternary system . . . . . . . . . 108 Linear variation of stress optical coefficient in the nepheline phase field of the Na 2 0 -Al 2 0 2 _Si 0 2 glass ........... . . . . . . . . . . . ' system . . . . . . 111 D-l. Apparatus for aligning and compressing the glass sample . 131 D-2. Detail of steel caps: 133 34. . 35. 36. 37. 38. (jT) in Figure D-l . . . . . . . . LIST OF. TABLES Table 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Page Chemical Composition and Critical Temperatures for Glasses Investigated in the D (Unknown) Phase Field of the Soda-Titania-Silica Ternary System ............ 43 Chemical Composition and Critical Temperatures for Glasses Investigated in the E (Na„0•TiO^•Si02) Phase Field of the Soda-Titania-Silica Ternary System . . . . . 44 Chemical Composition of the Five Commercial Laboratory . Melted Glasses in the Nepheline and Corundum Phase Fields of the Soda-Alumina-Silica Ternary System ......... 45 Chemical Composition and Critical Temperatures for Glasses Investigated in the Nepheline Phase Field of the Soda-Alumina-Silica Ternary System . . . .......... 46 Index of Refraction Values for Glasses in the D and E Phase Fields of the Soda-Titania-Silica Ternary System . . 52 Index of Refraction Values for Commercial Laboratory Melted Glasses in the Nepheline and Corundum Phase Fields of the Soda-Alumina-Silica Ternary System . . . . . 53 Calibration of the Optical System through Measurement of the Stress Optical Coefficient of National Bureau of Standards (NBS) Sample BF 588 ................... 74 Data Recorded in Measuring the Stress Optical Coefficient of Soda-Alumina-Silica Glass A3 . . . . . . ............. 75 Data Obtained in Measuring the Stress Optical Coefficients of Soda-Alumina-Silica Glasses .......................... 76 Data Obtained in Measuring the Stress Optical Coefficients of Glasses in the D Field of the Soda-Titania-Silica Ternary System . . . . . . . . . . . . . . . . . . . . . . 78 Data Obtained in Measuring the Stress Optical Coefficients of Glasses in the E Field of the Soda-Titania-Silica Ternary System ................. 79 x xi LIST OF TABLES— Continued Table 12. 13. 14. .15. 16. 17. 18. 19V . 20. A-l. B-l. C-l. Page Statistical Data on Glasses in the Nepheline and Corundum Phase Fields of the Soda-Alumina-Silica Ternary System ......... 85 Statistical Data on Glasses in the D Phase Field of the Soda-Titania-Silica Ternary System . . . . . . . . . . . . . 86 Statistical Data on Glasses in the E Phase Field of the .................... Soda-Titania-Silica Ternary System . 87 Average Stress Optical Coefficient Values for Commercial Laboratory Melted Glasses in the Nepheline and Corundum Primary Phase Fields of the Soda-Alumina-Silica Ternary .System .......................... . . . 91 Average Stress Optical Coefficient Values for Glasses in the Nepheline Primary Phase Field of the Soda-AluminaSilica Ternary System . . . . . . ........................ 93 Average Stress Optical Coefficient Values for Glasses in the Nepheline and Corundum Primary Phase Fields of the Soda-Alumina-Silica Ternary System ........... . 94 Average Stress Optical Coefficient Values for Glasses in the D Primary Phase Field of the Soda-Titania-Silica Ternary System . . . . . . , . . . . . . . . 95 Average Stress Optical Coefficient Values for Glasses in the E Primary Phase Field of the Soda-Titania-Silica Ternary System . ............ 96 Average Stress Optical Coefficient Values for Glasses in the D and E Primary Phase Fields of the Soda-TitaniaSilica Ternary System ....................... 97 Chemical Analysis of the Raw Glass Making Materials Melted in this Investigation ............................ 115 Computer Program Used for Linear Regression Analysis for Glasses in the Sodium-Titanium-Silicate Ternary System . . 118 Equipment Employed in Preparing Glass Samples 125 . . . . . . xii LIST OF T A B L E S - Continued Table D-l. D-2. Page Equipment Employed in Measuring the Stress Optical Coefficient of a Glass . . . . . . . . . . . ......... Itemized Description of Apparatus in Figure D-l . . . . . . . 129 132 ABSTRACT Refractive index, specific volume, Knoop hardness and fluidity have a linear relationship with composition in primary phase fields. This enables these properties to be computed from a glass' composition. The stress optical coefficient, another important property of glass, is used to evaluate a glass' residual strain. The objective of this study was to investigate whether the stress optical coefficient (1) also varied linearly j.n primary phase fields and (2) could then be determined from a glass' composition. Glass specimens were prepared from the D and E phase fields of the soda-titania-silica ternary system, and from the nepheline and corundum phase fields of the soda-alumina-silica ternary system. Each specimen, after being placed under uniaxial compression, then had its stress optical coefficient measured. The stress optic data obtained were subjected to linear regression analysis to determine whether they could accurately be described by the equation of a line. The stress optical coefficient was found to vary linearly with composition within primary crystallization phase, fields. CHAPTER 1 INTRODUCTION In 1669 Galileo pieced together a model of the telescope invented by the Dutch lensmaker Jacoma (Halsey 1963). Successive generations have since, with varied success, strived to better optical lens and mirrors. . A critical property of a lens or mirror is residual stress. When a glass is poured into a mold, the peripheral surface solidifies first, the center last. The outer crust is then in compression while the central portion, which tried to contract on cooling but was held by the solid crust, is under tension. If the glass is allowed to remain at and cool to room temperature after it has solidified, the internal strains will often cause it to become riddled with cracks and, at times, to explode into tiny fragments. This is why a glass is put into an annealing furnace just after it has solidified. The furnace slowly decreases the temperature and allows atoms to rearrange to a more stable configuration. A completely strain free glass is attained only by annealing for infinite time. At less than an infinite annealing, time, residual strain always remains in the glass. It should be noted that "residual strain" is actually . residual stress in a material engineering sense; however, in glass technology literature the two terms are used interchangeably. This thesis will follow the glass technology terminology. If residual tensile and compressive strain is high, a "flat" surface cut across the glass will buckle and be uneven like a rumpled shirt. It is impossible to polish this surface to the millionth of ah inch accuracy required on large telescope mirrors,— for once one "rumple" is polished away another appears. Since the strain in a glass must be small before an accurate flat surface m ay be obtained, it is necessary to be able to measure residual strain. So the question is, how may strain in glass be gauged? Strain in glass is detected by measuring its effect on polarized light or, in other words, by measuring its birefringence. Birefringence is the difference between the refractive indices of two plane-polarized light waves formed on traversing glass. A glass containing residual stress has a value of birefringence which is usually greater than zero. If the glass is isotropic (it has no residual strain) the birefringence is zero. However, isotropic glass placed under stress as in Figure 15 becomes birefringent: effect is called photoelasticity. this Brewster (1814) found that birefringence was proportional to the strain in a glass block. Birefringence thus serves as a direct measure of strain in glass. While birefringence represents strain in glass, the stress optical coefficient, a relation between the stress on and strain of glass, is, as an extension of birefringence, also used to analyze ' residual strain. The stress optical coefficient equals the 3 birefringence divided by the stress on the glass. Lillie and Hitland (1954) showed that the stress optical coefficient is needed to develop annealing schedules that will keep the amount of birefringence within prescribed limits. Van Zee and Noritake (1958) and McGraw and Babcock (1959) used the stress optical coefficient to determine the rate of stress release over the annealing range of soda-lime, potash-rbarium and borosilicate glasses. Babcock (1968, 1969) demonstrated that certain physical properties vary linearly in primary phase fields and so can be accurately predicted by computer if the composition of the glass is known. Data on stress optical coefficients is scattered throughout the literature, but it has not been determined whether birefringence varies linearly in primary phase fields. Since the stress optical coefficient reflects the degree of change of a glass' physical properties: when it is stressed, being: able to obtain the stress optical coefficient from a computer program, with the glass composition as input, would be handy. if this can be done. This study will determine CHAPTER 2 LITERATURE REVIEW . Since the first observation of the phenomenon of birefringence in 1669 there have been many studies examining its occurence and application. The purpose of this paper is to continue the investi gation by studying whether birefringence varies linearly with glass composition. As foundation to the study, it is desirable to review the evolution of birefringence. In recording this information an attempt was made to mesh historical developments with the theoretical background employed in analyzing.birefringence; hence, the following will briefly be considered: 2.1 Bartholinus Discovers Birefringence. 2.2 The Index of Refraction as a Measure of Birefringence 2.3 Huygens' Misconception of Light.Waves 2.4 The Nature of Light 2.5 Polarized Light and Birefringent Material 2.6 The Photoelasticity of Glass 2.7 The Theory of Photoelasticity 2.8 The Variation of Birefringence with Composition 2.9 Application of Data on the Variation of the Stress Optical Coefficient with Composition 2.10 In Review 2.1 Bartholinus Discovers Birefringence Erasmus Bartholinus, a Scandinavian physicist, mathematician and physician, was known for his observation and illustration of snow crystals. In 1669 he discovered what Huygens (Halsey 1963) called the "strange refraction" of Iceland spar, a type of calcite. Bartholinus saw that a light ray passing through calcite was divided into two rays. Even when the incident ray was perpendicular to the face of the calcite crystal, as in Figure 1, one of the rays deviated This effect was called birefringence. A transparent material which was isotropic, i.e., a material with equivalent optical properties throughout, would not have divided the ray. Calcite, an anisotropic material, i.e., a material with optical properties which varied with direction in. the crystal, split the incident ray. 2.2 The Index of Refraction as a Measure of Birefringence It is established that light traverses a vacuum with a uniform velocity of 3 x 10 10 ' cm/sec. When light travels into "transparent" matter as air or glass it is retarded to a degree characteristic of the matter. Air and other gases hinder light so little that we usually ignore the effect. On the other hand, the retardation of light by air is appreciable under some conditions, and is, for example, responsible for "mirage" images on deserts. Matter retards light, and we measure the retardation by the index of refraction, n, which is always greater than one and equals. L = light ray Le = extraordinary ray Lq = ordinary ray Iceland spar Figure 1. The double refraction of light by calcite. ' 7 Velocity of the light in a vacuum Velocity of the light in the material ^ In a birefringent crystal such as anisotropic calcite the velocity of light depends on the light's direction of travel and the ve locities, and hence the indices of refraction, of the ordinary and extraordinary waves differ. other. One of the waves is retarded behind the The difference between the index of refraction of the extra ordinary ray, n g , and the index of refraction of the ordinary ray, n Q , is also called birefringence. Birefringence 2.3 = n g - nQ (2) Huygens' Misconception of Light Waves Huygens (Halsey 1963) devised a geometrical construction which predicted the paths of the rays in a doubly refracting . (birefringent) medium like calcite. However, he mistakenly believed that the wave vibration of light was, as it was for sound waves, longitudinal (Fig. 2), and he could not explain the existence of the extraordinary and ordinary waves. One hundred years later Young (Halsey 1963) hypothesized that light waves were transverse (Fig. 3). wave it was a simple deduction that a From the transverse birefringent crystal separated light, represented by the vector E, into two vector components whose vibrations took place in mutually perpendicular planes (Fig. 4). S = sound wave: the arrow represents the direction of travel of the wave D = vector representing the amplitude of the sound wave f Tim® Figure 2. The longitudinal vibration of sound waves. Figure 3. L = light wave: the arrow represents the direction of propagation of the wave E = "electric" vector representing the amplitude of the light wave The transverse vibration of light waves. Figure 4. The two perpendicular components, and E^, of the electric vector, E. M o As shown.in Figure 4, both components of E--E^ and E — are perpendicular to the direction of travel of the wave. With tlie longitudinal vibration that Huygens assinned, it was not possible to obtain two perpendicular vector components without one of them being parallel to the direction of travel of the wave. Thus Huygens could not explain the splitting of the incident ray into the ordinary and extraordinary waves. 2.4 The Nature of Light Polarized light is necessarily employed in measuring the birefringence of a glass. To lead up to discussing the relation between polarized light and birefringent material in the next section this elementary background on the characteristics of light has been included. Light is electromagnetic radiation, and an electromagnetic wave like light has perpendicular electric and magnetic fields represented by the vectors E and M (Fig* 5). .Theoretical and experimental evidence indicates the electric vector, rather than the magnetic vector of light, is responsible for all the effects of polarization and birefringence. The magnetic vector is therefore not considered in this discussion. All light originates from the accelerated motion of electrons (Ernsberger 1970). An electron, being electrically charged, is always surrounded by an electric field. This field may be thought of as straight lines, radiating from the electron to infinity in all Figure 5. E = electric vector M = magnetic vector L = light wave The electric and magnetic vectors of a light wave.— The light wave, L, is traveling in a direction perpendicular to the page. 13 directions. A two dimensional representation of this is seen in Figure 6. In matter the electron oscillates, and, similar to ripples moving Outward from where a stone dropped into water, there is undulation of the electric field lines (Fig. 7). These waves or . "kinks" in the electric field lines are light (Ernsberger 1970). The kinks have a maximum amplitude in directions perpendicular to the oscillation axis> and this amplitude is represented by the vector Eq (Fig. 7). The direction of E q is (1) perpendicular to the direc tion of propagation of the light wave, and (2) parallel to the oscillation axis, YY. At any intermediate angle, theta, between the axis of oscillation and direction of propagation the amplitude of the vector will be E = E^sinO. E is called the electric vector. It should be noted that in Figure 7 there is no wave along the axis of oscillation, YY. This means the intensity of light emitted in the direction of oscillation is zero. An actual light source consists of many oscillating electrons which emit light waves with electric vectors oriented in various directions. At an instant in time the electric vectors of a beam of light, L^, coming toward you would appear as in Figure 8. Because light is a transverse wave, each electric vector is perpendicular to the w a v e ’s direction of travel (Fig. 3). has two components, E^ and E^ (Fig. 4). In Figure 8, each E vector All of these E^ and E^ components may be vectorally summed to arrive at single values for 14 Figure 6. Two dimensional representation of the electric field lines of an electron, e, after Ernsberger (1970). 15 Y Y Figure 7. Field of an oscillating electron after Ernsberger (1970).-E is the electric vector of a respective light w a v e . 16 II Figure 8. Reduction of the electric vectors of a light ray, L, to one equivalent vector, E^. 17 E x and E . y The sum values of E x and E y are the components of a r new vector, E^ (Fig. 8). If the E^ component of E^ were in some way removed, we would have polarized light. The E^ component--the electric vector of the polarized light--is represented by two components E^ and Eq . 2.5 Polarized Light and Birefringent Material In this study the objective of the experimental procedure was to measure the effect a birefringent material, stressed isotropic glass, had on polarized light. This section is intended to provide a sketch of polarized light and its interaction with birefringent substances. When a polarized light wave, L^, strikes a birefringent material it travels through the material as two wavefronts The electric vector, E ^ , (or E^) of the wave consists dicular components or vectors, E and E . r e o vectors of wavefronts E e and E o and Lq (Fig. 9). and Lq (Fig. 8). E e and E o oftwo perpen are theelectric The linear components are defined Ee = A esin(wt + ge) (3) Eo = Aosin(wt + S0) C4 ) Ae = maximum amplitude of Aq = maximum amplitude of Lq wave w = 27t (frequency of wave) t = time ge = phase of g = phase of L» wave o wave = 27rf wave (n, 2tt, etc.) o 18 Birefringent crystal / O z Figure 9. Division of an incident beam of polarized light, L^, into two wavefronts, and Lq , upon entering birefringent material. 19 If the Lg and Lq waves are (1) 0, 180 or 360° out of phase the light is linearly or plane polarized (Fig. 10), (2) 90 or 270° out of phase the light is circularly polarized (Fig. 11), (3) out of phase by any amount other than 0, 180, 360, 90 or 270° the light is elliptically polarized (Fig. 12). Basically, it is linearly polarized light that is utilized and analyzed in the experimental procedure of this study; hence, as an aside, mentioning everyday instances of this "type of light" might be beneficial. An interesting one is the navigation of the honeybee by linearly polarized light. Evidently, dependent on the position of the sun, areas of sky are linearly polarized in varying degrees throughout the day. The bee navigates through recognition of these areas as one may navigate by the stars, and the celebrated dance a scout bee does for his co-workers to point the way to the clover is based on this recognition. ,Another instance of this sort is the rainbow. As you look at a rainbow you view linearly polarized light. When, as occurs in the experimental procedure of this research, a beam of linearly polarized light--light with the and Lq waves in phase--passes through a birefringent crystal, the chance is small that the crystal will retard the or L ■ o wave exactly 180 or 360 behind the other so the light will leave the crystal linearly polar ized. Thus, linearly polarized light entering a birefringent crystal usually exits as elliptically polarized light (Fig. 13). 20 2 tt \ L e wave: E e = A sinfwt + g ) e &e E, = E + E 1 e o ge = go Ae ^ Ao 127T L wave: o E o = A sin(wt + g ) o °o L In the above example L and L are out of pfiase by 0, 180 and 360°. and L out e o of phase by °o 180 360° tttt End View Figure 10. Linearly or plane polarized light.--The top half of the page shows the relation between the perpendicular components, L and Lq , of the polarized light ray, . The bottom half e pictures the movement of the electric vector, E^, about the light ray, L^. The end view would be seen looking at straight on, as is the eye. 21 2tt 7 ^ L L e wave: E e = A sin(wt + g ) e v E 1 = E e + E o S0it/2 8e = A e ^ Ao L o w ave: E o = A sin(wt + g ) o o L In the above example Le and L0 are out of phase by 90°. and L out e o of phase by 90° 270 — IT End View A TT 2 Figure 11. 0 27T Circularly polarized light.--The top half of the page shows the relation between the perpendicular components, L and L , of the polar ized light ray, L^. The bottom half pictures ?he movement of the electric vector, E,, about the light ray, L^. The end . .. 11 1 straight on, as is the eye. view would be- seen looRing at 22 wave: Ee = \ s i n ( w t + ge) E 1 = Ee + Eo 8e ^ go A e ^ Ao 2 tt, wave -ir E o = A sin(wt + g ) o &o L In the above example Le and L are out of phase by 45°. 0 and L out e o of phase by Any amount except 0, 180, 360, 90 and 270°. TT -^TT -IT 2 tt End View A Figure 12. Elliptically polarized light.--The top half of the page shows the relation between the perpendicular components, L and L , of the polar ized light ray, L^. The bottom half pictures ?he movement of the electric vector, E^, about the light ray, . The end view would be seen looking at L1 straight on, as is the eye. 23 Birefringent crystal Elliptically polarized light Linearly polarized light Z Figure 13. Linearly polarized light ray, L^, becoming elliptically polarized by traversing a birefringent crystal. 24 2.6 The Photoelasticity of Glass Subsequent to Bartholinus' observation of birefringence in calcite, the next notable discovery towards understanding birefrin gence was made by Brewster in 1814. Ideally, "homogeneous glass" is (1) free of composition gradients and entrapped gas bubbles, and (2) has no residual stress. Homogeneous glass is optically isotropic and is not birefringent, and linearly polarized light entering a homogeneous piece of glass leaves the glass linearly polarized (Fig. 14). Sir David Brewster (1814) discovered that a piece of homo geneous glass placed in stress (Fig. 15) became anisotropic and birefringent. When an isotropic material like homogeneous glass becomes anisotropic and birefringent under stress it is termed photoelastic. The explanation of why this occurs is called the theory of photoelasticity. ■ i Figure 16 presents a physical representation of the photo elasticity of glass. The forces P compress the glass in the direc tion OY and extend it equal amounts in the directions OX and OZ, and the atoms are pressed closer together in the OY direction and spread further apart in the directions OX and OZ.. Plane polarized light entering the glass normal to the direction of thrust, OY, is separated into two perpendicular wavefronts, and L^. Because the glass is now birefringent, one of the wavefronts is retarded and the light . exits elliptically polarized (Morey 1954). 25 Y [omogeneous glass Linearly polarized light Linearly polarized light / O z Figure 14. Linearly polarized light ray, L^, passing through isotropic glass. 26 Homogeneous glass Elliptically polarized light Linearly polarized light Z Figure 15. Homogeneous glass acquiring the properties of a uniaxial anisotropic crystal by being subjected to uniaxial compression, P. 27 Y yHomogeneous glass Z! Linearly polarized light Elliptically polarized light jl r r z Figure 16. Deformation of a glass block under uniaxial compression, P. 28 In Figure 16 the cube has been distended in the OX and OZ directions. The light ray, L^, travels into a cube face which has b e e n 'compressed in one direction, OY, and distended in the other direction, OZ. If the linearly polarized light were to enter the top of the block, parallel to the OY axis and the force P, it would exit, as plane polarized light because the cube is distended equally in the OX and OZ directions; i.e., the glass is "optically isotropic" in the OX and OZ directions. A light ray entering the cube at any other angle would depart elliptically polarized. 2.7 The Theory of Photoelasticity Brewster's discovery led to the development of the theory of photoelasticity by Neumann (1841) and Maxwell (1852). As follows, the photoelastic theory is the basis of the experimental procedure and determination of the stress optical coefficient in this investigation. If r represents the retardation of one of the waves or L q behind the other as the polarized light travels through a birefringent crystal then, r = n = index of refraction of the 6 n n e - n (ne - n0)d o (5) extraordinary Le = index of refraction of the ordinary ray, L d = thickness of glass traversed by light O' = birefringence . ■ ■ o 29 2 (ne - no ) is proportional to the elastic stress, T (lbs/in ), (6) C"e " no) = CT C = C = a constant depending only upon the material and the wavelength, X, of the light the stress optical coefficient Combining Equations (5) and (6), r CTd (7) Equation (7) holds for compression and tension (Savur 1925) and is assumed valid up to the breaking stress of the glass (Filon 1936).. Mueller (1935), in explaining why glasses become birefringent under pressure, outlined two effects that occurred on the atomic level. They were (1) the lattice effect: a decrease in the spacing of the atoms in the direction of the stress, and (2) the atomic effect: a distortion of the atoms themselves (Fig. 17). birefringence (2) positive birefringence (8) negative birefringence (9) The lattice effect increases the atom density seen by the extraordinary ray and decreases the atom density seen by the ordinary ray; hence, it adds to positive birefringence. The atomic effect increases the atom density seen by the ordinary ray and contributes to negative 30 o o o o o o o o 0,0 fp Unstrained Strained o o Lattice effect. Distances between atoms decreased. Positive birefringence. o o o o o o Atomic effect. Spherical electron clouds are distorted into ellipsoids. Positive and negative birefringence. o o ° t p 0 Unstrained Figure 17. Strained Mueller concepts of "lattice effect" and "atomic effect" for glass under compressive force, P, after Ernsberger (1970). 31 birefringence. The atomic effect usually outweighs the lattice effect and glass in compression has negative birefringence. 2.8 The Variation of Birefringence with Composition Wertheim (1854) was the first to test the laws of photoelas ticity for a block of glass under simple tension or compression. He devised apparatus for producing uniform compression or tension in a glass block, something Brewster was unable to do, and demonstrated that the retardation, r, was a function of (Td), where T is the stress and d the glass thickness traversed by the light. Wertheim and following investigators measured birefringence with apparatus similar to or duplicating that utilized in this thesis (See Chapter 4: Optical System and Measurement Procedure). The first determination of the stress optical coefficient, C, was made by Kerr (1888). The effect of the chemical composition of glass on bire fringence was initially investigated by Pockels (1902). He concluded that an increase of lead oxide or boric oxide always lowered the stress optical coefficient. Filon (1907) and Adams and Williamson (1919) repeated Pockels' measurements and obtained equivalent results. Adams and Williamson compared their data with Pockels' as in Figure 18. Vedam (1950) measured the photoelastic properties of 18 optical glasses, but gave only approximate glass compositions. An 4 A Adams & Williamson Glasses O Pockels Glasses -4 * ' — 1— 30 40 1.1 50 W e ig h t % Figure 18. 60 Lead . ..I— .70 80 O x id e Comparison of Adams and Williamson's and Pockels' data of birefringence vs. composition for flint glasses after Adams and Williamson (1919). (X K) 33 inclusive study of photoelastic properties was made for 154 optical glasses by Schaefer and Nassenstein (1953). The glasses were . produced by Schott and Company and Schaefer and. Nassenstein did not include glass composition data in their paper. Finally, the stress optical coefficients of 27 optical glasses made at the National Bureau of Standards (NBS) were deter mined by Waxier and Napolitano (1957). data were listed. Definitive glass composition The NBS data was comparable to measurements made 50 years before by Filon (1907) and PoekeIs (1902)(Fig. 19). 2.9 Application of Data on the Variation of the Stress Optical Coefficient with Composition While it has been demonstrated that the stress optical coefficient varies with composition, birefringence has not been shown to vary linearly in phase fields, and the literature does not reveal a procedure which can accurately predict the stress optical coefficient of a given glass. Babcock (1968) segregated published data on refractive index into primary phase fields for various binary and ternary glass systems. He then subjected the data to least-squares analysis on a IBM 1620 computer. The computer presented an expression, an equation which allowed the refractive index to be calculated, of the form: Glass Property = A SiOg + B Na^O + C CaO + ... A, B, C ... = numerical constants, calculated by the computer for a given phase field, characteristic of the respective oxides (10) Brewsters in O NBS Glasses □ Pockels Glasses A Filon Glasses O 20 40 W e ig h t % Figure 19. 60 80 100 Lead O x id e Variation of stress optical coefficient, C, with glass composition after Waxier and Napolitano (1957). Cn 4^ 35 Si02 , N a 20, CaO amounts of these oxides, expressed in mole fractions The calculated indices of refraction showed the refractive index varying linearly within phase fields of the glass system. Figure 20 shows the linear variation of refractive index within phase fields of the Na20-Ca0-Si02 glass system. The standard error. (11) , ANp = difference between (1) published data submitted to the computer and (2) the data calculated by Babcock using Equation (10) n = number of data points was less than 0.0016 in all cases. Babcock (1968, 1969) applied this technique with.equal success to the glass properties of density, viscosity and thermal.expansion. Figure 21 pictures the linear variation of Knoop hardness within four phase fields of the Na20-Ca0-Si02 glass system. Equations employed to calculate data shown in the N2S and N2C3S phase fields of Figures 20 and 21 are given in Figure 22. The advantage of Babcock's formulation is as follows. Say a glass of the following specifications is required in the Na20-Ca0~Si02 system Refractive index = 1.623 Density 82.4 lb/ft Viscosity 100 poise 3 36 N3C6S N2C3S CaO (mole fraction) BCS N* N2S SiOa (mole fra c tio n ) Figure Linear variation of refractive index in phase fields of the Na^O-CaO-SiO^ glass system after Babcock (1968). 37 Beta CaO • S iO ) \ TRIDYMITE N a g O 3CaO 6 SiO (mole frac tio n ) u» CaO N a 20*2Ca0-3Si02 S i 0 2 (m ole fra c tio n ) Figure 21. Linear variation of Knoop hardness in phase fields of the Na20-Ca0-Si02 glass system after Georoff (1972). 38 A. B. In the N2S phase field: 1. Refractive index = 1.4739 Si02 + 1.7816. CaO + 1.5658'Na20 2. Knoop hardness In = 102.69 Si02 + 2633.0 CaO + 839.10 Na20 the N3C6S phase field: 1 1. Refractive index = 1.4715 Si02 + 1.7638 CaO + 1.5740 Na20 j 2. Knoop hardness 874.02 CaO - 49.460 Na20 j = 525.23 Si02 + . Figure 22. Equations used to calculate data plotted in the N2S and N3C6S phase fields in Figures 20 and 21 after Babcock (1968) and Georoff (1972). 39 Knoop hardness = 490 These glass properties along with the A, B, and C constants for each (1) primary phase field and (2) glass property, are submitted to the computer. The computer attempts to arrive at a glass composition of X mole fraction of Na20, Y mole fraction of CaO and Z mole fraction of SiOg, that will produce a glass with the above properties. In other words, the computer determines the composition per glass specifications. This procedure has advantage over melting, polishing and testing a series of glasses to obtain a sample to meet required properties. In addition, if a composition is known, the computer reverses the process, and, minus the usual laboratory investigation, predicts the physical properties of the glass. 2.10 In Review On the preceding pages we have seen 1. The discovery of birefringence 2. The discovery of the photoelasticity of glass 3. The development of the theory of photoelasticity 4. The variation of the stress optical coefficient composition 5. The prediction of physical properties of glass by Babcock1s formulation with In this paper the question at hand is whether the stress optical coefficient also has a linear relation within phase fields and can be added to the list of predictable glass properties. CHAPTER 3 PREPARATION OF GLASS SAMPLES USED TO MEASURE BIREFRINGENCE The majority of glasses h a v e .a stress optical coefficient of two to three brewsters, where a brewster equals a relative retardation (of the ordinary ray behind the extraordinary ray) of one angstrom unit per millimeter per bar. This small value, the retardation between the extraordinary and ordinary rays, is the quantity measured in this study. Because excessive residual strain or slight deviations in a glass' composition can mask a glass' true stress optical coefficient, it was necessary to produce glasses with negligible residual strain and with a high degree of compositional homogeneity. The factors affecting the compositional homogeneity Of the glasses used were (Babcock 1959), 1. Purity of the raw materials, and 2. their grain sizes and their distribution 3. Blending technique 4. Batch size 5. Extent of crushing 6. Extent of stirring 7. Time and temperature of melting 40 41 An additional factor, the annealing process, determined the amount of residual strain remaining in the glass at room temperature. Figure 23 demonstrates where the above factors came into play in the sample preparation procedure. A discussion of the preparation of glass samples and the role of the above factors therein follows. 3.1 Weighing of Glass Constituents A 1200-2000 gram batch was weighed on the Torsion balance (for further description of the equipment used in preparing glass samples, see Appendix C ) . each constituent used. See Tables 1-4 for the mole fraction of Reagent grade materials were employed (See Appendix B for chemical analysis of these materials). See the thesis by Georoff (1972) for photographs of the equipment employed in the preparation of glass specimens. 3.2 Mixing of Glass Constituents The v b atch was put into a plastic bag and mixed in a U.S. Stoneware porcelain ball mill. The reagents were in powder form of 50-100 mesh (300-150 microns) and further pulverization was not required. 3.3 Melting the Batch After blending the mixture was poured into a platinumrhodium crucible and placed in the Pereny furnace. This was done in increments so gases generated on heating would not cause the powder, to spill out of the crucible onto the furnace floor. Each . Step Sample Preparation Procedure 42 Key Factors in Glass Homogeneity Glass Constituents Weighed Purity of raw materials Glass Constituents Mixed Grain sizes and distribution Blending technique Batch Melted Batch size Glass Quenched, Washed and Dried Extent of crushing Extent of stirring. Time and temperature of melting Batch Remelted; Stirred 3.6 , Glass Poured into Mold 8. Glass Annealed Annealing process Composition Uniformity and Residual Strain of Glass Checked Overall Composition Checked 1 x 1 x 3 cm Samples Cut from Glass and Ground and Polished r- — r ----- 1 j 3 . 1 1 j Stress Optical Coefficient i I of Glass Measured i Figure 23. Preparation of a glass sample and important factors affecting glass homogeneity. 43 Table 1. Chemical Composition and Critical Temperatures for Glasses Investigated in the D (Unknown) Phase Field of the SodaTitania-Silica Ternary System D2 D3 04 0.650 0.600 0.600 0.670 0.150 0.150 0.200 0.120 0.200 0.250 0.200 0.210 1300 1340 1400 1350 Annealing temp (°C) corresponding temp to viscosity = 1 0 ^ poise 610 565 600 580 Liquidus temp (°C) 845 883 928 900 . Glass D1 . Composition (mole fraction) Si02 Ti02 Na20 Melting temp (0C) corresponding temp to viscosity = 10^ poise 44 Table 2. Chemical Composition and Critical Temperatures for Glasses Investigated in the E (NagO-TiO^'SiOg) Phase Field of the Soda-Titania-Silica Ternary System Glass El E2 E3 E4 0.500 0.450 0.400 0.550 Ti02 0.200 0.250 0.300 0.200 Na20 0.300 0.300 0.300 0.250 1280 1250 1300 1250 Annealing temp (°C) corresponding temp to viscosity = 10^^ poise 550 520 535 550 Liquidus temp (°C) 912 922 920 890 Composition (mole fraction) Si02 Melting temp (°C) corresponding temp tp viscosity = 10^ poise : 45 Table 3.. Chemical Composition of the Five Commercial Laboratory Melted Glasses in the Nepheline and Corundum Phase Fields of the Soda-Alumina-Silica Ternary System Glass A1 A2 A3 A4 . AS 0.620 0.620 0.620 0.620 0.620 0.100 0.150 0.190 0.205 0.220 0.280 0.230 0.190 0.175 0.160 Composition (mole fraction) Si02 A12°3 Nao0 46 Table 4. Chemical Composition and Critical Temperatures for Glasses Investigated in the Nepheline Phase Field of the SodaAlumina-Silica Ternary System Glass A6 A7 Composition (mole fraction) s i o 2 0.650 . 0.590 A 1 2 ° 3 0.125 0.125 N a20 0.225 0.285 1475 1500 600 625 1000 1050 Melting temp (°C) corresponding temp to viscosity -= 10^ poise Annealing temp .(°C) corresponding temp to viscosity = lO-*-^ poise Liquidus temp (°C) 47 glass was melted at a temperature equivalent to log 2 viscosity (Tables 1-4). Because glasses poured from 600 gram batches had composition gradients visible with the polariscope, larger batches of 1200 to 2000 grams were utilized. The larger batches minimized surface gradients resulting from (1) the interaction of molten glass with the sides of the platinum-rhodium crucible and (2) volatiliza tion of NagO into the furnace atmosphere. 3.4 Quenching, Washing and Drying the Glass Two or three hours following the final batch increment added to the Pereny furnace, the glass was poured between water cooled Fafnir aluminum rollers to form a thin sheet of glass which, upon passing through the rollers, fell into a pan of distilled water. This quenching technique caused the glass to fracture into tiny fragments and effectively mix itself. The glass was quenched in distilled water as tap water would have contaminated the glass with soluble salts which cause glass particles to adhere, hindering mixing. The distilled water was decanted and the glass mixture washed with alcohol and dried on a Corning laboratory hot plate. The highest degree of homogeneity from this m e 11ing-quenching technique was achieved after two or three repetitions. Furthbr melting and quenching did not improve the glass’ homogeneity. 3.5 Remelting and Stirring the Glass Subsequent to the final quench the batch was remelted, left in the furnace until free of bubbles, and, shortly before it was poured, stirred in an attempt to blend the surface layer that formed from.Na^O volatilization. In comparison with quenching, stirring was relatively ineffective in mixing the glass. The object was to leave the glass in the furnace for a time short enough to minimize surface volatilization and long enough to minimize seeds (bubbles). 3.6 Pouring the Glass Thirty to forty minutes after the final stirring, the glass was poured into a stainless steel mold which gave a cylindrical sample three inches in diameter and three-quarters of an inch thick. The mold was tapered from top to bottom to facilitate removing the glass disk. 3.7 Annealing the Glass Disk A minute or two after being poured, the glass would solidify such that it would not sag when removed from the m old. At this time it was transferred to the Lindberg annealing furnace on an asbestos board and cooled at 2.5°C per hour from its annealing temperature (Tables 1^4) to 345°C. At 345°C the furnace was shut down and allowed to cool to room temperature at about 20°C per hour. total annealing time per glass was. approximately 110 h o urs. The 49 When a glass is poured into a mold, the peripheral surface solidifies first, the center last. The outer crust is then in compression while the central portion, which tried to contract on cooling.but was held by the solid crust, is under tension. If the glass is allowed to remain at and cool to room temperature after it has solidified, the internal strains will often cause it to become riddled with cracks and, at times, to explode into tiny fragments. This is why a glass is put into an annealing furnace just after it has solidified. The furnace slowly decreases the temperature and allows atoms to rearrange to a more stable configuration. A completely strain free glass is attained only by annealing for infinite time. At less than an infinite annealing time, residual strain always remains in the glass. For the glasses examined in this paper, a cooling rate of 2.5°C per hour left residual strain only slightly visible when the glass specimens were examined with the polariscope. 5.8 Residual Strain and Compositional Uniformity Residual strain and composition gradients reflect a glass’ (in)homogeneity. As the measurement of the stress optical coefficient was extremely sensitive to nonuniform composition and strain, it was critical that the glasses be tested for these properties. A Bausch and Lomb vertical polariscope was employed in analyzing residual strain and compositional variation in the glass 50 disk. Those portions of the disk with the highest degree of homogeneity were cut out and polished as samples. As a final check prior to being measured for the stress optical coefficient, the sample was viewed in the polariscope. 5.9 Overall Chemical Composition of the Glass In 1893 F. Becke discovered that an optically transparent grain immersed in oil displayed a bright halo concentric with the grain's border when a microscope objective was focused slightly above the position of sharpest focus. This halo moved in a direction toward the medium (oil or crystal) having the higher refractive index as the microscope was adjusted upward from the correct focus. With these concepts in mind, the refractive index of the crystal could be determined by immersing it in oils of various refractive indices. When the refractive index of oil and crystal were equal the halo disappeared. The halo, after its German discoverer, came to be known as the Becke line and the procedure the Becke line method or method of central illumination. The Becke line method of measuring the index of refraction of a material is rapid and economical, and has long been employed by geologists and chemists in classifying transparent crystals. accuracy of ± 0.001 is easily achieved with this procedure. An Since the refractive index of glass varies directly with composition, measuring the refractive index by the Becke line method was chosen to verify the composition of glasses, melted for this paper. ■ 51 3.9.1 Procedure Followed in Measuring the Refractive Index of a Glass About one cubic centimeter of glass was ground to a fine powder which was sprinkled along a microscope slide. It is worthwhile to note that glass, because it is not. malleable like metals, does not have strain introduced during grinding. The slide was inserted on the stage of a Leitz polarizing microscope and a drop of oil placed on the powder. If the index of refraction of the oil was not equal to that of the glass powder, the Becke line appeared, and, when the microscope was adjusted upward, moved towards the medium (oil or glass) with the higher index of refraction. Through trial and error an oil with an index of refraction equal to the glass * was obtained (Bloss 1961). 3.9.2 Evaluation of Measured Refractive Indices Making use of a filter whose dominant wavelength was close to the sodium D line, the refractive indices of the (1) eight sodatitania-silica, (2) five commercially melted soda-alumina-silica and (3) two soda-alumina-silica glasses were measured. In Table 5 the data measured for the soda-titania-silica glasses is compared to values measured by Hamilton and Cleek (1958) and to values calcu lated by Babcock (1969). The indices obtained for the soda-alumina- silica glasses are listed with values calculated by Babcock (1968) in Table 6. All measured data was within ± 0 . 0 0 1 of the values of Ham ilton and Cleek (1958) and Babcock (1968; Babcock and Georoff 1973). 52 Table 5. Index of Refraction Values for Glasses in the D and E Phase Fields of the Soda-Titania-Silica Ternary System Fully Annealed Glass Samples Glass Measured Values NBS Values* Calculated Values** D1 1.602 1.6019 1.6025 D2 1.601-1.602 1.6016 1.6018 D3 1.642-1.643 1.6424 1.6421 : D4 1.657-1.658 El 1.636 1.6362 1.6360 E2 1.673-1.674 1.6739 1.6733 E3 1.710-1.711 1.7109 1.7105 E4 1.640-1.641 1.6402 1.6407 1.5786 ^National Bureau of Standards (NBS) values from Hamilton and Cleek (1958) **After Babcock (1968) 53 Table 6. Index of Refraction Values for Commercial Laboratory Melted Glasses in the Nepheline and Corundum Phase Fields of the Soda-Alumina-Silica Ternary System 'f Glass . Measured Values on Annealed Glasses Calculated Values* 1.5052 A1 1.506 A2 1.503-1.504 A3 1.501-1.502 1.5020-1.5021 A4 1.504-1.505 1.5043 A5 1.507 1.5066 A6 1.501-1.502 1.5013 .A7 1.507-1,508 1.5074 *After Babcock (1968) . 1.5035 3.10 At this Cutting and Finishing of the Glass Sample point the glass disk was annealed, relatively homogeneous and of known composition. The disk was submitted to the Optical Sciences Center Optics Shop and a sample approximately 1 x 1 x 3 centimeters prepared. Sample preparation, while the specimen is Figure 24 lists criteria used in the process used in polishing and grinding described in Figure 25. The specimens were made three times as long as they were wide to negate end effects: (Frocht 1948). The glass was polished on a standard glass polishing spindle and head. RPM. The spindle speed was 10 RPM, arid the head speed 25-30 The head weight of 10 pounds was reduced to two pounds, when polishing with milled barnesite. 3.11 The Finished Piece of Glass The succeeding chapter describes the measurement of the stress optical coefficient of the completed glass sample. 55 1 cm 1 cm Q Sample had two square end faces and four rectangular side faces Qzj End faces were parallel to +_ 0.01 inch fsj Opposite side faces were parallel to +_ 0.001 inch All six sides of sample were polished 3 cm Q5J Figure 24. The edges were slightly beveled to facilitate polishing Dimensions of the finished sample and criteria used in its preparation. Step Procedure © Sample of about 1.00 x 1.00 x 3,00 cm cut from glass, disk © Sample generated to about 0.995 x 0.995 x 3 cm by diamond cup wheel (142 micron grit), with water and water soluable oil as lubricant © Sample polished with ‘30 micron aluminum oxide slurry © Sample polished with 3 micron aluminum oxide slurry © Time(hr) 0.50 . r :_____ ......... ...... . Sample polished with one micron milled barnesite (cerium oxide 0.75 milled 200-300 hours) and distilled water I (Approximate Time ("stress optical coefficient of the sample determined L Figure 25. 3.00 — 1| j Procedure for preparing finished glass samples CHAPTER 4 OPTICAL SYSTEM AND MEASUREMENT PROCEDURE The optical arrangement described below was adapted from that employed by Waxier and Napolitano (1957). The quantity measured with this optical system was the retardation, in degrees, of the ordinary ray behind the extraordinary ray (Fig. 1). As discussed in the next chapter, the stress optical coefficient of a glass sample was calculated from this value. 4.1 The Optical System A schematic and photographs of the optical bench arrangement are pictured in Figures 26 and 27. The objective of the optical system was to (1) impinge a beam of linearly polarized light on the sample, (2) convert the elliptically polarized light leaving the sample (Fig. 15) to linearly polarized light and, (3) since the stressed glass and quarter wave plate in combination have the same effect as an optically active material that rotates the linearly polarized light, to measure the rotation of the plane polarized light by adjusting the analyzer to extinction. The difference, y degrees, between the position of the analyzer (1) before it was adjusted and (2) after it was rotated to extinction, represented the retardation of the ordinary ray with respect to the extraordinary ray. Figure 26. Optical bench arrangement. © © © °0 © 0 © © © © \ U ) , 11 ® © @ O p(> p Schematic of apparatus. H J = light source; = lens; ( ? ) compression; Q t J (?) = lens; (To) = = lens; ( T ) polarizer; ( ? ) = = = filter; ^ 3 ) photomultiplier; pin hole; glass sample under uniaxial Babinet compensator; ^ 8 ^ = = = = polarizer (analyzer); oscilloscope. Optical bench arrangement. above schematic. The numbers correspond to those in the in 00 Figure 27. Continued optical bench arrangement.--The numbers correspond to those in the schematic in Figure 26. in ID Figure 28 pictures the types of polarized light formed as the light makes its way through key parts of the optical system. The apparatus and the variation of the light in its transit of the optical system are further discussed below. 4.1.1 Light Source The original light source was a xenon arc lamp. Its beam proved unsteady, and a tungsten strip lamp was substituted in its place (for equipment details see Appendix D ) . 4.1.2 Monochrometer or First Lens The monochrometer in Figure 26 was removed because the light it produced was polarized. The focusing system of the monochrometer was then allowed for by the addition of a lens (number ( T ) in the schematic in Figure 26) with a radius of curvature equal to 70 milli meters. This lens, as part of the system which collimated the light before it passed through the sample, focused an image of the tungsten strip on the pinhole. The interference filter (number (jP) in the schematic in Figure 26), which was originally in the optical system to eliminate monochrometer light which did not have a wavelength "of 5461 angstroms, gave, by itself, a sufficiently monochromatic light beam. 4.1.3 Interference Filter The filter eliminated all wavelengths, X, contained in the white light produced by the tungsten strip bulb except for a yellow-green light with a wavelength equal to 5461 angstroms. © From lens. © p © ® Analyzer Adjusted 4 To lens. EXTINCTION: negligible amount of light Plane Polarized Light Figure 28 Elliptically Polarized Light 6 t 45° Plane Polar ized Light tf) ^ 45° Polarization of light traveling through the optical system.— f8j = glass sample under uniaxial compression; ^ 7 ^ = polarizer (analyzer). = = polarizer; Babinet compensator; ON 62 4.1.4 Pinhole The pinhole was employed in collimating the monochromatic light emerging from the filter. Light passing through the pinhole diverged at a constant rate, and a lens placed beyond the pinhole at a distance equal to the lens' focal length produced a collimated light beam. 4.1.5 The diameter of the pinhole was two millimeters. Second Lens This lens was used in conjunction with the pinhole to. obtain a collimated beam of light. millimeters. The lens' radius of curvature was 94 In selecting lenses for the optical system (1) the length of the optical bench and (2) the fact that shorter focal length lenses have more light gathering capability were taken into account. 4.1.6 First Polarizer A Glan-Thompson polarizer linearly polarized the monochromatic light exiting the collimating lens. Figure 29 details how light was linearly polarized by the polarizer. The optic axis of the glass was on the line along which the compressive forces, P, were applied, or, it was essentially vertical. The polarizer was set at 45° to the optic axis of the glass specimen. With the polarizer oriented in this fashion, linearly polarized light entered the stressed sample at 45° to the optic axis and split into two waves (the extraordinary and ordinary) of equal amplitude (Fig. 15). Wave vibration parallel to page Wave vibration parallel to page Calcite prisms " Lq = ordinary wave Le = extraordinary wave The ordinary light wave is reflected and absorbed by a blackened side of the polarizer while the extraordinary wave passes unaltered. Figure 29. Gian-Thompson polarizer 64 4.1.7 Glass Specimen Under Uniaxial Stress The method of applying a uniform compressive stress to the sample is described in Appendix D. As explained in Chapter 2, the linearly polarized light entering the stressed glass exited as elliptically polarized light. A piece of black cardboard with a six millimeter diameter hole was propped against this apparatus and adjusted back and forth so the light beam passed through the center of the specimen. 4.1.8 Soleil-Babinet Compensator Operation of this instrument is explained in Figure 30. Elliptically polarized light may be considered as composed of two linear components (1) parallel to the major and minor axes of the ellipse (Fig. 12), and (2) with a quarter wave phase difference between them. Hence, if a Babinet compensator is set as a quarter wave plate, and the horizontal axis of the compensator placed parallel to one of the axes of the elliptically polarized light, elliptically polarized light is, on traveling through the compensator, converted to linearly polarized light. . In this study the compensator was adjusted to act as a quarter wave plate for light with a wavelength of 5461 angstroms (Fig. 31). The compensator axis was then placed parallel to an axis of the elliptically polarized light by (1) putting the maximum load on the tare platform and (2) rotating the compensator alternately with the analyzer until the extinction position was identified. 65 Adjustable Quartz Wedges Quartz Plate Elliptically Polarized Lightx Linearly Polarized Lights OA = optic axis ordinary ray Lo = L = extraordinary ray Horizontal Axis of Compensator The extraordinary wave, because it is parallel to the optic axes of the two quartz wedges, is not affected as it passes through the wedges. However, it is retarded a fixed amount by the quartz plate. The ordinary wave, since it is not parallel to the the wedges, is retarded in traversing the wedges. retarded while passing through the quartz plate. optic axes It is not of To put the extraordinary and ordinary waves in phase, the quartz wedges were adjusted so the ordinary wave was retarded back into phase with the extraordinary wave. Figure 30. The Soliel-Babinet Compensator. 66 I- Light in: X = 5461 A u o <D 4-> cti to <D Ph E o u N •H r— ( © o Oh Light out = 0 (Extinctioi Linearly Polarized © 2 ) Circularly Polarized Mirror LinearlyPolarized ( T ) Circularly Polarized If, for the wavelength of light being used, a Babinet compensator is correctly set as a quarter wave plate, the situation pictured above occurs: (l) © © © Light passing through a polarizer is linearly polarized. Linearly polarized light is circularly polarized by the quarter wave plate (Babinet compensator). The electric vector of the circularly polarized light rotates in the opposite direction after the light is reflected by the mirror. The compensator transforms the circularly polarized light into linearly polarized light. The orientation of the linearly polarized light from the compensator is perpendicular to the orientation of the polarizer, and the light is extincted. To adjust the compensator so it acted as a quarter wave plate, the polarizer, compensator and mirror were arranged_as shown above and the compensator adjusted until the light ( at (V)) was extincted The Soliel-Babinet compensator had a scale of -20 to 0 to 20 (forty divisions), with a circular vernier readable to 0.01 of a division. It was set at 1.48 for the light (X = 5461 angstroms) employed in this investigation. Figure 31. Setting a Babinet compensator as a quarter wave plate. 67 4.1.9 Second Polarizer or Analyzer The polarizing axis of the analyzer was "crossed" (placed perpendicular to) with the axis of the first polarizer. The analyzer was mounted in a graduated circle rotator which had a 360° scale and a vernier readable to 0.05°. 4.1.10 Third Lens . ■ This lens focused light from the analyzer onto the. face of the photomultiplier tube. The radius of curvature of the lens was 94 millimeters. 4.1.11 Photomultiplier and Oscilloscope A voltage supply provided input for the photomultiplier while the output was exhibited as a straight line on the oscilloscope screen. Up or down movement of the straight line with the adjustment of the polarizer or the Babinet compensator defined points of minimum light intensity. Section 4.2 describes how these minimum light , intensities established the optic axis of the sample and determined the retardation of the ordinary ray behind the extraordinary ray. 4.2 Experimental Procedure As was the glass block in Figure 15, the glass specimens in this study were placed under a uniform compressive stress and acted as birefringent crystals, thus causing the ordinary wave to lag behind the extraordinary wave. This retardation, in degrees, was the quan tity measured by the procedure discussed below. 68 4.2.1 Alignment The interference filter used (number (T) in Figure 26) had a mirror surface on one side, and, as a base point from which to begin alignment, (1) the filter holder, pin and optical bench slide were removed from the optical bench, and (2) the mirror face put parallel to the front edge of the slide base with a bubble level. In other words, the mirror face was oriented perpendicular to the lengthwise horizontal axis of the optical bench. ' The mirror, pin and slide combination was then remounted on the optical bench and adjusted in tandem with the light housing (which did not mount on the optical bench) so: 1. When the filter mirror was at the end of the optical bench farthest from the housing, the beam of light emitted by the housing was centered on the mirror. 2. When the filter mirror was at the end of the optical bench closest to the lamp housing, the mirror's reflection of the tungsten strip image was centered on the iris of the housing. The circular opening of the iris was adjust able from two to forty millimeters. With the filter mirror one inch from the housing and the iris stopped down to two millimeters, an image of the tungsten strip about two millimeters wide and ten millimeters in length was visible on the black iris face. 3. If both (a) the reflected filament image was centered on the iris, and (b) the light beam was centered on the mirror when 69 the interference filter was at the far end of the bench, the light source was assumed on line with the optical bench. Other parts of the optical train were then added to the bench and adjusted to be perpendicular to the light path by noting that the reflected beam of light coincided with the incident beam. During the preliminary attempt at aligning the optical system, the filament of the tungsten light was not directly centered on the iris, but was slightly off center towards the upper part of the iris. This caused the light to "sink" as it progressed down the optical bench. So the light beam would travel parallel to the optical bench,.the light housing was disassembled and the filament centered on the iris. 4.2.2 Orientation of the Polarizers In order to position the polarizers at 45° relative to the optic axis, the glass sample was loaded to the maximum weight of 35.5 pounds, and both the polarizer and analyzer rotated until, as noted on the oscilloscope, a point of minimum intensity was reached. For each sample four such positions, each about ninety degrees from - the other, could be found. Only two of these, within 90 of each other, were required to position the polarizers. The two points of minimum intensity represented the positions, of the vertical and horizontal axes of the extraordinary and ordinary rays respectively. After the points were identified, the specimen was removed and the analyzer set at a position halfway between the two 70 points. Finally, the first polarizer was rotated to an extinction position, and, as was originally desired, the polarizers were crossed and at 45° relative to the optic axis. During the above procedure all readings were taken from the graduated circle rotator in which the analyzer was mounted. The extinction positions noted on the graduated circle scale were repro ducible to within ± 0.2 degrees. From Goranson and Adams' (1933) discussion on possible errors introduced by missettings, this appeared to be acceptable. The Babinet compensator was removed while the optic axis was located and the polarizers crossed with each other. After the compensator and glass specimen were reinserted, the apparatus was ready to measure the stress optical coefficient of the glass sample. 4.2.3 Measurement of the Retardation of the Ordinary Behind the Extraordinary Ray When the glass specimen was replaced, the maximum weight of 35.5 pounds was put on the loading platform, and the analyzer and Babinet compensator (set as a quarter wave plate)alternately rotated until the position of minimum intensity was determined, or, until the horizontal line on the oscilloscope screen was at its lowest point. This procedure was repeated six times. The six measurements were averaged to obtain a value, y degrees, for the extinction point. An increment of 5.5 pounds was then removed and the analyzer rotated to the minimum light intensity. made at this position. Six measurements were also Extinction positions were defined for four 71 more weight reductions of 5.5 pounds, giving a total of six extinction positions of y degrees. Table 8 shows how the differences. Ay, between these six points were averaged to give Ay, the retardation in degrees of the ordinary behind the extraordinary ray. Measurements were performed in a dark room at a temperature of about 20oC, and, after removing a 5.5 pound increment, five minutes were allowed to lapse before, the new extinction position was determined. During preliminary measurements, 5.5 pound increments were added to the platform and the extinction position noted. However, at lower weights residual strain and small composition differences masked the position of the axes of the elliptically polarized light emerging from the sample, and with each addition of 5.5 pounds the compensator had to be readjusted with the analyzer to get a true minimum, intensity. Data (y values) taken in this manner plotted more as a curved line (Fig. 32). This effect became negligible at higher weights; hence, the compensator was rotated into position with the maximum load applied. Once the Babinet compensator was positioned at the maximum load, re sults were independent of whether measurements were made by (1) be ginning with minimum weight on the platform and adding 5.5 pound increments or by (2) starting at the maximum load and removing 5.5 pound increments. 1. In summary: The Babinet compensator was set as a quarter wave plate as shown in Figure 31. Since the same wavelength of light was used throughout the study, the compensator remained at this setting for the entire investigation. 72 290 288 Babinet compensator adjusted at each extinction point---- 286 k 284 282 280 278 Babinet compensator positioned at the maximum load and kept in this position for the remainder of the measurements 276 I 13.5 24.5 19.0 W eight, Figure 32. 1 i in 29.0 1 34.5 pounds Affect of the positioning of the Babinet compensator on extinction points (y values) obtained.--The above measurements were made on glass A2. 2. Whenever a new sample was placed in the loading apparatus, (a) the optic axis had to be determined, (b) the polarizers crossed, and (c) the compensator repositioned at maximum load. 3. . For an individual sample: once the compensator was rotated into place at maximum load, it stayed at that position for the remainder of the measurements. If the glass specimens had had a residual tensile stress, this would have had to be overcome by an equal compressive force before the birefringence measured would have been due only to compressive stress. In accordance with the possibility of residual tensile strain, four 2 pound lead weights were laid across the loading platform before the five 5.5 pound weights were added. The credibility of the optical system was tested using a National Bureau of Standards glass with a stress optical coefficient measured by Waxier and Napolitano (1957). The. value obtained was within 1.51% of the stress optical coefficient arrived at by Waxier (Table 7). As an example. Table 8 illustrates all data taken during the measurement of a glass sample. Tables 9-11 list the stress optical data for the glasses investigated. The succeeding chapter outlines how, from the Ay values listed in these tables, stress optical coefficients were calculated and analyzed. 74 Table 7. Calibration of the Optical System through Measurement of the Stress Optical Coefficient of National Bureau of Standards (NBS) Sample BF 588 Data Relating to Measurement of the Stress Optical Coef ficient of NBS Sample BF 588 Measured stress optical coefficients, C, in brewsters Values Obtained from Optical System Employed in this Investigation ' 2.3134, 2.3120, 2.3206, 2.3119, 2.3149 NBS Values* --- Average stress optical coefficient, C, in brewsters 2.3145 2.35 Standard deviation among measured stress optical coefficients, C 0.0034 0.026 Wavelength of light, in angstroms, employed in measuring stress optical coefficients, C 5461 5893 Compressive stress in crements, in lbs/cm^, added to glass specimen during measurement of stress optical coef ficients, C 5.5 5.5 *After Waxier and Napolitano (1957) Table 8. Data Recorded in Measuring the Stress Optical Coefficient of Soda^Alumina-Silica Glass A3 Weight on Load ing Platform, in Pounds* 35.5 Retardation Values, y, in Degrees 287.150,287.050 287.150,287.000 287.125,287.095 Average Retar dation Values, y, in Degrees Ay Values, in Degrees Average Ay Value, Ay, in Degrees Calculated Stress Optical Coefficient, C, in Brewsters. = 287.095 2.524 30.0 284.625,284.650 284.525.284.525 284.575.284.525 284.571 2.488 24.5 282.125,282.075 282.150,282.025 282.075,282.050 . 282.083 2.495 19.0 279.575.279.600 279.575.279.600 279.575.279.600 2.5114 2.9535 279.588 2.546 13.5 277.050.277.025 277.075.277.025 277.075,277.000 277.042 2.504 ;8,o 274.600,274.475 274.450,274.625 274.625,274.450 274.538 - *The positions of the vertical and horizontal "optic" axes were 316.950 and 226.925° respectively. The Babinet compensator was set midway between these values at 271.940°. Table 9. Glass A1 A2 A3 A4 Data Obtained in Measuring the Stress Optical Coefficients,-of Soda-Alumina-Silica Glasses Ay, in Degrees Calculated Stress Optical Coefficient, C, in Brewsters Average Stress Optical Coefficient, C, in Brewsters Standard Deviation, s 2.1683 2.1375 2.1746 2.1477 2.1770 2.6290 2.6065 2.6367 2.6041 2.6396 2.6232 0.0150 2.2875 2.2780 2.2684 2.2838 2.3150 2.8050 2.7950 2.7811 2.8000 2.7910 2.7944 0.0082 2.5022 2.5324 2.5066 2.4901 2.5114 2.9427 2.9782 2.9479 2.9238 2.9535 2.9492 0.0176 2.4413 2.4463 2.4540 2.4320 2.4270 2.9792 2.9623 2.9948 2.9680 2.9618 2.9732 0.0125 Table 9--Continued Glass Ay/ in Degrees AS 2.6805 2.6708 2.6958 2.6842 2.6697 A6 A7 Calculated Stress Optical Coefficient, C, in Brewsters Average Stress Optical Coefficient, C, in Brewsters Standard Deviation, s 2.9939 2.9831 ' 3.0110 2.9980 2.9820 2.9936 0.0106 2.4275 2.3538 .2.3808 2.3283 2.4067 2.8510 2.7926 2.7962 2.7623 2.8266 2.8057. 0.0340 2.2988 2.4088 2.3488 2.4317 2.3275 2.6218 2.6671 2.6789 2.6328 2.6545 2.6510 -■ 0.0236 <! Table 10. Data Obtained in Measuring the Stress Optical Coefficients of Glasses in the D Field of the Soda-Titania-Silica Ternary System Calculated Stress Optical Coefficient, C, in Brewsters Average Stress Optical Coefficient, C , in Brewsters Standard Deviation, s .Glass Ay, in Degrees D1 2.4318 2.4605 2.4675 2.4390 2.4813 2.8292 2.8318 2.8398 2.8376 2.8557 2.8388 0.0104 2.2003 2.2200 2.2000 2.2129 2.2336 2.6500 2.6597 2.6495 2.6651 2.6759 2.6600 0.0111 2.4091 2.4318 2.3859 2.4493 2.4275 2.9316 2.9288 2.9034 2.9497 2.9541 2.9335 0.0247 2.3175 2.2850 2.3100 2.3005 2.3175 2.7371 2.7325 2.7283 2.7511 2.7371 2.7372 0.0086 D2 D3 D4 '-j 00 Table 11. Data Obtained in Measuring the Stress Optical Coefficients of Glasses in the B Field of the Soda-Titania-Silica Ternary System Calculated Stress Optical Coefficient, C, in Brewsters Glass Ay, in Degrees El 2.2820 2.2775 2.3188 2.2838 2.3110 2.6572 2.6842 2.7000 2.6915 2.6909 2.5008 2.4763 2.4788 2.4688 2.4838 2.9167 2.8909 2.8793 2.9132 2.5820 2.5963 2.5663 2.5713 2.5563 2.2863 2.2913 2.2700 2.3750 2.3700 E2 E3 E4 Average Stress Optical Coefficient, C, in Brewsters Standard Deviation, s 2.6848 0.0164 2.8993 0.0156 2.9903 3.0068 2.9720 2.9988 2.9813 2.9898 0.0138 2.7448 2.7508 2.7253 2.7363 2.7306 2.7376 0,0107 2.8966 CHAPTER 5 DATA ANALYSIS In the initial stages of this investigation, a fully annealed, relatively homogeneous glass sample was prepared and placed under a uniform compressive stress. Linearly polarized light was then im pinged on the sample, and the birefringence of the sample ascertained, by measuring the retardation of the ordinary behind the extraordinary ray. This retardation. Ay degrees, was tabulated for each sample in the preceding chapter. The calculation and analysis of stress optical coefficients from Ay values is examined below. 5.1 Calculation of Stress Optical Coefficients From the theory of photoelasticity, r = CTd (7) r = retardation of the ordinary behind the extraordinary fay in angstroms C = stress optical coefficient in brewsters T = stress in pounds per square inch d = thickness of, glass traversed by light in inches = (Aw)/bd Letting, T (12) 80 81 Aw = weight increment in pounds d = thickness of glass traversed by light in inches b = width of glass perpendicular path in inches (Fig. 33) to light And substituting Equation (12) into Equation (7), Ar = [1.752(Aw)Cd]/[bd] (13) Ar = change in retardation in angstroms with addition of Aw pounds 1.752 = conversion factor for the units employed The retardation may also be expressed in terms of the angle through which the plane of polarization is rotated in passing through the stressed glass and compensator: Ar = Equating Equations [2(Ay)A]/360.0 (14) 2 (Ay) = twice the rotation of the analyzer in degrees of arc A = wavelength of the light in angstroms (13) and (14) and solving for the stress optical coefficient, C, C = . [2b(A^)A]/[1.752(Aw)360.0] Since, in this study A = 5461 angstroms and Aw (15) = 5.507 pounds. 82 Glass Specimen Elliptically Polarized Lights Linearly Polarized Light^ J----- b equals about one centimeter d equals about one centimeter Figure 33. Definition of the "bM and "d" dimensions of the glass specimens. 83 ■ C = 3.1445 (Ay) b The stress optical coefficient for each specimen was calculated from Equation (16). This calculation procedure, which duplicates that em ployed by Waxier and Napolitano (1957), was used for the below reason. Equation (16) may be written. Ay = Q(Aw) Q = (17) the slope of a straight line I Increments of rotation,5 Ay degrees (2.430°, 2.426°, etc.) were considered in Equation (16) in place of accumulated total rotations (2.430°, 4.856°, etc.) because for a glass of given composition, the measurements performed at a specific load were all made on the same sample. As a result, errors of an accumulated rotation would have been cumulative. Hence, as shown in Table 8, the correct least squares estimate of slope Q, Ay degrees, was determined by averaging the five Ay values. Averaging the eight values recorded for each load to obtain a retardation, y degrees, also reduced errors in optical measurements. Waxier and Napolitano (1957) noted that, statistical analysis showed ' ■ ' •. . this reduced error to be negligible compared to errors resulting from the method of applying a uniform compressive stress to the sample. 5.2 Statistical Analysis , To give an indication of the data's precision, (1) the ■ standard deviation of the average stress optical coefficient, C, 84 was evaluated, and (2) the "t" test was employed to calculate a 95 percent confidence interval for the mean stress optical coefficient, C, of each glass. 5.2.1 Standard Deviation The mean stress optical coefficient, C, of each glass specimen was determined: C = (XCy/n = (C1 + C 2 + C3 + ...)/n C. = a measured value of the stress optical coefficient n = number of C. values summed i (18) Then the standard deviation, s, was: s = [ZCCh - C)2/(n - I)]0 '5 (19) Tables 12-14 list the calculated standard deviations. 5.2.2 Student's "t" Test If the mean stress optical coefficient, G, is assumed independent of the standard deviation, s, then the "t"statistic distributed as Student's "t" with (n - 1) degrees of freedom (Johnson and Jackson 1959). "t" is The "t" statistic is: = [(n°-5)(C - u)]/s C = sample mean u = population mean n = number of samples (20) 85 Table 12. Glass Statistical Data on Glasses in the Nepheline and Corundum Phase Fields of the Soda-Alumina-Silica Ternary System Average Stress Optical Coefficient, (SOC) G, in Brewsters Standard Deviation, s, of SOC 95% Confidence Interval of SOC Determined Using Student's "t" Test A1 2.6232 0.0150 2.6097 to 2.6367 A2 2.7944 0.0082 2.7870 to 2.8018 A3 2.9492 0.0176 2.9333 to 2.9651 A4 2.9732 0.0125 2.9619 to 2.9845 AS 2.9936 0.0106 2.9840 to 3.0032 A6 2.8057 0.0340 2.7750 to 2.8364 A7 2.6510 0.0236 2.6274 to 2.6746 -n 86 Table 13. Statistical Data on Glasses in the D Phase Field of the Soda-Titania-Silica Ternary System Average Stress Optical Coefficient, (SOC) C, in Brewsters Standard Deviation, s, of SOC D1 2.8388 0.0104 2.8294 to 2.8482 D2 2.6600 0.0111 ' 2.6500 to 2.6700 D3 2.9335 0.0247 2.9112 to 2.9558 D4 2.7372 0.0086 2.7295 to 2.7449 Glass 95% Confidence Interval of SOC Determined Using Student's "t" Test 87 Table 14. Glass Statistical Data on Glasses in the E Phase Field of the Soda-Titania-Silica Ternary System Average Stress Optical Coefficient, (SOC) C, in Brewsters Standard Deviation, s, of SOC 95% Confidence Interval of SOC Determined Using Student's "t" Test El 2.6848 0.0164 2.6700 to 2.6996 E2 2.8993 0.0156 2.8852 to 2.9134 E3 2.9898 0.0138 2.9774 to 3.0022 E4 2.7376 0.0107 2.7280 to 2.7472 88 s :. = standard deviation The probability of "t" lying between two values, t^ and t^ may then be expressed: P{t1 < (C - u)/(s/(n0 '5) < t 2> = 1 - a W The "t" distribution is symmetrical with its maximum ordinate at the center; hence, the shortest confidence interval would be obtained by allowing, (Xj = h = (%2 a/2 or (22) = -t2 (23) Solving Equation (21) for u produces the confidence limits: P{[C - (t2s/n°‘5)] < u < [C > P = (t1s/n0 '5)]} = 1 - a (24) probability of the interval, {} The level of significance, a, was, as is usually the case, chosen to equal 0.05. This meant there was a 95 per cent probability that the mean, C, would be within the calculated interval. The 95 per cent confidence interval of each mean stress optical coefficient, C, is tabulated in Tables 12-14. 5.5 The glasses (1) Linear Regression Analysis stress optical coefficients of the in the D soda-titania-silica phase field, (2)intheE phase field and(3) in both the D and E phase fields, were submitted to the Control Data 6400 computer for linear regression analysis (Appendix B ) . The computer utilized a standard catalogue program which solved the matrix AX = Y by searching for the largest element in a pivotal column. This procedure, called the Gauss-Jordan numerical analytical technique, produced coefficients A, B and C for the equation, C = A Si02 + B Ti02 + C Na20 C = (25) mean stress optical coefficient in brewsters SiC>2 , Ti02 , Na20 A, B, C = = respective oxide components in mole fractions coefficients of proportionality calculated by the computer The five commercially melted glasses in the nepheline and corundum phase fields of the soda-alumina-silica ternary system had a constant silica content of 0.62 mole fraction, and the linear regression program could not be applied. Instead, a Hewlett Packard 9100B computer was programmed to calculate the equation of a line by the least mean squares method. C The equation was of the form: = A + B A120 3 (26) C = mean stress optical coefficient in brewsters A, B = constants of proportionality calculated by the computer 90 AlgOg = mole fraction of aluminum oxide Soda-alumina-silica glasses A6 and A7 had variable silica content, and were submitted with Al, A2 and A3, which were also glasses in the nepheline phase field, for linear regression analysis. Finally, all seven soda-alumina-soda glasses were subjected to the GaussJordan numerical technique. The equations obtained for each glass phase field and the measured versus calculated stress optical coefficients are presented in Tables 15-20. 91 Table 15. Glass A1 A2 A3 A3 Average Stress Optical Coefficient Values for Commercial Laboratory Melted Glasses in the Nepheline and Corundum Primary Phase Fields of the Soda-Alumina-Silica Ternary System Measured SOC Calculated SOC Difference 2.6396 2.6259* 0.0137 2.6041 2.6259 0.0218 2.6367 2.6259 0.0108 2.6296 2.6259 0.0037 2.6065 2.6259 0.0194 2.8050 2.7889* 0.0161 2.7950 2.7889 0.0061 2.7811 2.7889 0.0078 2.8000 2.7889 0.0111 2.7910 2.7889 0.0021 2.9535 2.9519* 0.0016 2.9427 2.9519 0.0092 2.9782 2.9519 0.0263 2.9492 2.9519 0.0027 2.9238 2.9519 0.0281 2.9535 2.9498** 0.0037 2.9427 2.9498 0.0071 2.9782 2.9498 0.0284 2.9492 2.9498 0.0006 2.9238 2.9498 0.0260 (A) 92 Table 15--Continued Measured SOC Glass A4 A5 Calculated. SOC Difference (A) 2.9792 2.9720** 0.0072 2.9623 2.9720 0.0097 2.9948 2.9720 0.0228 2.9680 2.9720 0.0040 2.9618 2.9720 0.0102 2.9939 " 2.9942** 0.0003 2.9831 . 2.9942 0.0111 3.0110 2.9942 0.0168 2.9980 2.9942 0.0038 2.9820 2.9942 0.0122 Standard Error = 0.0145 * Calculated by Least Mean Squares Method from the equation: SOC = 2.177714 + 4.074750 A l ^ **Calculated by Least Mean Squares Method from the equation: SOC = 2.668600 + 1.480000 A1„0„ 2 3 93 Table 16. Average Stress Optical Coefficient Values for Glasses in the Nepheline Primary Phase Field of the Soda-AluminaSilica Ternary System Measured SOC Calculated SOC* Difference (A) Al 2.6232 2.6311 0.0079 A2 2.7944 2.8070 0.0126 A3 2.9492 2.9476 0.0016 A6 2.8057 2.7964 0.0093 A7 2.6510 2.6417 0.0093 Glass Standard Error = 0.0099 *Calculated by linear regression analysis computer program, from the equation: SOC = 3.25921 Si02 + 4.19831 A l ^ + 0.68054 Na^O 94 Table 17. Average Stress Optical Coefficient Values for Glasses in the Nepheline- and Corundum Primary Phase Fields of the Soda-Alumina-Silica Ternary System Measured SOC Calculated SOC* Difference (A) A1 2.6232 2.6414 0.0182 A2 2.7944 2.7981 0.0037 A3 2.9492 2.9234 0.0258 A4 2.9732 2.9705 0.0027 AS 2.9936 3.0175 0.0239 A6 2.8057 2.7972 0.0085 A7 .2.6510 2.6424 Glass 1 0.0086 Standard Error = 0.0170 *Calculated by linear regression analysis computer program, from the equation: SOC = 3.30796 Si02 + 3.86311 A l ^ + 0.72929 Na20 95 Table 18. Glass D1 D2 D3 D4 Average Stress Optical Coefficient Values for Glasses in the D Primary Phase Field of the Soda-Titania-Silica Ternary System Measured SOC Calculated SOC* Difference (A) 2.8376 2.8351 0.0025 2.8557 2.8351 0.0206 2.8292 2.8351 0.0059 2.8318 2.8351 0.0033 2.8398 2.8351 0.0047 2.6651 2.6595 0.0056 . 2.6759 2.6595 0.0164 2.6500 2.6595 0.0095 2.6597 2.6595 0.0002 2.6495 2.6595 0.0100 2.9034 2.9351 0.0317 2.9497 2.9351 0.0146 2.9541 2.9351 0.0190 2.9316 2.9351 0.0035 2.9288 2.9351 0.006:3 2.7371 2.7399 0.0028 2.7325 2.7399 0.0074 2.7283 2.7399 0.0116 2.7511 2.7399 0.0111 2.7371 2.7399 0.0028 .. Standard Error = 0.0125 *Calculated by linear regression analysis computer program, from the equation: SOC = 3.23707 Si0o + 5.23850 TiO_ - 0.27407 Nao0 A A A . 96 Table 19. Glass El E2 E3 ■ E4, Average Stress Optical Coefficient Values for Glasses in the E Primary Phase Field of the Soda-Titania-Silica Ternary System Measured SOC Calculated SOC* Difference (A) 2.6572 2.7054 0.0482 2.6842 2.7054 0.0212 2.7000 2.7054 0.0054 2.6915 2.7054 0.0139 2.6909 2.7054 0.0145 2.9167 2.8580 0.0587 2.8966 2.8580 0.0386 2.8909 2.8580 0.0329 2.8793 2.8580 0.0213 2.9132 2.8580 0.0552 2.9903 3.0105 0.0202 3.0068 3.0105 0.0037 2.9720 3.0105 0.0385 2.9988 2.0105 0.0117 2.9813 3.0105 0.0292 2.7448 2.7376 0.0072 2.7508 2.7376 0.0132 2.7253 2.7376 0.0123 2.7363 2.7376 0.0013 2.7306 2,7376 0.0070 /"Calculated by linear regression analysis computer program, from the equation: SOC = 2.28800 Si02 + 5.33880 Ti02 + 1.64560 Na20 97 Table 20. Glass D1 D2 D3 04 Average Stress 'Optical Coefficient Values, for Glasses in the D and E Primary Phase Fields of the Soda-TitaniaSilica Ternary System Measured SOC Calculated SOC* Difference (A) 2.8376 2.8065 0.0311 2.8557 2.8065 0.0492 2.8292 2.8065 0,0277 2.8318 2.8065 0.0253 2.8398 2.8065 0.0333 2.6651 2.6828 0.0177 2.6759 2.6828 0.0069 2.6500 2.6828 0.0328 2.6597 2.6828 0.0231 2.6495 2.6828 0.0333 2.9034 2.9474 0.0440 2.9497 2.9474 0.0023 2.9541 2.9474 0.0067 2.9316 2.9474 - 0.0158 2.9288 2.9474 0.0186 2.7371 2.6972 0.0399 . 2.7325 2.6972 0.0353 2.7283 2.6972 0.0311 2.7511 2.6972 0.0539 2.7371 2.6972 0.0399 98 Table 20--Continued Glass Measured SOC Calculated ’ SOC* Difference 2.6572 2.7002 0.0430 2.6842 2.7002 0.0160 2.7000 2.7002 0.0002 2.6915 2.7002 0.0087 2.6909 2.7002 0.0093 2.9167 2.8411 0.0756 2.8966 2.8411 0.0555 2.8909 2.8411 0.0498 2.8793 2.8411 0.0382 2.9132 2.8411 0.0721 2.9903 2.9821 0.0082 3.0068 2.9821 0.0247 2.9720 2.9821 0.0101 2.9988 2.9821 0,0167 2.9813 2.9821 0.0008 2.7448 2.8238 0.0790 2.7508 2.8238 0.0730 2.7253 2.8238 0.0985 2.7363 2.8238 0.0875 2.7306 2.8238 0.0932 El E2 E3 E4 (A) . Standard Error = 0.0449 ^Calculated by linear regression analysis computer program, from the equation: SOC = 2.87805 Si09 + 5.69731 TiO + 0.40564 Na 0 CHAPTER 6 RESULTS AND DISCUSSION Babcock (1968, 1969) denoted two terms describing, the structur al characteristics of glass. They were (1) the "substructure," which provided the foundation or framework of glasses and (2) the "microstructure, ". which related to structural details, as between different types of substructures. 1. Babcock (1968, 1969) further postulated that, Each primary crystallization phase of a glass had a unique substructure. 2. Some glass properties were principally determined by these substructures. 3. Silicate glasses whould be classified in terms of their substructures or "silica frameworks." From the above premises, (1) each primary phase field correlated with a unique substructure and (2) a substructure was, by its definition, directly affected by composition changes; therefore, evidence of the existence of substructures could be given by proving that some glass properties had a typical variation with composition in primary crystallization phases. Such evidence was presented when Babcock (1968, 1969) ahd Georoff (1972) showed refractive index, specific volume, fluidity and Knoop hardness varying linearly with composition 100 in primary phase fields of a number of silicate glasses (Figs. 20, 21). As an extension of these results, this study was initiated with the objective of discovering whether the stress optical coefficient also varied linearly with composition in primary crystallization phase fields. 6.1 Initial Investigation Before preparing a number of glasses, it was desired to obtain some indication of whether birefringence would vary linearly with composition in primary phase fields. For this reason, the preliminary step was to measure the stress optical coefficients of five commercially melted glasses from the corundum and nepheline phase fields of the soda-alumina-silica ternary system. The five glasses were characterized by a constant silica content of 0.62 mole fraction. When birefringence was graphed versus the alumina content of the five samples (Fig. 34), there was clearly an inversion point at the phase boundary between the corundum and nepheline phase fields. Both refractive index (Fig. 35) and Knoop hardness (Fig. 36) had an equivalent inversion point for the same five soda-alumina-silica glasses. Because these two glass properties also proved to linearly vary with composition in primary phase fields, a m o r e .extensive examination of composition's affect on birefringence was undertaken. This further examination involved (1) melting and annealing glasses, (2) cutting, grinding and polishing specimens, (3) measuring 101 3.00 2.95 C(brewsters) 2.90 2.85 AUO N a0O A I00-2SiO- NEPHELINE CORUNDUM 2.80 2.75 2.70 2.65 AloO^tmole Figure 34. fraction) Stress optical coefficient, C, versus mole fraction of aluminum oxide for the five commercial laboratory melted glasses in the nepheline and corundum phase fields of the soda-alumina-silica ternary system. -- Silica (Si02) content constant (0.62 mole fraction). 102 1.507 .506 1.505 1.504 Z 1.503 1.502 1.501 N a 0 O A I 0O v 2 S iO - AUO NEPHELINE CORUNDUM 1.500 AUO *! (mole Figure 35. fraction) Index of refraction, N, versus mole fraction of aluminum oxide for the five commercial laboratory melted glasses in the nepheline and corundum phase fields of the sodaalumina-silica ternary system after Babcock (1968). -Silica (SiO^) content constant (0.62 mole fraction). 103 510 505 500 CN E 495 i 42 490 Z z 485 480 N o nO-AUO«v2SiO CORUNDUM NEPHELINE 475 I .10 .12 .14 .16 AlgOg (niole Figure 36. .18 .20 .22 fractio n ) Knoop hardness number, K.H.N., versus mole fraction of aluminum oxide for the five commercial laboratory melted glasses in the nepheline and corundum phase fields of the soda-alumina-silica ternary system after Georoff (1972). -- Silica (SiO^) content constant (0.62 mole fraction). 104 the stress optical coefficients of the glasses, (4) statistical analysis of data and (5) linear regression analysis to resolve whether the data's variation with composition could be described by the equation of a line. . 6.2 Preparation of Glass Samples The greater the homogeneity of a sample, the the measured stress optical coefficient of the glass. more realistic Attaining homogeneous glass samples was the most difficult task of the study. 6.2.1 Melting The soda-titania-silica glasses containing lesser amounts of titania had fluid or "easy" melts. As the proportion of titania in the glasses increased, the melts acquired a water-like consistency, or were, in glass terminology, "fine." melts, seeds (bubbles) were minimal. Due to the fluidity of the The two soda-alumina-silica glasses (A6 and A7) were more viscous and required twenty to thirty hours in the furnace to disperse bubbles versus three to twelve hours for the soda-titania-silica glasses. Volatilization atmosphere resulted in of soda from the melt surface intothe furnace a thin layer with a composition deviating from the remainder of the molten glass. This layer gradually meshed into the entire melt as it was poured, producing composition gradients or streaks throughout the sample. Stirring with a platinum rod (1) introduced bubbles and (2) distributed the volatilization layer throughout the melt as streaks, and after allowing the bubbles to 105 disperse and time enough for diffusion to re-homogenize the glass, a new surface layer had formed. The most successful procedure in minimizing streaks was to take a platinum rod just before pouring and push this surface layer tip against one half of the platinum crucible and then pour using the lip of the other half of the crucible. Other solutions to the problem of volatilization would be (1) melting in a covered crucible or, (2) taking a sample, as did Waxier and Napol- itano (1957), from a 1000-2000 pound melt. 6.2.2 ' Annealing Cooling the glasses at 2.5°C per hour (Hamilton and Cleek 1958) through the principal annealing range left slight thermal strain around the edge of the disks that was visible with the polariscope. Waxier and Napolitano (1957) cooled their samples at a rate of 10C per hour through the transformation region. If a more accurate value of the stress optical coefficient (in the third significant figure) is desired and time permits, a 1°C per hour cooling rate through the transformation range is recommended. There were no instances, of phase separations, and none of the glasses was hygroscopic. 6.2.3 Grinding and Polishing The grinding and polishing criteria noted in Figures 24 and 25 appeared adequate. Any effect upon the measurements due to surface damage incurred from polishing was not observed by the author. ■ ■ ' This topic was not discussed in the. papers listed in the bibliography. ( 6.3 Measurement of the Stress Optical Coefficient The stress optic values calculated for each glass are listed in Tables 9-11. Birefringence tended to increase with increasing mole fractions of silica, titania and alumina, and decrease with increasing mole fractions of soda. It is of note that the stress optical, coefficient was measured using light with a wavelength of 5461 angstroms. How the stress optical coefficient varies with the wavelength of light employed, or how it "disperses" is not known. This would be an excellent topic for a dissertation. The accuracy of the measurements could be improved as follows 1. When the Babinet compensator was set as a quarter wave plate (Fig. 31), the minimum light intensity or extinction position was visually noted. Determination of the minimum light intensity with a photomultiplier and oscilloscope would be more accurate. 2.' In defining the optic axes and crossing the nicols, the first polarizer was adjusted by hand. Mounting it in a rotatable circle divider as was the analyzer would increase the reliability of the data obtained. 3. For a given load the extinction position was located by rotating the analyzer until the horizontal line on the .. oscilloscope screen had dropped to its lowest level. this minimum point, the line would not move for four At. 107 to eight turns (0.35 to 0.70 degrees) of the fine adjustment knob of the rotatable circle divider. Although centering the reading within these limits was attempted, this was considered a major source of error in the third significant figure of the stress optical coefficients obtained. Further sophistication of the electronic analysis of the signal, such as with a Faraday coil, would better identify the extinction position. Increasing the intensity of light entering the photomultiplier would also further clarify the line's minimum position. 6.4 Linear Regression Analysis The foremost objective of this study was to determine whether the stress optical coefficient varied linearly with composition in phase fields. Toward this end, all data obtained were examined to see if they could be represented by a linear equation. 6.4.1 Soda-Titania-Silica Glasses Table 18 lists the measured stress optical coefficients for the four glasses in the D field along with the stress optic values calculated by the equation formulated by the computer. The small standard error indicates a high probability of the data being linear. An equivalent result for glasses in the E field is found in Table 19. Figure 37 is a two dimensional representation of the linear variation of the stress optical coefficient in the E and D phase fields of the soda-titania-silica ternary system. The stress optical lines are continuous across the phase field boundary. 108 fraction) 3.011 D FIELD 2.858* TiOo (mole 2.935 2.835 * / 2.738 2.705 2 -660x 2.740 E FIELD V 05 .20 NagO (mole fraction) Figure 37. Lines of equal stress optical coefficient in primary phase fields D (unknown) and E (Na20.Ti02 .Si02) of the soda-titania-silica ternary system. 109 Table 20 lists the data returned when, phase field boundaries, both the D and E field to linear regression analysis. without regard to glasses were subjected The standard error shows that it is unlikely a linear equation could represent data from both phase fields. Stress optic data on D field glasses fell within the D field indicated in a phase diagram determined by Hamilton of the National Bureau of Standards (Levin 1964). Stress optic data on E field glasses were in the E field outlined on the same diagram. 6.4.2 Soda-Alumina-Silica Glasses Because glasses A1-A5 had a constant silica content, the linear equations method. (Table 15) were obtained via the least mean squares The negligible standard error noted in Table 15 implies the data are linear (Fig. 34). Glasses A6 and A7 had differing silica content and were submitted with glasses A1-A3, which were also in the nepheline field, for linear regression analysis. The measured and calculated stress optical coefficients are listed in Table 16. Again, the small standard error suggests that the data are linear. When, without regard to phase field boundaries, glasses from the corundum and nepheline phase fields were subjected to linear regression analysis (Table 17) the large standard error indicated a linear equation could not simultaneously represent data from both phase fields.. A two dimensional illustration of the linear variation of birefringence in the nepheline phase field of the soda-aluminasilica phase field is drawn in Figure 38. • ' Ill CORUNDUM CARNEGIEITE MULLITE NEPHELINE W ALBITE S1O 2 <m ole Figure 38. fra c tio n ) Linear variation of stress optical coefficient in the nepheline phase field of the N a ^ - A ^ O ^ - S i C ^ glass CHAPTER 7 CONCLUSIONS This investigation dealt with the determination and analysis of stress optical coefficients of glasses in two primary crystalli zation phase fields of both (1) the soda-titania-silica and (2) the soda-alumina-silica ternary systems. The following conclusions resulted: 1. Each primary crystallization phase field of a silica glass has a unique substructure. 2. This substructure is reflected through the relation between the stress optical coefficients of glasses within the phase field. a. ' The stress optical coefficients vary linearly with composition in the primary crystallization phase field. b. This linear relationship is continuous across phase boundaries. 3. Alumina glasses A1-A5 had equivalent silica contents. For these glasses, the stress optical coefficient Versus composition produced an inversion point at the boundary between the nepheline and corundum phase fields. was: 112 This point 113 a. Similar to that observed for both refractive index and Knoop hardness versus composition for the same glasses, b. Further evidence that primary crystallization phase fields have unique substructures. Increasing the mole fraction of: a. Titania, silica or alumina tended to increase the stress optical coefficient of a glass. b. Soda tended to decrease the stress optical coefficient of a glass. APPENDIX A CHEMICAL ANALYSIS 114 115 Table A - l . Chemical Analysis of the Raw Glass Making Materials Melted in this Investigation Constituents Per Cent (%) Silica (Si02) Quintus Quartz, Special Low Iron Silica Assay SiC>2 99.8 Loss on ignition 0.1 Alumina, Al^O^ 0.04 Iron oxide, Fe^O^ . 0.0038 Calcium oxide, CaO <0.001 Magnesium oxide, MgO <0.001 Sodium oxide, Na20 0.02 Potassium oxide, K20 0.01 Sodium Carbonate (Na2CO ) Anhydrous, Granular, Reagent Grade . 99.9 Assay N a ^ O g 0.003 Insoluble matter Loss on heating at 285°C v 0.27 Chloride (Cl) 0.001 Nitrogen compounds (as N) 0.0001 Sulfur compounds (as SO^) 0.001 Ammonium hydroxide precipitate 0.008 Arsenic (As) 0.00002 Calcium and magnesium precipitate 0.009 Heavy metals (as Pb) 0.0003 Iron (Fe) 0.0002 Potassium (K) 0.001 Silica (Si02) 0.003 116 Table A-1--Continued Constituents Per Cent (%) Titanium Dioxide (Ti02) Powder, Reagent Grade Assay Ti02 99.3 Acid-soluble substances 0.30 Arsenic (As) 0.0002 iron (Fe) 0.010 Lead (Pb) 0.002 Loss on.ignition 0.20 Water-soluble substances 0.15 Zinc 0.01 (Zn) Aluminum Oxide ( A ^ O , Powder, Reagent Grad Assay A ^ O ^ • Residue after ignition Chloride (Cl) 98.5 1.0 , 0.005 Silicate (Si02). 0.05 Sulfate (S04) 0.005 Heavy metals (as Pb) 0.001 Iron (Fe) 0.03 Substances not precipitated by NH^OH (as sulfates) 0.75 APPENDIX B COMPUTER PROGRAM USED FOR LINEAR REGRESSION ANALYSIS OF STRESS OPTIC DATA 117 118 Table B-l. 0003 0003 0003 0003 0003 0003 0003 0007 0015 0015 0017 0021 0022 0035 0035 0040 0043 0045 0047 0051 0053 0054 0056 0061 0064 0071 0106 0106 0107 0113 0115 0125 0125 0131 0133 0143 0143 0145 0147 0150 0152 0161 0164 . • Computer Program Used for Linear Regression Analysis for Glasses in the Sodium-Titanium-Silicate Ternary System PROGRAM TEST(INPUT,OUTPUT,TAPE60=INPUT,TAPE61=0UTPUT) COMMON DATA(8,100) INTEGER FMS(7),FMT(2) DIMENSION DELT(100) DIMENSION V(8),A(10,11) DATA KI/60/,KO/61/ EXTERNAL VSUB WRITE(KO,95) READ 2,KK 2 FORMAT(11) NOCOEF-KK+1 DO 80 K=l,100 DO 20 1=1,100 READ(KI,59)(DATA(KP,I),KP=1,NOCOEF) 59 FORMAT(8F10.0) IF(EOF,60)100,18 18 IF(DATA(1,I).EQ.O.O) G O T O 25 20 CONTINUE STOP 123 100 STOP 456 25 1=1-1 DO 30 11=1,110 A(II)=0.0 30 CONTINUE CALL LSMTX(KK,I ,A,VSUB,KK) CALL MATEQ(A,KK,A(1,KK+1)) .WRITE(KO,94)(A(KK,KK+1),11=1,NOCOEF) 94 FORMAT(*C0EF=*7FT2.5) DLTSUM=0.0 N Q = ((KK-2)*12+2) IF(KK.LE.2)NQ=1 ENCODE(68,89,FMS)NQ 89 FORMAT(57H(* CALC DEP VAR OBSERVED VAR INDEPENDENT 1 VARIABLES* 12,9HX,*DELT*)) WRITE(KO,FMS) NQ=KK+3 ENCODE(11,93,FMT)NQ 93 FORMAT(3H(IX,12,6HF12.6)) DO 40 11=1,1 CALL VSUB(II,V,KK) XX=0.0 DO 35 J=1,KK . XX=XX+V(J)*A(J,KK+1) 35 CONTINUE DELT(II)=DATA(1,II)-XX 119 Table B- l . 0170 0172 0213 0216 0220 0226 0233 • 0233 0237 0241 0241 0006 ■ 0006 0006 0007 0016 0021 0021 Computer Program Used for Linear Regression Analysis for Glasses in the Sodium-.Titanium-Silicate Ternary System-Continued CALL VSUB(II,V,KK) WRITE(K0,FMT)XX,V(NOCOEF),(V(IK),IK=1,KK),DELT(I1) DLTSUM=DLTSUM+DELT(II)**2 40 CONTINUE DLTSUM=SQRT(DLTSUM/(1-1)) WRITE(KO,91)DLTSUM 91 FORMAT(/1X*SQRT(SUM(DELT(I)**2)/(N-l)=*F12.7) WRITE(K0,95) 80 CONTINUE 95 FORMAT(1H1) END SUBROUTINE VSUB(I,V,KJ) DIMENSION V (10) COMMON DATA(8,20) DO 10 J=1,KJ 10 V(J)=DATA(J+1,I) V(KJ+1)=DATA(1,I) RETURN END 120 Table B-T. Computer Program Used for Linear Regression Analysis for Glasses in the Sodium-Titanium-Silicate Ternary System-Continued SUBROUTINE LSMTX(M>N , A >VSUBR,NUMB) 0010 0010 0012 0013 0014 0020 0023 DIMENSION A(10,ll),VEC(10) C M IS THE NUMBER OF FUNCTIONS IN THE POLYNOMIAL C N IS THE NUMBER OF DATA POINTS INVOLVED C A IS THE ARRAY INTO WHICH THE NORMAL EQUATION MATRIX WILL BE BUILT C C A MUST BE DIMENSIONED (10,11) M MUST BE LESS THAN OR EQUAL TO 10 C N MAY BE ANY SIZE LESS THAN 32,767 C C C C C C C A SUBROUTINE VSUBR(I,V,NUMB) MUST BE PROVIDED BY THE USER WHERE I IS THE INDEX OF THE CURRENT DATA POINT V IS A VECTOR WHICH MUST BE FILLED BY VSUBR WITH THE I TH VALUES OF THE M FUNCTIONS FOLLOWED BY THE I TH VALUE OF Y NUMB IS A NUMBER TRANSMITTED THROUGH THIS ROUTINE TO VSUBR C FIRST ZERO OUT THE MATRIX - : C 0025 0026 0037 0041 0042 0054 0057 " 0061 MAUG=M+1 DO 10 1=1,M DO 5 J=1,MAUG A(I,J)=0.0 5 CONTINUE 10 CONTINUE SECOND BUILD UPPER RIGHT MATRIX DO 25 IDATA=1,N CALL VSUBR(IDATA,VEC,NUMB) DO 20 IROW=l,M DO 15 ICOL=IROW,MAUG A(IROW,ICOL)=A(IROW,ICOL)+VEC(IROW)*VEC(ICOL) 15 CONTINUE 20 CONTINUE 25 CONTINUE 121 Table B-l. Computer Program Used for Linear Regression Analysis for Glasses in the Sodium-Titanium-Silicate Ternary System-Continued THIRD FILL OUT SYMETRIC MATRIX 0063 0065 0066 0070 0072 0103 0106 0110 0111 MSUB=M-1 DO 35 IR0W=1,MSUB ICOLS=IROW+l DO 30 ICQL=ICOLS,M A(ICOL,IROW)=A(IROW,ICOL) 30 CONTINUE 35 CONTINUE RETURN END : SUBROUTINE MATEQ(ARAY,ISIZE,ACOL) C GAUSS-JORDAN MATRIX REDUCTION ROUTINE C SOLVES THE MATRIX EQUATION A X = Y BY GAUSS-JORDAN TECHNIQUE USING LARGEST-PIVOT SEARCH THROUGH THE PIVOTAL COLUMN. C . C C C C C INPUT IS DEFINED AS FOLLOWS — ARAY - MATRIX TO BE REDUCED ACOL '- COLUMN VECTOR ASSOCIATED WITH -ARAYISIZE - ORDER OF MATRIX -ARAYISING - OUTPUT SWITCH - - 0 = MATRIX ISSINGULAR 1 = MATRIX IS NON-SINGULAR C. -ARAY- MUST BE DIMENSIONED IN CALLING PROGRAM AS ARAY '(10,10) C -ACOL- MUST BE DIMENSIONED IN CALLING PROGRAM AS ACOL (10) C C PROGRAMMED BY WM. RABKIN, ITEK CORP. DECEMBER 1965 C MODIFIED BY J. RANCOURT 6/2.1/67 TO CONSERVE TIME AND SPACE 0006 DIMENSION A RAY(10,10),ACOL(10) C C 0006 0007 SEARCH FOR LARGEST ELEMENT IN PIVOTAL DIAGONAL AND SWAP ITS.ENTIRE ROW WITH DO 65 IPIV=1,ISIZE IF(IPIV-ISIZE) 5,25,5 COLUMN BELOW THE PIVOTAL ROW. 122 Table B-l. 0010 0012 Computer Program Used for Linear Regression Analysis for Glasses in the Sodium-Titanium-Silicate Ternary System-Continued 5 IPP=IPIV+1 DO 20 I=IPP,ISIZE IF(ABSF(ARAY(I,IPIV))-ABSF(ARAY(IPIV,IPIV)))20,20,10 10 DO 15 J=l,ISIZE TARAY = ARAY(I,J) ARAY (I ,J) =ARAY (IP IV,J) ARAY(IPIVjJ)=TARAY 15 CONTINUE TARAY =ACOL(I) ACOL(I)=AC0L(IPIV) ACOL(IPIV) = TARAY ' 20 CONTINUE 0013 0023 0025 0031 0036 0041 0043 0045 0047 0051 C 0054 IF LARGEST PIVOTAL ELEMENT IS ZERO, MATRIX WAS SINGULAR 25 IF(ARAY(IPIV,IPIV)) 30,70,30 C 0057 00.63 0064 0071 0074 IF NON-ZERO, DIVIDE EACH ELEMENT IN PIVOTAL ROW BY PIVOTA 30 PIVOT=ARAY(IPIV,IPIV) DO 35 J=IPlV, ISIZE ARAY(IPIV,J)=ARAY(IPIV,J)/PIVOT 35 CONTINUE ACOL(IPIV)=ACOL(IPIV)/PIVOT C REDUCE EACH ROW EXCEPT THE PIVOTAL ROW 0076 oioo 0102 40 0106 0107 45 0111 0122 50 55 0125 0128 0132 0135 60 65 C 0137 0140 DO 60 1=1,ISIZE IF(I-IPIV) 40,60,40 CPE=ARAY(I,IPIV) IF(IPIV-ISIZE) 45,55,45 DO 50 J=IPP,ISIZE ARAY(I,J)=IPP,ISIZE CONTINUE CONTINUE ACOL(I)=ACOL(I)-CPE*ACOL(IPIV) CONTINUE CONTINUE REDUCTION COMPLETE -RETURN NON-SINGULARITY SWITCH ISING = 1 RETURN C MATRIX SINGULAR SWITCH OR NCOL = 0 AND INVSW = 0-- RETURN ERROR 123 Table B-l. 0140 0142 0144 0145 0146 Computer Program Used for Linear Regression Analysis for Glasses in the Sodium-Titanium-Silicate Ternary System-Continued 70 DO 75 I=1,ISIZE ACOL(1)=0.0 75 CONTINUE RETURN END APPENDIX C EQUIPMENT EMPLOYED IN THE PREPARATION OF GLASS SAMPLES 124 125 Table C-l. Equipment Employed in Preparing Glass Samples --- : . . . - .... .... Equipment [and the preparation step (from Figure 23) it was used in] Torsion optical reading laboratory balance [3.1] U. S. Stoneware porcelain ball mill [3.2] ’ Description Model PL-2. Operating on the torsion principle, it had no knife edges and was insensitive to out-of-level con ditions. Air currents and tempera ture changes had negligible affect on accuracy. The capacity was 2000 grams with an optical projection reading scale of 110 grams. A scale vernier allowed readings to 0.01 gram. Model BFO. 12" long x 6" diameter. Pereny furnace [3.3] Model IPKS1732*105. The furnace was lined with high purity-high density alumina brick. A second layer of more porous alumina was integrated with glass wool material for addi tional insulation. "Kanthai" heating elements were employed. These ele ments were made of 90% molybdenum disilicate and 10% ceramic binding and were highly oxidation resistant. Above 982°C the elements reacted with oxygen to form a silicon dioxide (quartz glass) coating which protect ed the MoSi.2 from further chemical attack (Giler 1970). The furnace melting space was three cubic feet with a maximum temperature capa bility of about 1600°C. Honeywell radiamatic head [3.3] Type RH, Model 569. The radiamatic ; head monitored the temperature in the Pereny furnace. PYRO optical pyrometer [3.3] The optical pyrometer was used to calibrate the radiamatic head. 126 Table C-l--Continued Equipment [and the preparation step (from Figure 23) it was used in] Honeywell electronic 15 inch circular chart . proportional controller (ECCPC)[3.3] Description Model Y15201116-01-17-(350)-025-00 with 6 tap-20 KVA Model 1985 trans former and a Model 1561-382 G01 Norbatrol SCR (silicon controlled rectifier) package. This unit pro vided a temperature control from 700 to 1600°C for the Pereny furnace. The ECCPC provided a temperature re cord on a circular chart via an ink pen receiving input from the radiamatic head. Fafnir A1 rollers [3.4] Model RAS 1. Corning laboratory hot plate [3.4] Model PC-35. 12" long x 6" diameter. Lihdberg heavy-duty rodoverbend furnace [3.7] Model 11-R0-122412-20. This elec trically heated box type furnace was . used to anneal glass. It was lined with insulating brick and block type insulation. Base metal resis tance type elements heated the fur nace. Maximum temperature was 1091°C. Honeywell electronic strip chart proportional controller [3.7] Model CY15301116-01-3-(350)-002-07. The ESCPC provided temperature con trol for the annealing furnace. This controller used a thermocouple vs. the radiamatic head employed by the Pereny furnace. A "Data Trak" pro grammer automatically lowered the annealing temperature at 2.5°C through the annealing range. Thermolyne portable pyrometer [3.7] Model PM-1K50. The pyrometer was used to calibrate the thermocouple in the Lindberg furnace. 127 Table C-l--Continued Equipment [and the preparation step (from Figure 23) it was used in] Description Bausch and Lomb polariscope [3.8] Catalogue No. 31-52-62-60. Leitz-Wetzlar polarizing microscope [3.9] Model DIALUX - Pol. APPENDIX D EQUIPMENT EMPLOYED IN MEASURING THE STRESS OPTICAL COEFFICIENT OF A GLASS SAMPLE 128 129 Table D-l. 'Equipment Employed in Measuring the Stress Optical Coefficient of a Glass Equipment [and the identification number in Figure 26] Description GE tungsten strip lamp + DC power source [i] GE. lamp: microscope illuminator, 18A/T10/2P-6V, ASA Code EDW, The lamp housing was an EG I G Model 590-20. Lens [1] Two inch diameter; radius of curvature was 70 millimeters. Interference filter [2] Oriel 2" square interference filter, blazed for 5461 angstroms. Product No. G-572-5461. Pinhole [3] Two millimeter diameter pinhole. Oriel Catalog No. B-26-36-03. Lens [4] Two inch diameter; radius of curvature was 94 millimeters. Polarizer [5] UV transmitting polarizing GlanThompson prisms, 0.3’ beam devia tion, wavelength range to 2300 angstroms, 25° field of view, length to aperture ration of 3.0; Yardney Razdow Product No. PGT-UV5121. Mounting tubes were Product No. RMT-PPR-15C. The calcite comprising the polarizer was very soft and extreme care had to be used in cleaning. Apparatus for applying uniform compression to glass sample [6] See Table D-2 and Figures D-l and D-2. Soliel-Babinet compensator [7] Adjustable through four waves; Yardney Razdow Product No. SBC12Q-UVV. Polarizer [8] Same as Polarizer [5] ___ __________ _____ _______________ 130 Table D-l--Continued Equipment [and the identification number in Figure 26] Divided circle rotator [8] Description Precision angle divider, 360°, 120 millimeter outer diameter, vernier readable to 0.05°; Yardney Razdow Product No. DCR120-360. Lens [9] Two inch diameter; radius of curvature was 94 millimeters. Photomultiplier [10] RCA Model No. IP21. The power supply was a Keithely Model 245 with a maximum output of 2000 volts. 300 to 600 hundred volts were used in the operation of the photomultiplier. Oscilloscope [ii] Hewlett Packard Model HP 108A; time base plug in Model HP 1821A; differential amplifier plug in Model H05 1801A. Optical bench 200 centimeters long; Precision Tool and Instrument Company Ltd., Thorton Heath, Surrey, England. Weights Plastic bottles, filled with sand were used as weights in compres sing the glass sample. These weights were accurate to ± 0.25%. . 131 0.25 1.75 0.25 7.25 2.25 2.375' Figure D-l. 2.75 Apparatus for aligning and compressing the glass sample. 132 Table D-2. Part Number (Fig. D-l) Itemized Description of Apparatus in Figure D-l. Dimension Part . Length Material 16' Steel (T) Yoke (^2^ Pivot Pin (7) Guides (2) 0.75" OD; 0.0625" wall thickness 0.25" Brass (7^ Inserts (2) 0.625" OD; 0.125" wall thickness 0.25" Steel (7) Shaft 0.375" D 4" Steel @ Cylinder 1" OD; 0.125" wall thickness 7" Steel @ Window (2) 0.75" wide x 2.25" high Ball Bearing (2 ) 0.125" D Steel Plates (2) See Figure D-2 Steel 10J Glass Spec imen 1 x 1 x 3 cm 11) Base 4.5" D to 2" D to 0.75" D 12) Screws 8) 0.5" wide x 0.. 1875" thick (3 at Steel 0.25" + 0.25" + 2.5" = 3" 0.25" D Steel Aluminum 120°) 13) Table 7.75" high x 14" wide 8" Aluminum 14) Platform 4" wide 16" Aluminum is) Wire 4 feet from yoke to platform Steel I ------1 0.519 0.1875" 0.0625 0.0625" Figure D - 2 . Detail of steel caps: (?) in Figure D - l . A piece of rubber tape or a piece of rubber gasket was cut out and placed between the cap and the top of the glass specimen. Otherwise, small machining ridges on the steel cap would have fractured the glass. SELECTED BIBLIOGRAPHY Adams, L. H . , and E. 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