I m 0 61939 ABSTRACT MECHANICAL STRENGTH AND FRACTURE STUDIES OF PARTIALLY CRYSTALLIZED SELENIUM BY Anwar Rahman Daudi Partially crystallized selenium samples having various size and volume fractions of crystallites were obtained by heat treating amor- phous selenium samples at 620, 820 and 100°C for different lengths of time. Polarized-light microscopy was used to determine the size and volume fractions of the spherulites present in each specimen. Mechanical tests were carried out under three-point bending. Flexure strength of partially crystallized selenium decreases with increasing size and volume fraction of the crystallites. Scanning electron microscope studies of the fractured surfaces revealed that the frac- ture always started from spherulites. The fracture nucleates in the weak peripheral regions of the spherulites. Amorphous selenium shows inherent flaw size of about 5 pm. The flaw size in partially crystal- lized selenium varies from 10 to 100 um, depending on the crystallite size. The flexure strength of amorphous and partially crystallized selenium can be determined from.the mirror radius of fracture, provided the crystallites are less than 14 volume percent. The theoretical expressions of Maxwell and Eucken agree well with the experimentally determined variation of elastic modulus of partially crystallized selenium as a function of volume fraction of crystallites. MECHANICAL STRENGTH AND FRACTURE STUDIES OF PARTIALLY CRYSTALLIZED SELENIUM ANWAR RAHMAN DAUDI A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics, and Materials Science 1974 To my Parents .1 .1 ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to his advisor, Dr. K. N. Subramanian, whose guidance and assistance were invaluable throughout the course of this study. The interest, counsel, and time generously extended by Dr. D. J. Montgomery is gratefully acknowledged. The writer appreciates the help and guidance offered by Dr. W. Hartmann (Physics) and Dr. G. Martin (Mechanical Engineering). Thanks are also extended to Dr. R. Summitt, Chairman, Department of Metallurgy, Mechanics, and Materials Science for financial assistance. Finally, he wishes to thank his wife, Rafat, without whose help and consent this would not have been possible. iii TABLE OF CONTENTS List of Tables . . . . . . . . . . . . . List of Figures . . . . . . . . . . . . I. II. Introduction . . . . . . . . . . . 1.1 Morphology and Structural Details of Amorphous and 1.2 Crystallized Selenium . . . . . . . . . 1.1.1 Structure of Amorphous Selenium . . . . 1.1.2 Structure of Crystalline Selenium 1.1.3 Morphology of Trigonal Selenium . . . 1.1.4 Physical States of Amorphous Selenium General Considerations of Fracture and Strength of _Amorphous and Partially Crystallized Selenium . 1.2.1 Strength and Fracture Studies on Amorphous Materials . . . . . . . . . 1.2.2 Strength and Fracture Studies on Glass- Crystalline Materials . . . . . . 1.2.3 Mode of Fracture in Glassy and Glass- -Crystalline Materials . . . . . . . . . . . . . . . . . . . 1.3 Objectives of this Research . Experimental Procedure . . . . . . 2.1 Sample Preparation . . . . . . . . 2.2 2.3 2.1.1 Safety Precautions . 2.1.2 Melting of Selenium . . . . . . . . . . . . . 2.1.3 Quenching Technique . . . . 2.1.4 Cutting Bulk Amorphous Selenium Samples for Mechanical Testing . . Heat Treatment of Specimens . 2.2.1 Crystallization Study . 2. 2. 2 Heat Treatment of Mechanical Test Specimens Quantitative Metallographic Analysis of Specimens . 2.3.1 Volume Fraction and Size of Crystallites . iv Page vii ix 11 11 14 16 21 27 33 36 36 36 36 4O 4O 41 41 42 43 43 III. IV. 2.4 Mechanical Testing of Specimens . . . . . . . 2.4.1 Experimental Method to Determine Flexure Strength of Amorphous and Partially Crystallized Selenium . . . . . . . . . . . 2.4.2 Experimental Method to Determine Elastic Modulus of Amorphous and Partially Crystallized Selenium 2.4.3 Experimental Method to Determine Fracture Surface Energy of Amorphous and Partially Crystallized Selenium . . . . . . . . . . 2.5 Fractograph Studies . . . . . . . . . . . . . 2.5.1 Preparation of SEM Fractographs . . . . . . 2.5.2 Measurement of Mirror Radius from the Fracture Surface of Amorphous and Partially Crystallized Selenium Specimens . . . . . . . . . . . . . . Results 0 O O O O O O O O O O 3.1 General Observations . . 3.2 Microstructural Studies . . . . . . . 3.2.1 Qualitative Analysis . . . . . . . . . . 3.2.2 Quantitative Analysis . . . . . . . . . . 3.3 Bend Test Results . . . . . 3.3.1 Flexure Strength of Amorphous and Partially Crystallized Selenium . . . . . 3.3.2 Elastic Modulus of Amorphous and Partially Crystallized Selenium . . . . . . 3.3.3 Fracture Surface Energy of Amorphous and Partially Crystallized Selenium 3.4 Fractograph Studies . . . . . . . . . 3.4.1 Observations on SEM Fractographs . . Discussion . . . . . . . . . . . . . . . . . . . 4.1 Microstructure of Partially Crystallized Selenium . . 4.2 Elastic Properties of Partially Crystallized Selenium . 4.3 Mechanical Properties of Partially Crystallized Selenium . . . . . . . . . . . . . . . . . . . . 4.3.1 Effect of Thermal Expansion Differences of Amorphous and Crystalline Selenium . . 4.3.2 Effect of Elastic Moduli Differences of Amorphous and Crystalline Selenium . . . 4.3.3 Effect of Size of Crystalline Phases on the Flexure Strength of Partially Crystallized Selenium . . . . . . . . . . . . . . . . . . . . 4.3.4 Effect of Volume Fraction of Crystalline Phases on Flexure Strength of Partially Crystallized Selenium -- A Statistical Approach . . . . . . . V Page 44 45 46 46 48 48 48 50 50 52 52 53 54 54 55 56 57 57 106 107 110 116 116 118 123 133 "to. ,.,_~. 4.4 Fracture Analysis of Partially Crystallized Selenium . . 4.5 4.4.1 Flexure Strength of Amorphous Selenium as 4.4.2 Determined from the Dimensions of Fracture Mirrors . . . . . . . . . . . . . . . . . . Flexure Strength of Partially Crystallized Selenium as Determined from the Dimensions of Fracture Mirrors . . . . . . . . Comparison of Flexure Strength and Elastic Modulus of Amorphous and Partially Crystallized Selenium with Glass and Glass-Crystalline Composites . . . . . 4.5.1 Comparison of Flexure Strength and Elastic 4.5.2 Modulus of Amorphous Selenium with Oxide Glasses. Comparison of Flexure Strength and Elastic Modulus of Partially Crystallized Selenium with Glass-Ceramics and Glass-Crystal Composites . V. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . List of References . Appendix A.. vi Page 144 144 152 160 160 162 170 173 177 Table 10 11 12 13 LIST OF TABLES Volume Fraction of Crystallites in Heat-treated Selenium Specimens . . . . . . . . . . . . . Size of Crystallites in Heat-treated Selenium Specimens . Flexure Strength of Amorphous and Partially Crystallized Selenium for Melts A, B, and C . . . . . . . . . . . . A Comparison of the Flexure Strength of Partially Crystal- lized Selenium Determined Using 1" Span and 1/2" Span in Three-point Bend Test . . . . . . . . . . . . . . Flexure Strength and Elastic Modulus of Partially Crystal- lized Selenium Versus Volume Fraction of Crystallites for Constant Size of Crystallites . . . . . . . . Flexure Strength and Elastic Modulus of Partially Crystal- lized Selenium Versus Size of Crystallites for Constant Volume Fraction of Crystallites of Selenium . . . Elastic Modulus of Selenium and Partially Crystallized Selenium Specimens for Melts A, B, and C. . . Fracture Surface Energy for Amorphous and Partially Crystallized Selenium . . . . . . . . . . . . Elastic M lus Calculated by Hashin's36, Paul'sAO, Kingery's , and Maxwell and Eucken's66 Relations . Flaw Size as Determined by Griffith's Equation for Amor- phous and Partially Crystallized Selenium . . . . . . . Flexure Strength of Partially Crystallized Selenium as a Function of (Size of Crystallites)‘1/2 . . . . . . . . Growth Rate and Number Per Sq. mm. of Spherulitic Selenium at 100°C, 82°C, and 62°C. . . . . 1v Mirror Radius, Function f(D,r), Breaking Stress and om(p)2 Values for Amorphous Selenium Specimens . vii Page 61 62 63 64 67 7O 73 »74 114 129 134 137 151 Table '14 15 16 Page Mirror Radius, Function f(D,r), Breaking Stress and om(p)35 Values for Partially Crystallized Selenium Specimens . . . . 153 Flexure Strength and Elastic Modulus of Certain Glasses . . 161 Pertinent Physical Pr0perties of Certain Oxide Glasses and Crystalline Phases . . . . . . . . . . . . . . . . . . 163 viii 10. 11. 12. 13. 14. LIST OF FIGURES Periodic table showing elemental glass formers Cis-trans model of selenium prOposed by Foss. (A) CIS- MODEL, (B) TRANS‘MODEL. o o o o o o o o o o o o (A) The trigonal selenium lattice (B) The chain structure of selenium . (A) Ring structure of selenium (B) The monoclinic selenium lattice . . . . . . . . . . . . The spiral structure of selenium spherulites (shown by X) . C 0 U . C O O O O O O . O O C O O C O O O O O O O O (A) The physical states of amorphous selenium (B) The elastic modulus plotted as a function of time for amorphous selenium when heat treated at 32°, 50°, and 80°C . . . . . . . . . . . . . . . .,. . . . . . . . . Schematic illustrating the four prominent regions of fracture . . . . . . . . . . . . . . . . . . . . . . Diagram of mirror surface of glass fracture slggwing inner and outer loci and axes of measurement (Shand ). . Plot of mirror radius versus breaking stress for glass and glass-ceramic specimens 0.457" in diameter. Curves repre- sent cggresponding mean values of am(p)25 for flexure. (Shand)...................... Distribution of variances of individual test points from curves of Figure 9 . . . . . . . . . . . . . . . . Shape of quartz tube for melting selenium . . . . Furnace set-up for melting selenium . . . . . . . . . Load-deflection curg§ for a typical brittle material (Davidge and Tappin ) . . . . . . . . . . . . . Reproducibility of surface microstructure of specimens heat treated at 100°C . . . . . . . . . . . . . . . . . . . . . ix Page 10 12 15 29 34 34 34 37 39 47 75 Figure Page 15. Reproducibility of surface microstructure of specimens heat treated at 82°C . . . . . . . . . . . . . . . . . . . . . . 76 16. Microstructures of specimens heat treated at 100°C. Polar- ized light micrographs of partially crystallized selenium obtained by: (A) Heat treating at 100°C for 45 min (B) Heat treating at 100°C for 1 hr ,and (c) Heat treating at 100°C for 1 1/2 hr. . . . 77 l7. ‘Microstructures of specimens heat treated at 82°C. Polar- ized light micrographs of partially crystallized selenium obtained by: (A) Heat treating at 82°C for 4 hr , (B) Heat treating at 82°C for 6 hr , and (C) Heat treating at 82°C for 8 hr . . . . . . 78 18. Microstructures of specimens heat treated at 62°C. Polar- ized light micrographs of partially crystallized selenium obtained by: (A) Heat treating at 62°C for 200 hr , (B) Heat treating at 62°C for 250 hr , and (C) Heat treating at 62°C for 311 hr . . . . 79 19. Plot of flexure strength as a function of volume fraction for melts A, B, and C . . . . . . . . . . . . . . . . . . . 80 20. Plot of flexure strength as a function of volume fraction for specimens heat treated at 62°C. . . . . . . . . . . . . 81 21. Plot of flexure strength as a function of volume fraction for specimens heat treated at 82°C. . . . . . . . . . . . . 82 22. Plot of flexure strength as a function of volume fraction for specimens heat treated at 100°C. . . . . . . . . . . . 83 23. Plot of flexure strength as a function of volume fraction for constant-size crystallites (ds = 12.75 and 15.78 pm). . 84 24. Plot of flexure strength as a function of volume fraction for constant-size crystallites (ds = 19.06 and 22.22 um). . 85 25. Plot of flexure strength as a function of volume fraction for constant-size crystallites (dS = 25.14 and 50.80 um). . 86 26. Plot of flexure strength as a function of size of crystal- lites for constant volume fraction of crystallites (Vs = 2.09 and 3.73 70) O O O O O O O O O O O O O O O O I O 87 27. Plot of flexure strength as a function of size of crystal- lites for constant volume fraction of crystallites (V8 = 6.55 and 8.15 %) . . . . . . . . . . . . . . . . . . 88 Figure ‘ Page 28. Plot of flexure strength as a function of size of crystal- lites for constant volume fraction of crystallites (v8 = 14016 7o) 0 I o o o o o o o o o o o o o a o o o o o o 89 29. Plot of elastic modulus of partially crystallized selenium as a function of volume fraction of crystallites for constant-size crystallites (d = 12.75, 15.78, 19.05, 22.22, 25.14 and 50.80 um) .3. . . . . . . . . . . . . . . 90 30. SEM fractographs of amorphous selenium. (A) Total field of fracture (30x). (B) The fracture propagates with no preferred direction shown at location X. (C) The fracture consists of coarse and mirror regions shown at location Y . . . . . . . . . . . . . . . . . . 91 31. SEM fractographs of partially crystallized selenium obtained by heat treating at 62°C for 200 hr (A) Total field of fracture (30x) (B) Origin of fracture (200x) (C) Spherulites in hackle region of fracture (500x) . . . . 92 32. SEM fractOgraphs of partially crystallized selenium obtained by heat treating at 62°C for 250 hr (A) Total field of fracture (30x) (B) Origin of fracture (200x) (C) Spherulites in hackle region of fracture (500x) . . . . 93 33. SEM fractographs of partially crystallized selenium obtained by heat treating at 62°C for 311 hr (A) Total field of fracture (50x) (B) Origin of fracture (500x) (C) A spherulite in hackle region of fracture (ZOOOx) . . . 94 34. SEM fractographs of partially crystallized selenium obtained by heat treating at 100°C for 45 min (A) Total field of fracture (50x) (B) Origin of fracture (lOOOx) (C) A spherulite in the mirror region (lOOOx) (D) A spherulite in the coarse region (lOOOx) . . . . . . . 95 35. SEM fractographs of partially crystallized selenium obtained by heat treating at 100°C for 1 hr (A) Total field of fracture (50x) (B) Origin of fracture (lOOOx) (C) A spherulite in the mirror region (1000x) (D) A spherulite in the coarse region (lOOOx) . . . . . . . 96 xi Figure Page 36. SEM.fractographs of partially crystallized selenium obtained by heat treatment at 100°C for 1% hr (A) Total field of fracture (30x) (B) Origin of fracture (500x) (C) Spherulites in the mirror and coarse region (500x) (D) Spherulites in the hackle region (500x) . . . . . . . . 97 37. SEM fractographs of partially crystallized selenium obtained by heat treating at 82°C for 4 hr (A) Total field of fracture (50x) (B) Origin of fracture (500x) (C) A spherulite in coarse region of fracture (lOOOx) (D) Spherulites in hackle region of fracture (500x) . . . . 98 38. SEM fractographs of partially crystallized selenium obtained by heat treating at 820C for 6 hr (A) Total field of fracture (50x) (B) Origin of fracture (2000x) (C) Spherulites in coarse region (1000x) (D) Spherulites in hackle region (lOOOx). . . . . . . . . . 99 39. SEM fractographs of partially crystallized selenium obtained by heat treating at 82°C for 8 hr (A) Total field of fracture (30x) (B) Origin of fracture (500x) (C) Spherulites in coarse region of fracture (500x) (D) Spherulites in hackle region of fracture (500x) . . . . 100 40. SEM fractograph showing the details of fracture surface in and around a spherulite (2000x) . . . . . . . . . . . . . . 101 41. SEM fractographs showing (A) A spherulite in the mirror region of fracture for specimen heat treated at 82°C for 6 hr (2000x) (B) Spherulites present at the origin of fracture for specimen heat treated at 82°C for 6 hr (2000x) . . . . 102 42. SEM fractographs showing (A) The origin of fracture for specimen heat treated at 82°C for 8 hr (500x) (B) The hackle region of fracture for specimen heat treated at 82°C for 8 hr (lOOOx) . . . . . . . . . . . . . . . 103 43. SEM fractographs showing (A) A spherulite in the mirror region of fracture for specimen heat treated at 82°C for 8 hr (1500x) (B) A spherulite in the coarse region of fracture for specimen heat treated at 82°C for 8 hr (1500x) (C) Two spherulites in the mirror region of fracture for specimen heat treated at 82°C for 8 hr (1500x) . . . . 104 xii Figure Page 44. SEM fractographs showing (A) A spherulite in the mirror region of fracture for specimen heat treated at 100°C for 45 min (2000x) (B) A pocket left by spherulite in the mirror region of fracture for specimen heat treated at 100°C for 45 min (2000x) (C) A spherulite in the mirror region of fracture for specimen heat treated at 100°C for 1 hr (2000x) . . . 105 45. Polarized-light micrograph of spherulites of selenium illustrating the absence of separation at the glass- crystal interface. . . . . . . . . . . . . . . . . . . . . 111 46. Plot of calculated valuzs of elastic modulus (according to the expressions of Paul and Hashin36)as a function of volume fraction of crystallites . . . . . . . . . . . . . . 115 47. Arrangement of lamellas in the spherulite . . . . . . . . . 120 48. (A) Schematic representation of discontinuous crystal phase and continuous glass phase. (B) Schematic representation of continuous crystal phase and discontinuous glass phase . . . . . . . . . . . . . 124 49. Schematic diagram showing path of fracture in a spherulite. 132 50. Plot of flexure strength of partially crystallized selenium versus (size of crystallites)‘% . . . . . . . . . . . . . . 136 51. Plot of frequency of fracture as a function of flexure strength of amorphous and partially crystallized selenium (for constant size of spherulites). . . . . . . . . . . . . 140 52. Plot of half band width of Weibull's curves of Figure 51, as a function of size of spherulites . . . . . . . . . . . 141 53. Plot of frequency of fracture as a function of flexure strength of amorphous and partially crystallized selenium (for constant volume fraction of spherulites) . . . . . . . 142 54. Plot of half band width of Weibull's curves of Figure 53, as a function of volume fraction of spherulites . . . . . . 143 55. Stresseconcentration factor ka and kb. All loads applied in tension . . . . . . . . . . . . . . . . . . . . . . . . . . 146 56. Plot of the function f(D,r) versus mirror radius for amor- PhOUS and Partially crystallized selenium . . . . . . . . . 148 57. Plot of the breaking stress as a function of mirror radius for amorphous selenium . . . . . . . . . . . . . . . . . . 149 xiii Figure 58. 59. 60. 61. 62. 63. 64. 65. 66. Page Plot of the breaking stress as a function of mirror radius for partially crystallized selenium . . . . . . . . . . . . 155 Plot of function f(D,r) versus mirror radius for small fracture cracks, with the ggdition of curve corrected for flexure conditions. (Shand ) . . . . . . . . . . 156 Mirror areas of partially crystallized selenium (A) Specimens heat treated at 62°C for 200 hr, 250 hr, and 311 hr (70x). (B) Specimens heat treated at 82°C for 4 hr, 6 hr, and 8 hr (70x). 0 (C) Specimens heat treated at 100 C for 45 min, 1 hr, and 1% hr (70x) . . . . . . . . . . . . . . . . . . . . . . 157 Plot of o (p)!5 values as a function of volume fraction of crystalliges of selenium . . . . . . . . . . . . . . . . . 158 Plot of flexure strength z§rsus volume fraction of crystal- ) lites (Frey and Mackenzie 164 Plot of elastic modulus vggsus volume fraction of crystal- lites (Frey and Mackenzie ) 164 Plot of (A) Uniaxial and biaxial strength of a soda borosilicate glass containing alumina spheres 60p in diameter (Hasselman and Fulrath45). (B) Uniaxial strength of a soda borosilicate glass con- taining spherical pores 60p in diameter (Hasselman and Fulrath45). (C) Uniaxial strength of sodium borosilicate glass con- taining spherical pores (Bertolotti and Fulrath7°) . . 168 Plot of function f(c/d) versus old, as calculated from results of Gross and Srawley by Corum (Davidge and Tappin62) . . . . . . . . . . . . . . . . . . . . . . 179 The general form of stiffness (K) versus crack area (A) plot (Davidge and Tappinéz) . . . . . . . . . . . . . . . . 179 xiv I . INTRODUCTION Many technologically important materials consist of more than one phase. The mechanical prOperties of such polyphase materials depend on the properties of the individual phases and the characteristics of the boundaries that exist between them. The size, shape, and distribution of the constituents, which depend on the boundary characteristics, play important roles in determining the behavior of the composite. These constituent phases could be crystalline or glassy. The crystalline phase has long-range order, whereas the glassy phase has definite first- order coordination but lacks long-range periodicity. In glasses, the crystallization required by equilibrium conditions has not taken place. As a result, the structure is similar to the liquid state. The poly- phase materials containing crystalline materials in glassy matrices have some attractive physical properties. However, the basic mechanisms that control the mechanical behavior of such materials is not fully under- stood at present. The purpose of the present work is to study the mechanical properties of such a composite through a fairly simple glass- crystal model system, with the hope that the results may shed some light on the mechanical behavior of the glass-crystal materials in general. In practice, the crystalline phases can be incorporated in a con- tinuous glass matrix by one of the following methods: 1) by suitable heat-treatment schedules as in glass-ceramics, and .o u‘ o\ 2) by hot pressing mixtures of glass and crystalline powders, as in glass-crystal composites. The glass-crystal composites are ideal materials to verify the theoreti- cal predictions for mechanical strength, since one can produce definite size, shape, and distribution of various crystalline phases in a given glassy matrix. On the other hand, intricate shapes can be produced with glass-ceramic, because objects can be shaped in the glassy state and then heat-treated to produce the crystalline phases. However, one does not have much control in the crystals that will crystallize out of a given glass matrix, with respect to their size, shape, or distribution. Although the glass-ceramic process is not ideally suitable for checking the theoretical predictions, it is the most useful means of producing objects for practical requirements. It would be ideal to have a system which can be formed by glass-ceramic means but can have all the controls on size, shape, and distribution of crystals as in glass-crystal comb posites. The material selected for the present study, selenium, satis- fies these requirements. The structure of amorphous selenium differs from the structure of glasses used in studies on glass-ceramics and glass-crystal composites. One can group glasses according to their internal structure; such a classification will include, a) glasses with continuous network structure, such as in silicate and oxide glasses, and b) glasses with randomly-distributed linear-chain structures such as organic polymers and elemental glass-formers like sulfur, selenium, and tellurium. Mechanical-strength studies carried out so far in glasses, glass- ceramics, and glass-crystal composites have been on continuous-network structured glasses. However, the inorganic glasses having linear chain structure have not received any attention. The only materials having linear-chain structure, whose mechanical properties have been studied extensively, are the organic polymers, such as polystyrene and poly- ethylene. These studies would be of great help in understanding the mechanical behavior of inorganic polymers such as sulfur, selenium, and tellurium. Selenium, having a linear-chain structure in glassy state, is an ideal model for studying the effect of crystallization on mechanical strength of a glass-crystalline material. Partially crystallized selenium contains two crystalline forms of selenium distributed in the amorphous matrix. Further, the transformation from completely amorphous to partially crystallized state is extremely important, since it pro- duces excellent physical, electrical, and magnetic properties in selenium. The electronic industries have used amorphous selenium in a large number of applications such as switching, memory, light-sensing, and photoelectric devices. A very important area, however, the. mechanical properties of selenium, has not been explored so far. The object of the present work is to study the role of crystallization on the mechanical strength and fracture behavior of partially crystallized selenium. 1.1 Morphology and Structural Details of Amorphous and Crystallized Selenium. In the periodic table shown in Figure l, selenium (atomic number 34) appears as a group VIA element. Its electronic configuration is Fig. 1. Periodic table showing elemental glass formers. (shown by dark area) o 13 at! 88 L: o a: 33 3: L:- 35 83's? a3 :3 3 35 32 I 53 9‘ t= a} :3 k8 s5 ##121 6 #3 gt {3’ :1 2" a“ 8“ 35 a“ 3" '33 = 3 a; 88 a“ s“ ‘l:=|= 3 =§ ”at B: :8 5 1322822p63823p64824p4. It is a well-known small-band-gap semiconductor. Sulfur (atomic number - 16) and tellurium (atomic number = 52) also belong to this group. Sulfur, selenium, and tellurium are well-known glass-formers. 1.1.1 Structure of Amorphous Selenium The structure of glassy selenium is found to be similar to that of molten selenium at temperatures slightly above the melting point. Caldwell and Fan1 have shown by infra-red spectroscopy that the struc- ture of glassy selenium and liquid selenium is a mixture of Se rings 8 and long polymeric chains with the atoms approximately equally distri- buted. There is some indication, however, that the structure is strongly dependent on the preparation procedure. When selenium glass is dissolved in cold 082 solution, only part of the glass can be dissolved. Briegleb2 attributes this phenomenon to the existence of two different molecular species of selenium that could be present in the glassy state. The soluble portion was believed to be composed of Se rings, as in 8 ‘monoclinic selenium, which is also soluble in C82; the insoluble species was thought to be long tangled chains, as in trigonal selenium, which is almost insoluble in CSZ' On this basis, Briegleb concluded that the relative amount of rings and chains in the selenium glass depend on its thermal history. Briegleb showed that only 45 weight percent of the selenium glass exists in the ring configuration when quenched from the melting point of trigonal selenium (220°C). Only 15 weight percent, however, exists in the ring configuration when quenched from 600°C. The break-up of Se8 rings and subsequent polymerization into long chains at a higher temperature depends on the thermodynamic equilibrium that can be established between rings and chains. A knowledge of thermodynamic 6 quantities for the above equilibrium state allows one to evaluate the average chain length and the concentration of Se8 rings. The two impor- tant factors, then, that control the average number of chains and rings in glassy selenium are: a) the temperature from which the liquid selenium is quenched to produce the glass, and b) the time allowed for the melt to reach equilibrium in the molten condition. 1.1.2 Structures of Crystalline Selenium In its crystalline allotropes, selenium--like sulfur-~forms sp hybrid bonds with its nearest neighbors. This bonding can lead to either long helical chains or to rings containing either six or eight atoms. Two distinct molecular units are present in the crystalline states, the ring (known as nonoclinic selenium) and the helical chain (known as trigonal selenium). KrebSB’4 has discussed the structure of both sulfur and selenium in terms of cis-trans model of Fosss. In this bonding scheme, as shown in Figure 2, the first three atoms of both arrangements are positioned in a plane perpendicular to the plane of the paper. In the gig-configuration the fourth and fifth atoms are added in , Se such a way as to lead to ring molecules (S6, S 8’ etc.) whereas 8 Egang-bonding results in long helical chains. In a lighter element like sulfur, gig-bonding is favored, whereas in the heavier, more metallic group VIA elements, like selenium, tellurium, and polonium, there is an increasing tendency toward Ergng-bonding. Therefore in sulfur, the eight-membered ring characteristic of rhombic sulfur is more stable (although an unstable chain structure has also been observed)6. An unstable form of crystalline sulfur, composed of 8 molecules, has also 6 Fig. 2. ‘Qig-trans model of selenium proposed by Foss. (A) CIS- MODEL, (B) TRANS-MODEL. (See text for additional description.) (A) CIS PLANE .L TO PAPER (8) .. o. - .__ -. 3“. “'5; been observedy. 0n the other hand, monoclinic selenium, formed by loose packing of Se8 rings, is unstable and on heating above 70°C transforms to the trigonal form, which is composed of parallel helical chains. In agreement with the findings of Krebs3 and Tunistra6, no ring modifica- tions of tellurium have been observed. The trigonal selenium lattice is shown in Figure 3. It shows the parallel packing of helical chains. Each selenium atom within a chain is bonded to two nearest neighbors, and the Se-Se interatomic distance is 2.374Ao. There are six next-nearest neighbors, resulting from atoms in neighboring chains, at a distance of 3.426Ao. This is somewhat shorter than the distance8 expected for van der Waals bonding (about 4.0Ao), and hence implies a fairly weak chemical bond. 3 8 ’ The structures of Selenium can exist in two monoclinic forms these two allotropes, o-selenium and B-selenium, are essentially the same, as shown in Figure 4. The difference exists in the crystalline packing of rings, the individual selenium rings being almost identical in both formsg. The average Se-Se bond length within a ring is 2.34Ao. The next-nearest-neighbor distance, for atoms in neighboring rings, varies from atom to atom, the average distance being 3.48AO. The shortest inter-ring distance is 3.48A9, the largest is 3.99Ao. The latter is of the magnitude expected for van der Waals bonding. The two a and B monoclinic forms of selenium are unstable and will readily convert to trigonal selenium on heating above 70°C. According to Burnbanklo, the transformation begins at the points of closest packing where the structure is highly polarized by p- and d-electron interactions between neighboring molecules. Fig. 3. (A) The trigonal selenium lattice. (B) The chain structure of selenium. 10 Fig. 4. (A) Ring structure of selenium. (B) The monoclinic selenium lattice. 11 1.1.3 Morphology of Trigonal Selenium The shape, size, and distribution of the trigonal selenium (hexa- gonal selenium) would be shown later to be important in determining the mechanical strength and fracture properties of selenium. The crystallization of spherulites of selenium from amorphous selenium is now well-known11’12’13. From previous works on the morph- ology of these spherulites, their crystallization can be considered similar to those found in organic polymers. To check this possibility, a detailed structural examination of spherulites of selenium was under- taken by Fitton and Griffithsl3. This study showed that crystallization of amorphous selenium occurs by the formation of lamellas in which the molecular axis lies across the wide face and perpendicular to the growth direction. At temperatures below about 200°C the lamellas grow radially from a common nucleus, branching noncrystallographically to produce a spherical crystalline mass. The theoretical selenium chain length in the quenched amorphous phase is much longer than the average lamella observed at large supercoolingsl3. This difference suggests that at the edges of the lamellas the chains must fold back to avoid stress accumu- lation due to interconnection of the two phases of different density. Fitton and Griffiths suggest that such an array of chain folds intro- duces a dilation of the lattice which is accommodated if the lamella twists. Where the effect is cooperative, such twisting within the radial array of lamellas is responsible for the spiral structure of selenium spherulites as shown in Figure 5. 1.1.4 Physical States of Amorphous Selenium Amorphous selenium, as described in the previous sections, has a long-chain molecular structure, and as a result possesses unique 12 Fig. 5. The spiral structure of selenium spherulites (shown by X). r... *a n .‘p...-- ,. e. .. . .u n.... - v ‘M 0.. v o», F v ""--n— w.~ , 4.)!” K .. U I I1. '1' m l J .» ~s 13 physical and mechanical properties. A single molecule of this inorganic polymer consists of repeated units linked by primary valence bonds, the number of repeating units being typically of the order of 105 atoms. The atoms bonded in the chain can rotate, the degree of freedom of rotation being dependent on the nature of atoms present in the chain. Besides the restriction to rotation within the molecule itself, the presence of other molecules in the bulk amorphous selenium also imposes restrictions on the molecular motion. These may be caused by the field forces of a van der Waals nature. The restrictions are maximized at low temperatures, when internal rotation is almost com- pletely inhibited and the materials are hard and glassy. The glassy selenium is brittle, and has a high elastic modulus, approximately 0.7 x 1011 dynes per cm2 at 32°C. In amorphous selenium, the response to an applied stress is believed to be analogous to that found in other solid materials; it is associated with the change in internal energy caused by a stretching of interatomic bonds, or by the distortion of bond angles. In the ”ideal" glassy state of selenium the molecules are considered to be randomly disposed. The random configuration is very difficult to realize in practice, and probably the deviations from it can be considered as a flaw in the glassy selenium. As the temperature is raised, the thermal energy of the molecules is increased to the point where the barriers to motion can be overcome. As seen in Figure 6, in the rubbery region, the elastic modulus is low (107 dynes per cm2 at 70°C) and the extensibility is high. Because of the thermal energy of the molecules, the large deformation is accom- panied by essentially free rotation about the primary valence bonds, so that the molecules adopt an extended configuration. In the l4 rubbery region, the stress in a sample held at constant elongation increases with increasing temperature, but in the glassy region, the stress decreases with increasing temperaturela. The transition between the two extremes is marked by a region of visco-elastic behavior as seen in Figure 6, where the properties are predominantly time-dependent. It is important that the glass-transition temperature (Tg) be defined, so that the mechanical strength determined in this study is independent of time of test. The glass—transition temperature in amorphous selenium is much lower than that in oxide glasses because in the former case the transition involves the breakage of low-energy van der Waals forces, rather than the breakage of the primary covalent bonds. From studies of Properties of amorphous selenium by Eisenberg and Tobolsky15 the glass-transition temperature is found to be 31.00C. The macroscopic elastic properties and the molecular response of glasses possessing long molecular chain structure such as amorphous selenium, are analogous to those of other solid materials. 1.2 General Considerations of Fracture and Strength of Amorphous and Partially Crystallized Selenium To understand the state in which amorphous selenium exists in this investigation, a study of the physical states of the amorphous selenium was presented, in the preceding section 1.1.4. In the following sec- tion, 1.2.1, a review of the existing theories of mechanical strength and fracture properties of inorganic network-structured glasses is presented. 15 Fig. 6. (A) The physical states of amorphous selenium. (B) The elastic modulus plotted as a function of time for amorphous selenium when heated to 30°, 50°, and 80°. (A) _ (owns/CM?) .. O MODULUS OF ELASTICITY d GLASS/Y REGION ” : VISCO-ELASTIC REGION I I 1 / RUBBERY REGION "’ I ~ I VISCOUS ~ I i ; REGION I I '- : I : / L———r| I: ;'¢— - L 1 l 4 1 L1 L 1 J 32°C C TEMPERATURE -—-> I°CI GLASSY REGION IS THE PHYSICAL STATE IN WHICH MECHANICAL TESTS WERE PERFORMED IN THIS WORK (B) 5 q MODULUS OF ELASTICITY 5 (omes’ M‘) ‘. TIME —> (HRS) THE MECHANICAL PROPERTY TESTS WERE CARRIED OUT AT 25-26°C 16 1.2.1 Strength and Fracture Studies on Amorphous Materials Experimental studies on fracture and strength Of amorphous materials have normally been carried out in silicate glasses having continuous net- ‘work structure. As a result, the existing theories have been developed mainly from the results of silicate-glass studies. Fracture and strength of amorphous materials are closely related to each other in a brittle material like glass. Fracture of a material implies separation into at least two integral parts; the strength of the ‘material refers to the value of the externally applied stress necessary to achieve this result. In the fracture process, the bonds between the atoms present in tne newly-created surfaces are ruptured; the number of bonds involved can be computed from the structure of the materials. If the nature of bond and the force required to rupture one such bond are known, then the theoretical strength of the material can be calcu- l lated 6’17. There have been several attempts to calculate the theoreti- cal strength of amorphous materialsl6’17. Orowan18 assumes that the stress (a) in a rod under tensile loading changes with increasing interatomic spacing according to the relation 0 = om sin (2nx/A), ‘where am is the maximum of o, the theoretical strength of the material, and x is the increase in interatomic spacing. He equated the work done by fracture to the surface energy of the two new surfaces and derived the strength of glass to be O = (2Ey/a )25 [1] m o where E, y, and a0 are Young's modulus, fracture surface energy, and the interatomic distance, respectively. The fracture strength Om obtained after substituting suitable values if 1000 kg/mm2 or 17 106 lb/sq.in. The strength values obtained in these studies are usually two or more orders of magnitude greater than those realized experimen- tally with conventional samples. One of the reasons for this discre- pancy is that the method Of calculation assumes that the stress is shared equally among all the interatomic bonds present in the cross- section. Then the fracture process would result in instantaneous separation of two planes in the material. A reasonable assumption, therefore, is to consider that the stress applied is not uniformly distributed over the entire cross-section, and hence that there are regions of relatively high stress. When the applied stress creates a stress large enough to cause rupture of the interatomic bonds, the fracture will be initiated. Therefore, the dis- crepancy between the theoretical and observed values of the ultimate strength is explicable in terms of stress concentrations present in the samples tested. Another approach to determine the strength of the material is to consider that physical defects exist in an elastic continuum. This con- sideration disregards the atomistic nature of the materials, and assumes that the material is completely isotropic except for areas of defects which act like stress raisers. Inglis19 originated such an approach, and Griffithzo’21 developed the theory. Griffith assumed that the system could be represented by an infinite plate containing a central crack of length 2C, and with an energy criterion for instability derived the condition of applied stress at which the crack would increase in size and lead to fracture. According to Griffith, the critical tensile stress (T), that will cause fracture is given by T = (Tammi [2] I I I. r sew .uc '“o 5 Q 1 u u ‘M. N .__‘ 24. (I? 18 where E is Young's modulus, and Y is the specific surface energy, i.e., the energy required to create unit area of surface. Breaking strength T, therefore, depends upon E, Y, and C. Griffith showed that for artificially-made cracks in a silicate glass, the product TC35 equals 240 when T is in lb/in2 and C in inches. From this value of ’5 TC , and the observed values of breaking strength, Griffith calculated that the glass contains cracks Of length 2u. It has been pointed out by Orowan that by equating the interatomic spacing "a0" in Equation 1 to the radius of curvature p at the tip of a crack, and assuming a0 = nC/2 and C = 2u, Orowan's expression becomes the same as that of Griffith. The two-dimensional model discussed above was later developed into a three-dimensional mode122’23. A further development of this theory is to relate the flaw or defect of the material to the atomistic nature of real materials. These extensions and refinements, however, merely change the numerical constants in Griffith's expression, usually by a small factor, without influencing the functional relationship between the predicted critical stress (T), the material constants (E,y), and the size of the defect (C). According to the flaw theories, we can say that the ultimate strength observed in brittle material is not entirely an intrinsic property Of the material. From Griffith's equation, the strength of a sample is determined by two factors: 1) the properties of the material, as defined by elastic modulus and the surface energy, and 2) the size of the defect which it contains. The size of the crack does not depend on the property of the material I»- .us .‘e L. .- 19 itself; it depends on the processing variables. Several distribution functions have been proposedza’25 to account for the variations in the tensile strength of specimens that have undergone the same processing treatment. This approach provides a convenient way of representing the extent of variability of the data, but the distribution functions them- selves do not have any fundamental significance, since they are believed to be caused by unknown or uncontrollable factors during production of the samples tested. A review26 Of the statistical treatment of strength data has been presented very recently, as discussed later in this section. One serious Objection that has been raised against the flaw theo- ries is the representation of a real material with an atomistic struc- ture as an elastic continuum. This representation is unreasonable, since the fracture process which occurs at the tip of the postulated flaw must involve the rupture of interatomic bonds, and on a molecular scale. As a result, the concepts of classical elasticity have no meaning. To treat fracture phenomena directly in molecular terms, there exists an entirely different approach. Under an applied stress, the interatomic bonds in a body will be subjected to forces which will increase the mean interatomic distance. The characteristic relation of Potential energy versus displacement for the atom pair will be so modi- fied that less energy will be required to effect rupture Of the bond. As a result, the probability of bond rupture will be increased. To relate the effect to observable quantities, some relation has then to be assumed between the interatomic forces and the applied stress. The problem of defining an instability condition at this point generates #0 o a ‘. a In. "4.. ‘x 20 different theories. The instability is attributed to differences in es27,28,29 30,31 rat , and nucleation32. The final equations , probability provide relations between stress, time to failure, and temperature. These theories do not give the absolute value of fracture stress. In the earlier portion of this section, it is shown that the frac- ture strength of brittle solids is dependent on the distribution of Griffith flaws in the material and on the probability that a flaw capable of initiating fracture at a specific applied stress is present. Various statistical theories for the strength of brittle materials have been proposed33’34. These theories assume that the number of the critical-size flaws present is related to the volume or surface of the specimen. Most of these theories have been devised to explain the fracture behavior of the materials when prepared by specific techniques. As a result, none of the theories can explain the behavior of all the materials. The best known Of these statistical theories was developed by Weibu1133’34. According to Weibull, the risk of rupture R is pro- portional to the function of stress and the volume of the body, and is given by R = ff(o)dv [3] where f(o) is a stress function and the dv is an infinitesimal volume. It is assumed that f(0) = woof“ [4] Where do is a "characteristic strength" dependent on the distribution function best fitting the data, and m is a constant related to the material homogeneity. The applied tensile stress f(o) was integrated over the volume of the body. 21 In conclusion, the two types of theories, based on atomistic or statistical basis, that have been proposed to explain the strength and fracture phenomena of amorphous materials, are not completely satis— factory in the light of the experimental evidence. But the flaw theories utilizing the statistical considerations appear to be widely accepted at present. 1.2.2 Strength and Fracture Studies on Glass—Crystalline Materials All theories of strength and fracture properties of materials con- taining glass and crystalline phases have been experimentally verified with glass-ceramics and glass-crystal composites. If a material has a glass phase and a crystalline phase with a relatively small volume frac- tion of the latter, the glass forms a continuous matrix in which iso- lated crystals are dispersed. McMillan35 noted that the strength of such materials would be dependent on the properties of the glassy phase. At high volume fractions of crystalline phase, the glassy phase may take the form either of a thin layer between adjacent crystals or,in some cases,of small isolated pockets. The strength of such materials would be independent of properties of the glassy phase. For all intermediate volume fractions, the strength of glass-crystalline materials would be dependent on the property of the individual phases. Only a few factors, such as volume and density of individual phases, can be averaged either on a weight or volume basis to give the properties of the composite. MEChanical strength Of the glass-crystalline material is a complex func- tion of a number of factors such as elastic moduli, thermal-expansion coefficients, size, shape, and distribution of the phases. These proper- ties do not follow the simple additive rule. ‘ .. "a"! ,....-v o u. I -- 0 c. o . . a; .-.‘ -. .__‘ ‘Vou. ." I- "n... "Vs.- . I. -. .. ‘h I (7' [In (7": ,. 2’.- 22 Generally glass-ceramics are stronger than the conventional glasses, when tests are carried out on specimens that have received the samua'pre-abrasion treatment. Theoretical studies by Hashin36, Hashin 37 .38,39 and Strikhman , R031 , and Pauli}o have predicted increase of strength in glass-ceramics because of the increase in Young's modulus of the glass-crystalline material through the introduction Of the crystal- line phases having higher rigidities than the glassAl’az. However, the observed increase Of the elastic moduli is insufficient to account for the increase of mechanical strength. McMillan et a143 experimentally verified this, by working on LiZO-ZnO-SiO -P 0 system and showed that 2 2 5 strength of glass-ceramics increases from 1800 to 3500 kg/cm2 when heat- treated at 600°C for one hour, whereas the Young's modulus increases by a much smaller factor from 7.2 x 105 to 9.2 x 105 kg/cmz. Another possibility for the increase of strength of glass-ceramics may be due to differential thermal expansion of the crystalline and glassy phases. The difference of thermal-expansion coefficients could cause a favorable system of microstresses to develop there by enhancing the mechanical strength. This factor was verified experimentally by McCOllister and Conradaa. Studies of the rough path of fracture in an intercrystalline material indicate that the fracture toughness and Umchanical strength of the glass-ceramics are higher than that in the transgranular or fairly smooth fracture in completely crystalline and completely glassy materials, respectively. Hence the mode of fracture Propagation in a glass-crystal composite could determine the strength Of the glass-crystalline material. If the thermal expansion of the crystalline phase is lower than that Of the glassy phase, the stress at the glass-crystal interface in 23 the radial direction is compressive in both phases. In the circumfer- entiail direction the stress is tensile in the glass phase and compres- siana in the crystalline phase. In this case, the fracture is trans- granular; that means a crack approaching the crystal tends to propagate across the interface because the stresses normal to the glass-crystal interface are compressive. If the expansion coefficient of the crystal is higher than that of the glass, however, the stresses in the radial direction in both phases are tensile; in the circumferential direction, the stress is compressive in the glass, and tensile in the crystal. In this case, intergranular fracture is favored since the stress normal to the glass-crystal interface is tensile. McCollister and Conrad44 have investigated the influence of micro- stresses caused by differences in thermal-expansion coefficients on the fracture propagation in glass-ceramics, and have experimentally verified the general ideas discussed above. These investigations dealt with LiZO-AlZO3-Si02 and Can in glass matrix. The expansion coefficient of the crystalline phase was 6 x 1077 per 0C, and that of the glassy phase was 41 x 10"7 per oC 44. An average compressive stress of 17000 lb/in2 1:151 was measured by birefringence techniques, at the glass-crystal interface. As predicted on examination of the fracture surfaces, trans- granular fracture was predominant in these materials. In the second system, calcium fluoride crystals were present in the glassy matrix. The crystalline phase had an expansion coefficient Of 195 x 10.7 per C, and the glassy phase had an expansion coefficient of 93 x 10'.7 per 0C 44 The birefringence technique showed tensile stress of 43000 lb/in2 : 15% at the glass-crystal interface. In this case, as expected, the fracture was almost exclusively intergranular. 24 'The stress concentration caused by the shape, size, and distribu- tixni of the crystalline phase in the glassy matrix may be of the greatest . . .. 45 unportance 1n determ1n1ng the fracture stress. Hasselman and Fulrath postulated that the effect of stress concentrations on the strength is governed by the relative size Of the Griffith flaw and the volume of material over which the stress concentration acts. On this basis the effect of porosity on the strength of material can be analyzed in three distinct ways: a) b) C) It when the pore size is much larger than the flaw size (as in Hasselman and Fulrath, case I), flaws located near pores will be entirely within a stress-concentration field. A precipitous decrease in strength would be expected with the introduction of the first pore into the loaded area under such loading condi- tions. as the size of the pore approaches the flaw size, (Hasselman and Fulrath, case 11), the flaw will not be entirely located in areas of high stress concentration. A smaller decrease in strength would be expected than in case I. when the pore size is much smaller than the flaw size (Hassel- man and Fulrath, case III), the stress concentration field will no longer be large enough to affect the strength of the material appreciably, so only slight decreases of strength with porosity will beobserved. is possible that the mechanical strength of glass-ceramics, like that of glasses, is dependent on the density and distribution Of micro- cracks 46 . If this is the case, the mean sizes of the microcracks, and ~ , - .n.~- Q " I. . 0“. -. ..'... I .. ‘e. I an . ., . I ., e > . ' v ’. se,‘ . \ v 1,_ ‘P \ \I ‘. . we 5'. 25 thereby the mechanical strength, might be expected to be influenced by the microstructure of the glass-ceramic. . . 46 . According to th1s approach the mechanical strength of glass- ceramic (o) is given by [5] where K is a constant and d is the mean diameter of the grain. This suggests that crack length "C" of Griffith equations 1 ("Ev/2C)2 Q II or O = Kc-% [61 to be prOportional to or equal to the grain diameter. The critical flaws are therefore present within the crystalline grains and do not extend into the glass-crystal interface, and are therefore proportional to the circumference of the grains. Another theory explaining the strengthening due to crystalline phase dispersed in glassy matrix is prOposed by Hasselman and Fulrath47. They suggest that in glass-ceramics containing strong crystalline dis- Persions, the fracture will be initiated within the glass matrix. For such composites at a sufficiently high volume fraction Of the crystal- line phase, the maximum size of the flaws present in the glass may be restricted because of the presence of the dispersion. Hence the conclu- siOn from this theory is that the flaws present in the glass matrix are terminated at the glass-crystal boundaries. The spacing between the crYStals, or the mean free path in the glass phase, will therefore be a critical parameter in determining the mechanical strength. Let the mean free Path be represented by "p", and 26 p = d(l-V)/V [7] where V is the volume fraction of the crystalline phase and d is the diameter Of the grain. There are two possible cases: Case I. For small volume fractions and large grain diameters, the value of the mean free path will be larger than the flaw size, C, in the original glass, and therefore little effect Of the crystalline dispersion on mechanical strength would be expected; Case II. For higher volume fractions and small particle size, the mean free path becomes smaller than the original flaw size, and therefore the flaw size in the glass-crystal composite is controlled by the interparticle spacing in the glass matrix. As a result, two regions exist--one in which the strength is not greatly affected by the crystalline dispersion since the interparticle spacing is too large, and a second in which the strength will be prOpor- -% tional to p , if p, the mean free path, and c the flaw size, can be equated. Freiman and Hench°° determined the mechanical strengths Of a series of glass-ceramics derived by heat-treating a glass Of the molecular per- centage composition Li 0 (33%) - SiO2 (67%). The glass was given 2 nucleation treatments at 475°C for times ranging from three to forty- eight hours, and was then heat-treated at 575°C to develop lithium disilicate crystals. The crystal sizes varied from 4.2 to 61 um, the volume fractions ranged from 0.05 to 0.95, and the mean free paths varied from 2.8 to 178 um. With Griffith's expression, they calculated a Critical flaw size of 85 um, and showed that in the early stages of 27 crystallization when the strengthening effect was maximum, fracture is controlled by the initiation of flaws in the glassy phase rather than in the lithium disilicate spherulites. The mechanical strength was found 45 to be prOportional to p where p is the mean free path. Hence there are five existing theories to explain the strengthening due to the crystalline phases present in a glassy matrix. These theo- ries have explained the strengthening of glass by the introduction Of crystalline phase in the following manner: i) the strengthening is due to difference in elastic moduli of the crystalline phase and the glassy matrix. ii) the strengthening is due to the difference in thermal-expansion coefficients of the crystalline phase and the glassy matrix. iii) the strengthening is due to stress-concentration effects of the crystalline phase in the glassy matrix. iv) the strength is affected by the number and size Of the micro- cracks in the crystalline and the glassy phases. v) the strengthening is due to the limiting of the size of the microcracks by the hard crystalline phase dispersed in the glass matrix. 1.2.3 Mode of Fracture in Glassy and Glass-Crystalline Materials Experimental studies of fracture in glassy materials indicate that fracture originates at flaws or cracks of critical size, most of which are at the surface. The fracture propagates through the glass when the 1Ocal stress at the tip of the crack exceeds a minimum value. The rate °°’°° claims that this of propagation increases with crack length. Shand rests of propagation of fracture reaches a limiting crack velocity at a Péiaticular crack length when the stress at the crack tip reaches a 28 critical value. This limiting condition has been identified with the boundary of the mirror surface Of the fracture. The surface of fracture when examined under the microscope shows four prominent areas of fracture. Figure 7 illustrates diagramatically the type of surface observed. 1) The origin of the fracture which shows the flaw at which the fracture starts, 2) the mirror region of fracture which appears as a brightly polished region, 3) the rough region of fracture surrounds the mirror region, and 4) the hackled region of fracture, which clearly shows grooves radiating out from the origin of fracture. These regions of fracture have been discussed in detail by Shand°°’5°, SmekalSI, Murgatroyedsz, and Barson53. Smekal pointed out that there was no exact boundary between the rough zone and the mirror-like zone. More recently G'Olz54 studied the boundary between the mirror and rough regions of fracture, under the electron microscope, with the technique of Mahl, and concluded that the mirror zone also contained hackles, and that the roughness became increasingly small towards the origin of the fracture. The observance of the mirror zone was attributed to the inability of the instrument to resolve the fine-scale roughness. Smekal51 has also noted that occasionally the glass rod under uniaxial tension may break from a sub-surface flaw, which results in a circular mirror area with hackle regions radiating from it. Shand55 has shown that there exists a close relationship between the fracture velocity and the topography of fracture surface. A typical fracture surface of borosilicate glass broken in bending is shown in 29 Fig. 7. Schematic illustrating the four prominent regions of fracture. F 4. HACKLE @( ——---~ 3. COARSE I e REGION a; _ J; , 5m”:- EDGE OF SPECIMEN \ 2.MIRROR REGION I. ORIGIN OF FRACTURE 30 Figure 8. The fracture surface generally consists of four regions -- i) (Irigin of fracture, ii) mirror, iii) rough, and iv) hackle regions. As the crack begins to propagate from the originating flaw, the fracture front moves along D - D and passes into a section in which the nominal stress decreases with distance. The extremities, however, follow the tension surface of the bar where initially the nominal stress remains constant. In both cases, the local stress concentration increases with the crack dimensions. The local stress distribution around the semicir- cular boundary of the crack is such that the stress is a maximum along the axis D - D. This results in a corresponding velocity difference, so that the crack spreads more in the lateral directions, and the crack boundary therefore is no longer semicircular, but tends to become semi- elliptical, as indicated by the "intermediate crack front." The local stresses at the fracture extremities soon reach the critical value, resulting in the transition to the violent phase of fracture as evi- denced by triangular shattered areas. As a result of a certain segment's being shattered, the cross-sectional area that supports load is reduced. This reduction permits accelerated rates of deflection of the bar, and consequently a pronounced relaxation of the applied bending moment. The ndrror surface shows the stage when fracture velocity is increasing; the boundary of the mirror and rough region is the stage when fracture velo- City reaches the maximum value, which Occurs when the stress reaches a critical value. The rough and hackle regions represent the latter Stages of fracture when the applied bending moment is reduced, resulting in a reduction of fracture velocity. Walner lines, as defined by Walner57, are lines on fracture surface formed by intercepts made by the elastic wave travelling at a constant velocity (as determined by the 31 PhYSical properties Of glass) and the fracture front. The closing of the Walner lines near the final crack front is characterized by a slowing down of the crack velocity. Hackles are produced by the frac- ture fronts from two different planes meeting in the plane of fracture. It is generally accepted that there is some relationship that encists between the dimensions of the mirrored section of a glass frac- Inxre and.the normal stress causing the fracture. SmekalS8 and his co- workers found that when circular glass rods are broken in tension, the rumninal breaking stress in the section outside the mirror is essentially constant over a fairly wide range of mirror sizes. However, Terao59 found that the breaking stress of glass rod '1b is approximately propor- tional to the inverse square root of the mirror radius % 1'; Ob = K/r [8] where K is a constant of proportionality. This relation has been con- firmed by Lavengood6C'and Orr61 for specific types of fractures. 49,50,51,56 Shand has experimentally determined the breaking stress by utilizing the dimensions of the mirror area in the fracture process. According to Shand55, during the fracture of glass a critical velo- city Of crack propagation and a critical value of local stress are reached at the boundary of the mirror region. The nominal breaking stress (ob) is the critical stress (am) of the glass divided by the stress-concentration factor of a crack with dimensions of the mirror surface; i.e., Ob - om/K [9] and K = f(D,r)/(D)° [10] 32 f(D,r) = V'— = /11.27/?/D-_r I11] .272r + (D-r) where p is the effective radius of the crack tip, D is the diameter of the rod, r is the mirror radius, and f(D,r) is a function depen- dent on the stress-concentration factor K. This stress—concentration factor K is determined by the size and geometrical form of the exposed surface of the fracture crack, and also by the characteristic deformation which occurs at the crack tip. In the case of glass it is estimated that if the flaw extends to a depth of only a few atom spaces, the effect on the size of the mirror surface, and consequently the change in the value of p, will be very small. For some glass-ceramic bodies, the extent of this plastic flow is apparently much greater, and therefore the change in the value of p will be very large. f(D,r) The factor can be computed from the geometrical form of the crack. a quantity am p can then be determined. The Two possible limitations exist in taking the mirror radius as an indication of breaking stress: 1) the glass specimen should be free from residual stresses, and ii) the dimensions of the mirror should be small in relation to those of the section of the specimen. In Shand's bending tests, in the limiting case, the mirror radius is fifteen percent of the diameter of the specimen. A typical mirror region of a glass fracture is shown in Diagram 8. Measurements of mirror areas were made by Shand55 by optical microscopy. Mirror radii were measured along two axes OA and OB displaced 45° from the center line of the mirror. This procedure tends to compensate for local irregularities of the mirror locus and for the lack of symmetry. 33 This asymmetry tends to increase with the angle 6 where 8 is the angle shown in Diagram 8. The mirror areas of glass fractures are formed because of the homo- geneity and continuity of the body. Such considerations may be carried to the structural levels. The mirror areas of the glass-ceramics which are composed of large crystalline particles in the glassy matrix, lack homogeneity and continuity of the body. Consequently, the detailed record on the mirror surface is lost, and the fracture Of the material extends over a large area. For certain ceramic materials, such as sintered alumina, no mirror region is formed at all. Shand55 has plotted the mirror radius versus breaking stress for glass and glass- ceramic specimens in Figure 9. Distribution of variance of individual test points is shown in Figure 10. For eighty percent of the glass speci- mens the variance for fracture-stress values Obtained from flexure tests are within i 47.. It is much larger in the case of glass-ceramic speci- mens (approximately fifteen percent). This method is probably adaptable to some glass-ceramic bodies, but the results will be less accurate than in materials which are completely glassy. 1.3 Objective of this Research The foregoing review of the present state Of the theoretical and experimental studies of materials containing glass and crystalline phases shows that the factors affecting the strength and the mechanism of fracture of glass-crystalline materials are not clearly understood. In the present research an attempt was made to understand some of the basic factors that affect the mechanical properties of glass-crystalline materials with the hope that the results may elucidate the basic mechan- ism 0f fracture of brittle materials. The material chosen for the present 34 Fig. 8. Diagram of mirror surface of glass fracture showing inner and outer loci and axes of measurement (Shand58) Fig. 9. Plot of mirror radius versus breaking stress for glass and glass-ceramic specimens 0.457" in diame er. Curves repre- sent corresponding mean values Of om(p) for flexure (Shand55). Fig. 10. Distribution of variances of individual test points from curves of Figure 9. / STIPP ORIGIN - O STRESS =¢mcos 9 LED NEUT a O. a; D O I .— I U) GD U C 5.. (n 0 .2. X ( U C O 8 .0I oz .03 .04 .05 .05 MIRROR RADIUS P-INCH +Is / / +Io , p7 [L’Gfass 7740 “+5 ’6 U 0"” z I q/, S o 5 ' A/ > -s ,2 In I 3 / /GLASS CERAMIC 9606 “'I0 / / I; ‘ / / -I5 5 I0 20 40 60 80 90 95 99 CUMULATIVE PERCENT 35 investigation, moreover, is an elemental glass-former, whose mechanical properties are not well documented. As a result, one of the aims was to understand the mechanical properties of this elemental glass-former, selenium, a commercially-useful material, especially in the electronic industries. The first objective of this investigation was to determine the mechanical properties of an inorganic linear-chain-structured glassy material, such as selenium, and to compare the results with those reported on glasses having continuous-network structure, such as oxide or silicate glasses. The second objective was to investigate the effects of the microstructure of partially crystallized selenium on its mechanical properties, and to correlate the results with reported inves- tigations on glass-ceramic composites having various crystalline phases diapersed in the network-structured glassy matrices. Since varied microstructures can be produced in selenium by different heat-treatment SChedules, it is possible to determine the effect of volume fraction, Size, and distribution of various crystalline forms present in an amor- phous matrix, on the mechanical properties of the material. The third objeet:ive was to determine the flexure strength of glassy and partially crystallized selenium from the dimensions of the mirror region of frac- ture surface. A further objective was to obtain an understanding of the basic fracture mechanism in such a glass-crystalline material. The fourth ob.‘lective was to study the mechanical properties of selenium in bulk form rather than in thin-film form. I I . EXPERIMENTAL PROCEDURE 2.1 Sample Preparation The selenium for this investigation was obtained from Kawecki Chemical Company in pellet form 99.99+°. pure. 2 . l . 1 Safety Precautions Selenium is an extremely hazardous material. According to Encyclo- pedia of Chemical Technology, one part per million of selenium hydride in the human body is fatal. Hence, the selenium was stored in sealed bottles, and was handled with extreme care. Melting was carried out in evacuated quartz tubes in a special furnace set-up described in detail later, This furnace was housed in an exhaust hood with exhaust fan Operating when melting was in progress. Specimens were ground and p011Sihed under water to prevent the selenium dust from floating in the air, During handling, rubber gloves were worn; nose and mouth were covered by breathing-protection devices. gdi Melt ing of Selenium The selenium was melted in special quartz tubes of the shape shown in Figure 11. These tubes were cleaned with a 107. solution 0f hydro- fluotic acid, washed with distilled water, and then dried in a warm Oven. This procedure removed the silicon dioxide particles present in thQSe tubes. A white powdery film of silicon dioxide forms on both the 1 “her and outer walls of the quartz tube during shaping with an 36 37 Fig. 11. Shape of quartz tube for melting selenium. 0.9 cm IO.I6cm———> 11, 20. 32cm. 38 oxy—acetylene torch. This film is easily soluble in the dilute hydro- fluoric acid. About 300 grams of selenium was put in each cleaned quartz tube. Approximately two inches of the tube length was left empty to accommo- date selenium vapors. The tubes were attached to the evacuating system to achieve a vacuum of 10.3 mm Hg, and then sealed. Melting of selenium was carried out under the following schedule. The sealed tubes were rotated for 24 hours at 300°C in the horizontal position in the tube furnace set-up shown in Figure 12. A motor mounted on the frame of the furnace provided the rotary action of the quartz tube. The entire fur- nace set-up was housed in an exhaust hood and all operations were carried out by remote control. Upon completing rotation of the quartz tube con- taining selenium for 24 hours at 300°C, the furnace was made vertical. The furnace temperature was raised to 600°C and held for 4 hours in this Position. At this temperature the liquid selenium was fairly fluid. The furnace was made horizontal and then vertical a number of times at this raised temperature, in order to drive locked vapors out of the melt. The temperature was taken back to 300°C, and the tube was held in the vertical position for another 20 hours. When the melt cools to 3000C, it becomes more viscous than the melt at 600°C. This lowering of temperature to 300°C helped in reducing the length of pipe that formed in the bulk glass obtained by quenching in ice water. Holding for 20 hours at 300°C helped in achieving an equilibrium structure for the melt before quenching. This step was essential to obtain bulk glass with reproducible structure and properties. 39 Fig. 12. Furnace set-up for melting selenium. _ I L v.24... LEE—mam g oziozuno mmfiw, «0.5.2 025.com . 842mg... .nI w >aac3m .6528 30.2mm . . a $38 mo... $.65 W t 84sz $58 Hi 3 625m I mm:... 83 oztfiom 1“... moezmaa . macsooozmmxt n5 .me moecimen, b is the breadth of the specimen, and d is the depth of the specimen . £§E_:;éu3 Experimental Method for Determining Fracture Surface Energy of Amorphous and Partially Crystallized Selenium In the present investigation, the effective surface energy (VI) was determined by the following method. Three amorphous selenium speci- tnem. with c/d = .1875, .1067, .3333, and five of different volume frac- ‘i ion of crystallites with old = .2365, .2136, .2143, .3679, and .3670, ‘E'Sarre selected. Values of OF’ the fracture stress, c, the notch depths, and E, the elastic modulus as determined from the notched-bar tIlreenpoint bend test, were substituted in Equation 28 (Appendix A) to (v ). ‘(313tain the effective fracture surface energy I 47 Fig. 13. Load—deflection curve for a typical brittle material (Davidge and Tappin62). 7 .: b 1 lb 1‘ ' T —————— -- 3"“ " .73 j: O’ 3 L o=—P —— F 2 dez 0 . U 8': 25 (d- c) LOAD P OEFLFCTION 6' 2.5 Fractograph Studies The fractured specimens were studied with a scanning-electron- microscope fractograph, by the method described in Section 2.5.1. To compute the breaking stress, the mirror radius was measured on each amorphous and partially crystallized specimen. The details of this nmthod are described in Section 2.5.2. 2.5.1 Preparation of SEM Fractographs One representative specimen for each temperature and time of heat treatment was mounted on the scanning-electron-microscope specimen Tholder with the fracture surface on top. The fractured surface was coated with carbon and then with Au-Pd alloy in an evacuated chamber. CEhe specimens were continuously rotated so that a uniform layer of the (boating was applied on the fractured surface. Generally a 100Ao thick layer was deposited to increase the secondary emissions from the surface Of the specimens. The fractographs were prepared under the following scheme. A lower (IESOX) magnification picture was taken to show the entire fracture sur- 2Ezace. Then three fractographs were made at higher magnification to ‘ITQeveal the finer details of the regions of the same fracture surface. TITluese areas were: i) origin of fracture, ii) mirror region, thii) rough region, and iv) the hackled region. 5gL;,5.2 Measurement of Mirror Radius from the Fracture Surface of Amorphous and Partially Crystallized Selenium Specimens After the flexure tests, the fractured specimens were mounted on ‘llne vertical stage of an optical microscope. The location of each frac- ‘ilare origin was determined. TWO fracture fronts, more or less distinct, 49 were observable on the mirror surface. The inner fracture front corre- sponded to the appearance of a stippled roughness on the surface. The outer fracture front represented the change-over from smooth-mirror region to a rough and shattered region showing deep furrows and highly irregular surface features. The method suggested by Shand55 was to rneasure the mirror radius of the specimens from the origin to the inner :Eracture front. Measurements of mirrors were made with a 50-power rnicroscope fitted with a calibrated reticule. III. RESULTS During the present experimental work, the mechanical-test specimens were heat-treated to preselected times and temperatures to obtain dif- ferent volume fractions and sizes of crystallites in the amorphous matrix. The first section of this chapter contains general observations on the heat-treated specimens. The second section presents microstruc- tural studies which are divided into two sub-sections, namely, qualita- tive and quantitative analysis. The third section contains the results of the bend tests. The fourth section presents SEM fractographs of the fractured surfaces. 3.1 General Observations The specimens that were heat-treated at 100°C developed large, deep surface undulations and exhibited a high degree of softening during heat treatment. The specimen shapes were maintained by externally wrapping the specimens with strips. The specimen heat-treated at 62°C showed very few wrinkles on the surfaces. This tendency to form wrinkles during heat treatment can be explained on the basis of the difference in densities of the crystalline selenium (4.82 gm/cm3 at 25°C) and the amorphous selenium (4.26 gm/cm3 at 25°C). With heat treatment, the amorphous selenium transforms to the higher-density crystalline form and, as such, the specimens shrink. The undulation on the surface of specimens cannot be caused by difference in thermal-expansion coeffi- cients (a) of the amorphous and crystalline forms of selenium, 50 51 because the difference is extremely small (aamorphous Se = 3.73 x 10-5 cm/cm.per oC , m w v N O O_ _ q comm 4 t . 8. (awn/5).) 141603113 amxald 82 Fig. 21. Plot of flexure strength as a function of volume fraction for specimens heat treated at 82°C. ON ex; mwtqnmemm no ZOFU m. w. v. N. O. m w v N — '1 a 1 J u .11 o 9 8 H19N3 HIS aanxau 00m ( .m/ 6») 00v 83 Fig. 22. Plot of flexure strength as a function of volume fraction 0 for specimens heat treated at 100 C. 3.. V mutunmuxam no zoFo ww we N¢ 0v mm mm in Nn On wN wN wN NN ON 9 w. 1 N. O. m w c N O 1 q u 1d1 _ q — a _ u a — q u a q T u . q - 8 J .8. H19N381$ NIXB'IJ 1 Con 0600. tun/On) § 84 Fig. 23. Plot of flexure strength as a function of volume fraction for constant-size crystallites (d8 = 12.75 and 15.78 pm). 3L mwkfismwrmm “.0 20:05“... 9231.0) w. v. N. O. m m v 0 ll! - . _ n n . — . . . a / I .0/ 1. 813.9 1 .6 -1... . é:~m.m_u.uulx 1 00. CON oon § (gum/bx ) o 8 HIONBHIS 380x 31:1 85 Fig. 24. Plot of flexure strength as a function of volume fraction for constant-size crystallites (d8 = 19.06 and 22.22 pm). 1'. x111 I"?! 1 1‘11 II): If! 7]. ITO/l / ,1 x/ / 414’ / //l o//. /, /- / / / S L S 1 .8 I 5 (tum/6111 HIONBHIS 380x313 l2 I4 16 1%) IO VOLUME FRACTION OF SPHERULITES 86 Fig. 25. Plot of flexure strength as a function of volume fraction for constant-size crystallites (d8 = 25.14 and 50.80 pm). CO. OON. 00m on 0.. on 8 o. a0\ov mwthmeQm “.0 202.0135. MEDJO> m. V. N. O. m 0 ¢ N O """"" .'-'IIIIII'II'"' * 1+, llll .l'", l o 1|II+T 1 +1! 1.1 1 . I I ' l 7' In 1 ’ ' I . J .2188"; 111 1. 5.3mm 1.6 .. 1... l 00¢ 00m HIONBBLS 38 OX 31:! (,un/ 5» ) 87 Fig. 26. Plot of flexure strength as a function of size of crystal- lites for constant volume fraction of crystallites (VS = 2.09 and 3.73 X). wN .VN NN :53 mutqammxam 11.0 mm...ws.<.o ON m. m. w. N. $2.6. 164.71... O\om0.N u a) II. x O. 00. B .1 3 w 1 CON 3 1 S I. a 3 1. 00» W l H 1 00¢ Anya W 1 wt 1, 00m L 88 Fig. 27. Plot of flexure strength as a function of size of crystal- lites for constant volume fraction of crystallites (VS = 6.55 and 8.15 Z). .23 mutuammxam to 552.45 mm 8 mm ow m. m. a N_ o. q a q 4 q q q 4 J u a q 00— l . 08 I 1.. I I. I I I .1 00m I Ik ” L. 1 cow L o\.. n. m u .> 111.. o\o mmd ..... 6) 1| 6 . com 330x313 H19N38 iS (3119/ 611 ) 89 Fig. 28. Plot of flexure strength as a function of size of crystal- lites for constant volume fraction of crystallites (VS = 14.16 %). .53 $535.18 so $5.245 Om mN ON vN NN ON m. w. v. d T _ 1 a — a T q _ q q 1 q T q 8— 1+1 111111111 +1.11 108 + 1+7, 1 + 111 1.1. 1.1.1+11 M / 18m +\\\\ I ,’+ \* +” J 11. ..00¢ a ..OOm A«62.41.1511... 1 aanxa'u HISNEHJ. S (imx 611 ) 90 Fig. 29. Plot of elastic modulus of partially crystallized selenium as a function of volume fraction of crystallites for constant-size crystallites (d = 12.75, 15.78, 19.05, 22.22, 25.14 and 50.80 11111). 3 (Kg/C1112 x I05) OF ELASTICITY MODULUS —-.-‘--.- KINGERY'S CURVES l7 -1 "0 0— MAXWELL AND EUCKEN ‘ CURVES '5 , -_.... EXPERIMENTAL CURVES 1 9 l3 " sup 9’ 2: x ‘1» .. \ II 11 93 1 / / 4 - ,o’ 1"/' ,0’ 9 '1 3‘ A; d, I ,1), 1» .’ ‘0 “t; \Vq’ 1 ‘7. q '0 . I 7 It a? 60*; _ / 6‘; ‘9 1” b" x 9 5 < X / " \3 <5 to/ 6 fi ‘5 1 1 O 2’ ‘\ 1‘ T C 9 / $ ’1 ‘1 V1: 0 I; ¥\ ‘\ up ., x \1 3 d 1: o o .x‘m‘fi 6 r” ’6,’ . l/ x, .I/ ’ 0’ , . f / , / | h ‘,o.'::./ __,.-"' __ 0 IO 20 30 4O 50 60 7O 80 90 IOO VOLUME FRACTION OF SPHERULITES (°lo) 91 Fig. 30. SEM fractographs of amorphous selenium. (A) Total field of fracture (30x). (B) The fracture propagates with no preferred direction shown at location X. (C) The fracture consists of coarse and mirror regions shown at location Y. 92 Fig. 31. SEM fractographs of partially crystallized selenium obtained by heat treating at 62°C for 200 hr (A) Total field of fracture (30x). (B) Origin of fracture (200x). (C) Spherulites in hackle region of fracture (500x). 93 Fig. 32. SEM fractographs of partially crystallized selenium obtained by heat treating at 62°C for 250 hr (A) Total field of fracture (30x). (B) Origin of fracture (200x). (C) Spherulites in hackle region of fracture (500x). 94 Fig. 33. SEM fractographs of partially crystallized selenium obtained by heat treating at 62°C for 311 hr (A) Total field of fracture (50x). (B) Origin of fracture (500x). (C) A spherulite in hackle region of fracture (2000x). Fig. 34. 95 SEM fractographs of partially crystallized selenium obtained by heat treating at 100°C for 45 min (A) Total field of fracture (50x). (B) Origin of fracture (lOOOx). (C) A spherulite in the mirror region (lOOOx). (D) A spherulite in the coarse region (lOOOx). Fig. 35. 96 SEM fractographs of partially crystallized selenium obtained by heat treating at 1000C for 1 hr (A) Total field of fracture (50x). (B) Origin of fracture (lOOOx). (C) A spherulite in the mirror region (lOOOx). (D) A spherulite in the coarse region (lOOOx). 97 Fig. 36. SEM fractographs of partially crystallized selenium obtained by heat treating at 100°C for 1% hr (A) Total field of fracture (30x). (B) Origin of fracture (500x). (C) Spherulites in the mirror and coarse region (500x). (D) Spherulites in the hackle region (500x). 98 Fig. 37. SEM fractographs of partially crystallized selenium obtained by heat treating at 82°C for 4 hr (A) Total field of fracture (50x). (B) Origin of fracture (500x). (C) A spherulite in coarse region of fracture (lOOOx). (D) Spherulites in hackle region of fracture (500x). Fig. 38. 99 SEM fractographs of partially crystallized selenium obtained by heat treating at 82°C for 6 hr (A) Total field of fracture (50x). (B) Origin of fracture (2000x). (C) Spherulites in coarse region (lOOOx). (D) Spherulites in hackle region (lOOOx). Fig. 39. 100 SEM fractographs of partially crystallized selenium. obtained by heat treating at 82°C for 8 hr (A) Total field of fracture (30x). (B) Origin of fracture (500x). (C) Spherulites in coarse region of fracture (500x). (D) Spherulites in hackle region of fracture (500x). 101 Fig. 40. SEM fractograph showing the details of fracture surface in and around a spherulite (2000x). 102 Fig. 41. SEM fractographs showing (A) A spherulite in the mirror region of fracture for specimen heat treated at 82°C for 6 hr (2000x). (B) Spherulites present at the origin of fracture for specimen heat treated at 82°C for 6 hr (2000x). 103 Fig. 42. SEM fractographs showing (A) The origin of fracture for specimen heat treated at 82°C for 8 hr (500x). (B) The hackle region of fracture for specimen heat treated at 82°C for 8 hr (lOOOx). 104 Fig. 43. SEM fractographs showing (A) A spherulite in the mirror region of fracture for specimen heat treated at 82°C for 8 hr (1500x). (B) A spherulite in the coarse region of fracture for specimen heat treated at 82°C for 8 hr (1500x). (C) Two spherulites in the mirror region of fracture for specimen heat treated at 82°C for 8 hr (1500x). Fig. 44. SEM (A) (B) (C) 105 fractographs showing A spherulite in the mirror region of fracture for specimen heat treated at 100°C for 45 min (2000x). A pocket left by spherulite in the mirror region of fracture for specimen heat treated at 100°C for 45 min (2000x). A spherulite in the mirror region of fracture for specimen heat treated at 100°C for 1 hr (2000x). IV. DISCUSSION As discussed in Section 1.3, the objectives of this research were to evaluate the flexure strength and elastic properties of partially crystallized selenium as a function of: i) the elastic properties, ii) the thermal-expansion coefficients, iii) the size, shape, and distribution, and iv) the volume fraction of the component phases. The results of the flexure strength and the elastic properties Of partially crystallized selenium were presented in Section 3.3. In this chapter, Section 4.1 discusses the microstructure features observed in the heat-treated specimens. The theoretical predictions of the elastic properties are discussed in Section 4.2. The factors that control the flexure strength are examined in Section 4.3. In Section 4.4, the frac- tographs are analyzed, and the flexure strength as determined by the mirror radius is compared with the experimental results. The last section, 4.5, compares the flexure strength and elastic properties of partially crystallized selenium obtained in this work with the flexure strength and elastic properties of certain well-known glass-ceramics and glass-crystal composites. 106 107 4.1 Microstructure of Partiallprrystallized Selenium The photomicrographs of heat-treated samples were shown in Figures 16, 17, and 18 in Chapter II. The size, shape, and distribution of crystallites can be discussed on the basis of these photographs. The effect of internal stresses (due to differences of thermal-expansion coefficient between glassy phase and crystalline phase), the stress con- centration (due to differences in elastic properties), and the crystal- glass bonding (due to difference in wettability of the glass and the crystalline phase) on the flexure strength and fracture of the partially crystallized selenium can also be analyzed. The flexure strength and elastic properties are dependent on the microstructure of partially crystallized selenium. The microstructure is defined by the size, shape, and distribution of the crystalline phase in the glassy matrix. From Figures 16, 17, and 18, the shape of the spherulites is invariably spherical in every specimen. Hence shape of the crystallites is not a variable in the present work. The spherical shape of the crystalline phase helps considerably for making valid com- parisons with the theoretical studies where the shape of the crystalline phase is generally assumed to be spherical. The distribution of the crystalline phase, as can be seen in detail in Figures 16, 17, and 18, can be considered uniform for specimens heat-treated at both 62°C and 82°C. Clustering of spherulites occurs only for the large-size crystals (approximately 30 to 50pm). These large-size crystals were present in specimens heat-treated at 100°C. Higher growth rate and lower number of crystallites per sq. mm. at this temperature contributes to such clus- tering. The heat-treatment schedules were selected to obtain widely scattered crystalline phases and to have very few clusterings. 108 Frey and Mackenzie63 have found that the flexure strength and the elastic modulus of glass-A1 O3 and glass-ZrO composites decrease dras- 2 2 tically if internal stresses due to the difference in thermal expansion of the glass and crystalline phase were sufficient in magnitude to cause cracking of the glassy matrix during the preparation of the specimens. Binns64 reported that glass-A1 O3 composites with angular Al O grains 2 2 3 cracked on cooling in cases where the thermal-expansion coefficient of the glass was higher than that of A1 These cracks have originated 203. from the internal stresses generated in the composite as it cooled from its fabrication temperature. High tangential stresses at the glass-A1203 interface are considered to be responsible for these cracks in the matrix. If the crystalline phase has a higher thermal expansion coeffi- cient than the glassy phase, however, the cracks should originate around the crystalline phase at the glass-crystal interface. High radial stresses at the glass-A1 interface are considered to be responsible 203 for these circumferential cracks in the glass. The microstructures of partially crystallized selenium shown in Figures l6, l7, and 18, do not display any evidence of cracks in the glassy region or at the glass- crystal interface. This result can be explained by determining the radial and tangential stresses due to the difference in thermal-expansion coefficients of glassy and crystalline selenium. These stresses, according to Selsing's65 equation, for the case of a spherical shape inclusion, are given by: O = -2 ‘ -(am - OR 55: r °t 1 + v 1 - 20 3 111+ 2 E E m p 109 Thermal expansion coefficient of matrix; 81 m H m Q s u up = Thermal expansion coefficient of crystalline phase; vm = Poisson's ratio of crystalline phase; Em = Elastic modulus of matrix; Ep = Elastic modulus of crystalline phase; R = Radius of crystals r = Radial distance from center of crystal to a point in glass matrix; or = Radial stress; and at = Tangential stress. If the thermal expansion coefficient 01p) of the trigonal selenium in this equation is taken to be the thermal-expansion coefficient of spherulites, then. up (37.79 x 10-6 cm/cm/OC) is slightly greater than am. (37.73 x 10.6 cm/cm/OC). In this case, a very small tensile radial stress would exist at the glass-crystal interface, and if failure occurs, the crack will have to propagate circumferentially around the spherulite. In microscopic studies there was no evidence of any circumferential cracks around the spherulites. The magnitude of internal stresses was not high enough to produce premature cracks in the specimen, as can be verified from the microstructures given. If the spherulites are consi- dered to be made up of lamellas of trigonal selenium arranged annularly, as suggested by Fitton and Griffithsla, then by the expected value of thermal-expansion coefficient of spherulite shown in Section 4.3.1, the magnitude of internal stress in the radial direction should have increased. This increase again would result in a circumferential crack at the glass-crystal interface. The microstructures do not show the existence of any cracks in the glassy matrix or in the glass-crystal 110 interface. It can be concluded that premature cracks caused by internal stresses do not exist in partially crystallized selenium. Evidence of good interfacial bonding is shown in Figure 45 by the adherence of the glassy selenium layer onto the spherulites of selenium. The fractured surface of the specimens, when observed under the micro- scope, do not show any evidence of separation of spherulites from the glassy matrix. All these facts suggest that the crystalline phase could be the weakest area in partially crystallized selenium. 4.2 Elastic Pr0perties of Partially Crystallized Selenium The elastic properties of partially crystallized selenium could be calculated from the elastic properties of its components by using models (Hashin'336, Paul's4o, Maxwell and Eucken'g? Hashin and Strikhman'337and Kingery's).The elastic moduli of heterogeneous materials were theoreti- cally determined by Hashin36, (1962) with a concentric-sphere model. The upper and lower bounds for the elastic moduli of two-phase composite materials were obtained by an approximate method based on variational theories in elasticity. Hashin36 determined these bounds for the effec- tive elastic moduli of the two-phase composite by considering the change in strain energy in a loaded homogeneous body due to the insertion of nonhomogeneities. It is assumed that (a) the particles are spherical, and (b) the action of the whole heterogeneous material on any one inclusion is transmitted through a spherical shell lying in the matrix. For those composite materials consisting of a matrix in which per- fectly spherical inclusions are embedded, the analysis does not involve 111 Fig. 45. Polarized-light micrograph of spherulites of selenium illustrating the absence of separation at the glass- crystal interface. 112 any approximations. The upper bound of the bulk modulus of a two-phase composite, k is given by the expression 1, K m 1 7 (X -K)(4G +3K)c 1+ 111 p m m 1((40 +3K)-4G(K-K)c m m. p m m p and the lower bound of the bulk modulus of a two-phase composite, k2, is given by (4G + 3K )c K ) m m m 4G + 3K + 3(K - K )c m P m P k = Km + (Kp - where c is the volume concentration of inclusions, K.In is the bulk modulus of matrix, KP is the bulk modulus of inclusion, and Gm is the bulk shear modulus of matrix. With the results obtained for the bulk modulus, Young's modulus for the composite can be calculated from the expressions: E = - ’ '-"- —————m E 3(1 2v)K , Gm 2(Vm + 1) where K is the bulk modulus, E is the Young's modulus, and v is Poisson's ratio = 0.3 (assumed for amorphous phase) = 0.25 (assumed for crystalline phase). The treatment becomes exact if the distributed phase occurs as perfect spheres. 40 Paul (1960) has also determined the bounds of Young's modulus of two-phase composites by the Variational theories of the theory of elas- ticity. He assumes that the stresses are the same, and are both tensile 113 or compressive in matrix and particles. Both the matrix (material 1) and the dispersed particle (material 2) are assumed to be linearly elastic and isotropic. The upper and lower bounds of elastic modulus of two-phase composite are given by the following expression. 1 f + 1-f E E g LE1f+ME2(l-f) LE1 M E2 1 - v + 2m(m - 2v ) 1 1 'where L = 1 _ v _ 2v 2 , 1 l l - - _ v2 + 2m(m 2v2) M - . 2 , 1 - v2 - 2v2 m _ v1(l + v2)(l - 2v2)fE1 + v2(1 + v2)(1 - 2v1)(l - f)E2 (l +-v2)(l - 2v2)E1f + (l + v1)(l - 2v1)(l - f)E2 ’ v1 is Poisson's ratio of the matrix, v is Poisson's ratio of the inclusion, f is the volume fraction of inclusion, E1 is the elastic modulus of the matrix, and E2 is the elastic modulus of the inclusion. These upper and lower bounds are generally too far apart to give a good estimate of the effective Young's modulus. With Young's modulus of glassy selenium taken as E8 = l x 105 kg/cm2 and that of crystallized selenium as Ec = 14 x lOSkg/cm2 (determined from Section 3.3.2), the upper and lower bounds of elastic modulus of the composite as suggested by the model of Hashin and of Paul were calcu- lated. These values are presented in Table 9. In Figure 46 the calcu- lated values of elastic modulus are plotted as a function of volume 114 oo.qH oo.¢H oo.¢N oo.¢N oo.¢H oo.¢H OOH mam.aa moa.e NeH.oH oe maa.oH eam.m Haw.aa om oo.m emm.m eew.m ea.oH on om.m oN.e om.~ ow.m oe oo.e Ne.a oe.ma oo.~ om.e HANN.~ meH.ea on oo.m mm.a oa.oH me.H mN.e Humm.a meae.m oe emN.N oe.H mm.a om.a oo.n teem.a mmme.~ om me.a emm.a mm.m m~.H me.m Hemm.a «mme.a om mmm.a eoH.H om.m OH.H om.~ eNmH.H emaa.a oH oooo.a oooo.a oooo.a oooo.a oooo.a oooo.~ oooo.a oooo.a o «mace «meme mazes meme: amass meme: mazes «wee: oneoeme A se\wa meme Amsu\wa moHv Amae\ma moHv Am66\wa oHv zme em 8 aamzxez eemosz asee szm mzeuo> mcowumHom www.coxosm pom HHoaxmz pom oom.%pomcwx .oqm.Hsmm .omm.sw&mm: mo muowumfiom use he woumHsono msasuoz owummfim m MAm x .\.o.m_v .> . 00¢ no $5233 1 00.. 1 CON 4 00V 1 00m H19N381$ 380x313 (gm/6») 137 TABLE 12 Growth Rate and Number per sq. mm. of Spherulitic Selenium at 100°C, 82°C, and 62°C. (A) GROWTH RATE OF SPHERULITES Tem erature Size after Different Average Growth Efiirtfi: p Heat Treatment Times Rate for Each Melt a a ( m/hr) Each Temp. 0 (Hm/hr) 45 min 1 hr 1% hr 15.89 29.55 50.80 42.50 100°C 12.75 21.16 44.45 46.64 48.78 15.87 22.22 50.82 57.20 4 hr 6 hr 8 hr 19.05 22.22 25.4 1.59 82°C 19.05 25.40 31.75 3.175 2.118 19.05 22.22 25.40 1.59 200 hr 250 hr 311 hr 12.75 15.80 25.4 .1574 62°C 8.46 15 87 25.4 .1562 .1570 12.75 15.80 25.4 .1574 138 TABLE 12 (Continued) (B) NUMBER OF SPHERULITES PER SQ. MM. IN MELT A Number of Spherulites Temperature per sq. mm. after Different Heat-Treated Times 45 min 1 hr 1% hr 100°C 17.24 22.99 40.23 4 hr 6 hr 8 hr 82°C 20.11 28.74 48.85 200 hr 250 hr 311 hr 62°C 66.09 77.59 100.57 139 of the amorphous and partially crystallized selenium specimens. The amorphous selenium specimens show a higher value for the most probable flexure strength (approx. 450 kg/cmz), and a large scatter in the results as shown by the larger half band width of the peak (250 kg/cmz). With increasing size of spherulites the flexure strength decreases, reaching the lowest value for 50.80pm size of spherulites. In Figure 51 it is also seen that the most probable flexure strength of 0.00 size of spherulites agrees with the most probable flexure strength evaluated from Weibull's curve for amorphous selenium. Another important fact can be obtained by plotting the half band width of the Weibull's curves as a function of various sizes of spherulites. In Figure 52 such a plot is shown. The trend of the curve shows that half band width decreases with increasing size of crystallites in partially crystallized selenium specimens. Another approach would be to see how the flexure strength changes with increasing size of spherulites for different volume fraction of spherulites. In Figures 32, 33, and 34 the flexure strength is plotted as a function of size of spherulites for 2.09, 3.73, 6.55, 8.13, and 14.16 volume percent of spherulites. As can be seen from the figure, the flexure strength decreases for volume fractions 2.09, 3.73, and 6.55, with increase of size of spherulites of selenium. Weibull's curve has been plotted for this case also, and is shown in Figure 53. The most probable flexure strength is also plotted. The flexure strength decreases with increase of volume fractions. The most probable flexure strength, as predicted from the intersection of the curve with 0.00 volume fraction of crystallites, agrees with the most probable flexure strength determined from Weibull's curve for amorphous selenium. The 140 Fig. 51. Plot of frequency of fracture as a function of flexure strength of amorphous and partially crystallized selenium (for constant size of spherulites). I, constant 12 75 15.87 19 05 22 22 3. 56’ _ g a 8 a a 9 8 b 8N 3 a :3 [-N I 555 s s E e 9" " x’ 8 I’ ‘60 II '. ,+ ,.~ - 1’ . ' l/ O I, s../ ’ ‘8A 0' .0. .’../” “CE: 0.0. ./ I ~ 3 ,".A.’ g I. y ” -8 . I. / \\o V 00;. . ’ "' \\ E '7’" /’.4 ’0 ‘k - 0 tr: ’lv" \\ "" Z Xx;‘ ___,,.—f' _O&-’ ---- .---’-.‘.-o“:_--=.d-_.¢-‘-'-- ‘ 3*. z.-- ’\ 0° .‘ + . w w... ‘t ‘5‘: '\ ' _ ~~~ ‘s- ‘ . “J .‘~. -‘~:‘\ \ K .... ~‘~§-\ .\ 83 Q... ' . . - X ~~~ 58“.& NW “Q ..’0 ‘ 4 k“ 0°:\Q§ _ “- ‘3‘ 8 |_ n l 1 n I l I n l ' o ‘ ‘ o P. 3 8 8. 8 3 (’N) 380101783 .10 AONBOOEHJ 141 Fig. 52. Plot of half band width of Weibull's curves of Figure 51, as a function of size of spherulites. O .0 [O L. ,' 0 + +8 0” OJ." " x ----- 3 L l i 1 L 1 1 l 1 O 53 8 B 8 2 ° (“1") SBlIWflBBHdS :10 BBIBWVIO O __O 10 #0 .23 '0- ,’ 9’ A8 0" ’ m x ............... -‘a‘ r _.L l 1 l l 1 1 1 1 8 .9 8 8 8 9 0 ~ (“1") SBll‘lflBBHdS :10 BBIBWVIG WIDTH HALF BAND MOST PROBABLE FLEXURE STRENGTH (kg/cm?) ( kg /cm?‘) 142 Fig. 53. Plot of frequency of fracture as a function of flexure strength of amorphous and partially crystallized selenium (for constant volume fraction of spherulites). + 0 X 0 d '3 O E )2 .2 32 R m 0’ n '0 $2 3 > CQFT'Q d N m ‘0 v- < a. O O O \3 8 w 5316 o F) N (‘0 N '3' 'U - c o O 0 ’ — m O EE§§ :9 s A x I 1” I o ’l .0 I"’ -. I 0 I, ,6 I ..° I’ '3'. 0" 'I ‘0'. .’ 1 ,.°'o ./ O a 1 ,.° ./ ‘,4’ \..-' pr' av .0... .‘ ’0’ ’ I x ..--"o \\ .z" / o' 9000...... ,A‘./ ’ O. .00 .’.’.’ ) .0 .’O’ A, \‘ fl .0 .’o’ ’ ‘ . to.’ ’ ~ ' O ...'000 ’ ~ 4"...Oo...... . ~~~ ’ 00...... .x ~‘ ~. . .o""-... ‘\ ‘0‘. .# °. 4‘ \ \'\' 0' ‘ \ ’0‘ 'Q B ‘\ C o \‘ .x. \.. .- \ . \, . \ .. \ o \ . . .. O \ o ‘. I. ‘ \ ..x \ e \ 0. '9 \ ,K '..., . q o ‘T -€f\. . a e\ \ \ L J 1 1 1 l 1 l l l 1 l 1 '0 0 ID 10 0 ID '°. '0 N 8 —. "- Q (fN) sanlovua 30 ADNBOOEHJ 600 700 400 500 (kg Icm'i) 300 STRENGTH ZOO FLEXURE IOO 143 Fig. 54. Plot of half band width of Weibull's curves of Figure 53, as a function of volume fraction of spherulites. 3:9.\ 9: FEuxoxv Ibo—3 oz v. N. o. m m i. m o q q d 4 d d d A: d 4 q d1 on d .i...‘....'..'.'.."..” -'. "00' A C'OOUIOOOIOIICIIOUI'UI- -III’" III " I, .. z - 2. ”I OI I I I I I I I I [IL _ I I I I 159 the partially crystallized selenium specimens. The flexure strength of the crystallized specimens is lower than that of the amorphous specimens, and they have a lower value of' om(p)15 = 66 to 70. In Figure 61, the 35 values of anyp) are plotted as a function of volume fraction of spherulites. The curve indicates that the amorphous selenium has the highest value of amfip)% and, with increase of crystalline phase in the 3 amorphous matrix, the value of’ cufip) decreases non-linearly. The specimens with 45 to 50% crystallinity do not show any mirror area. With increased volume fraction of crystallites, the strength decreases and the mirror radius increases to large values. As a result, no mirror area is formed. From results shown in Table 14 it can be seen that specimens having greater than 14% volume fraction of spherulites do not possess mirror areas. The actual flexure strength determined by bend test is compared with the flexure strength determined from the dimensions of fracture mirror. The i 10% error noted may be due to the following factors: a) the partially crystallized selenium specimens may not possibly be free from residual stress, because of the complex stresses introduced owing to the crystalline phase. b) the mirror dimension increases with increase of volume fraction of the crystalline phase. Shand55 has noted that the relaxation of the applied force occurs when the fracture of the crack becomes large. For the bending tests, the limiting dimensions of the mirror radius was taken as 15% of the diameter of the specimen. In the present work, the ratio (r/D) varies from 5 to 20%, and hence the error due to relaxation of applied force does affect the flexure strength calculated from the mirror 160 radius. The loading rate was so selected to make the time of test as short as possible (15-30 seconds in the present work). The rate of loading minimizes the error created owing to relaxa- tion of breaking load resulting from large fracture crack. c) the partially-crystallized selenium specimen may not record the extent of mirror region, sharply owing to the presence of dif- ferent size and volume fractions of crystalline phase in the glassy matrix. This factor may introduce a large margin of error, as compared with the results of the amorphous selenium specimens. 4.5 Comparison of Flexure Strength and Elastic Modulus of Amorphous Selenium and Partially Crystallized Selenium with Glassy_Materials and Glass-Crystalline Materials Desirable physical properties not provided by single-phase materials ' often can be conveniently attained by multiphase composites. For example, oxide glasses can be strengthened by the introduction of spherical crys- talline inclusion363. Unlike this case, the introduction of spheru- lites in the glassy selenium matrix decreases its flexure strength. The elastic modulus of glassy selenium, as in the case of oxide glasses, increases with the introduction of the spherical crystalline phase. 4.5.1 Comparison of Flexure Strength and Elastic Modulus of Amorphous Selenium with Oxide Glasses In Table 15, the flexure strength and elastic modulus of a number of oxide glasses are presented from the work of Davidge and Tappin62 and from Frey and Mackenzie63. The flexure strength and the elastic modulus of linear-chain-structured amorphous selenium as determined in this 161 TABLE 15 Flexure Strength and Elastic Modulus of Certain Glasses FLEXURE ELASTIC MATERIAL STRENG H MODULUS (kg/cm ) (kg/cmz) ' 6 Glass (soda-lime type) 0.71 x 10 Class I (6810) 6 (SOda-zinc type) 504 0°658 X 10 Class II (7740) 6 (borosilicate type) 490 0.637 x 10 Class III (1990) 6 (potash-soda-lead type) 441 0'588 X 10 (ICI Perspex sheet) 0-0303 X 10 Amorphous Selenium 480 to 520 .07 to 0.10 X 106 162 investigation are exactly the same range of some of the network- structured oxide glasses, such as soda-lime, soda-zinc (glass I), borosilicate (glass 11), and potash-soda-lead glasses (glass III). 4.5.2 Comparison of Flexure Strength and Elastic Modulus of Partially Crystallized Selenium with Glass-Ceramics and Glass-Crystal Composites An interesting comparison can be made with the results of the pre- 62 sent investigation and those of Frey and Mackenzie , who worked with glass-Al O3 and glass-ZrO composites. Table 15 lists the pertinent 2 2 physical properties of all glasses and the crystalline phases studied. The size of spherical inclusions varied from 125 to 150nm, and volume of the crystalline phase varied from 20 to 40 volume Z. In the present work, the size of spherulites of selenium varied from 12.75 to 50.8um, and the volume fraction of crystallites varied from 0 to 50 volume %. It is worth noting in Figure 62, reproduced from Frey and Mackenzie62, that both glass I and glass II were strengthened by the A1203 inclusions even though internal stresses of considerable magnitude were present. In the tangential direction of these inclusions, a tensile stress will develop, since Ao is positive but small. Class III is a typical example where A0 is very large and op positive, where up is the thermal expansion coefficient of the crystalline phase, oml is the thermal expansion coefficient of the glassy phase, and do = (op — am). The large value of Ad results in a large tensile stress in the tengential direction at the glass-crystal interface. As a result, the flexure strength tends to decrease up to 20 volume percent of crys- talline phase, and then increases. The lack of strengthening in this composite series is attributed to the cracks present in the glassy matrix 163 TABLE 16 Pertinent Physical Properties of Certain Oxide Glasses and Crystalline Phases GLASS I GLASS II GLASS III A1203 ZrO2 Amorphous Hexagonal 6810 7740 1990 Selenium Selenium Density . . . . 5.6 . . (g/cm3) 2 65 2 23 3 47 3 91 5 4 3 4 81 Strain Point (QC) 490 515 330 Softening Point (QC) 770 820 500 40-50 Melting Point (CC) 2000 2550 217 217 Elastic Modulus 2 0.658 0.637 .588 3.85 1.42 0.07 0.588 E(106 kg/cm ) Linear Coefficient of Thermal 6.9 3.3 12.4 5.47 8.85 37.73 37.79 Expansion (00 x 10-6) 164 Fig. 62. Plot of flexure strength versus volume fraction of crystallites (Frey and Mackenzie62). Fig. 63. Plot of elastic modulus versus volume fraction of crystallites (Frey and Mackenzieéz). AVERAGE MODULUS or RUPTURE (PSI I I6”) EFFECTIVE ELASTIC MODULUS IPSIDG") LEGEND COMPOSITE go: g; g 2 0 GLASS I’Alzos +L43 -8,540 44,270 D GLASS II—AIz o, -2.I7 “3,850 45,975 A GLASS m—AI, 0, +6.93 44,800 “2,400 o.__ J l l I 0 0' 0.2 0.3 0.4 0’ VOLUME FRACTION OF DISPERSED INCLUSION / / / 6 / ’ LEGENQ ’ GLASS 1 - AI203 _— THEORETICAL 0 EXPERIMENTAL 8’ GLASS I - 2:0, — — - THEORETICAL 0 EXPERIMENTAL l— 1 6 1 ‘ ‘ 0 m 0.2 0.3 0.4 VOLUME FRACTION OF DISPERSED INCLUSION 0.5 165 before flexure-strength tests were performed. In the case of glass 1, there was tensile stress in the tangential direction, but it did not cause premature cracking in the glassy matrix; thus the flexure strength was unaffected. Hence, internal stresses were found to reduce flexure strength only if they were large enough to cause premature fracture of the glassy matrix. In the case of partially crystallized selenium, the microstructure shows no premature fracture in the glassy matrix for volume fraction of crystalline phase varying from 0 to 50%. The linear thermal-expansion coefficient (op) for trigonal selenium is slightly higher than that for amorphous selenium. The spherulites of selenium, because of the internal arrangements of the trigonal selenium lamellas, show a mechani- cally weak peripheral region, where invariably the fracture originates. The tangential direction would have a tensile stress, and would tend to produce initially radial paths of fracture. The flexure strength of partially crystallized selenium decreases with increase of volume frac- tion of crystallites. This decrease can be explained by the same reasoning used to explain the behavior of the oxide glasses containing crystalline inclusions having higher expansion coefficient. The only difference in the present system is that the crystalline phase is weaker instead of the glassy phase. In Figure 63 the elastic modulus of glass-A1 and glass-ZrO has 203 2 been plotted as a function of volume fraction of crystalline phase. The plot of elastic modulus of partially crystallized selenium versus volume fraction of crystallites is shown in Figure 29. The trend of the elastic- modulus curve for partially crystallized selenium is similar to that for oxide glasses containing crystalline inclusions. For low volume fraction 166 (less than 10%) and small size of crystals, the elastic modulus varies linearly and rises sharply. A similar comparison cannot be made in glass-crystal composites since no results exist for this size and volume fraction of crystals. However, the elastic-modulus curve of partially crystallized selenium containing larger-size crystals (50.80m) with volume fractions greater than 10% is similar to that of glass-A1203 composite. The results of the present investigation can be compared with results of Hasselman and Fulrath45, and of Bertolotti and Fulrath70. Hasselman and Fulrath45 plot flexure strength of glass--A1203 and glass- pore composites, as shown in Figure 64(A) and (B). Bertolotti and Fulrath70 plot flexure strength of glass-pore composite, as shown in Figure 64(C). The variation of flexure strength of partially crystallized selenium.as a function of volume fraction of crystals (Figure 23) agrees with the plot of Bertolotti and Fulrath70 of flexure strength versus volume fraction of pores (pore size varying from 20 to 186nm). The reduction in flexure strength with increasing volume fraction of pores (60pm) in Figure 64(B) has been explained by Hasselman and Fulrath45, as attributable to the stress concentration developed around the pores. Flexure strength of glass-pore composites (A) decreases severely when pore size (p) is much greater than Griffith's flaw size of glass, (B) is unaffected when pore size (p) is much less than Griffith's flaw size of glass, and (C) decreases slightly between case (A) and case (B) when pore size (p) is approximately equal to Griffith's flaw size of glass. 167 In case of partially crystallized selenium specimens, however, the size of Griffith's flaw present in the amorphous region is always much smaller (less or equal to Sum) than the size of Griffith's flaw in the spherulites (greater or equal to 30pm). Consequently, flexure strength of partially crystallized selenium decreases with increasing volume fraction of crystallites. The decrease of flexure strength of partially crystallized selenium is dependent on the flaw size in the spherulites, which in turn is determined by the size of spherulites. Pore sizes of 5 to 10um strengthen the glass, as seen in Figure 64(C). Contrary to this observation, increasing volume fraction of small-size spherulites (dsjg 15.0um) tend to drastically reduce the flexure strength of partially crystallized selenium. Bertolotti and Fulrath have explained the strengthening of glass by addition of small-size pores (5 to loum) to be due to nickel spheres (the agent for creating the pores) not separa- ting out from the glass matrix. In lithium-aluminosilicate glass-ceramic the flexure strength decreases with increasing volume fraction of crystallite71. In this glass-ceramic the crystalline phase has a lower thermal-expansion coefficient compared with that of the residual glassy phase. As a result, the flexure strength decreases with increasing volume fraction of beta-spodumene crystals. In partially crystallized selenium specimens, the crystalline phase can be considered to have thermal-expansion coeffi- cient greater than that of the glassy phase; in this way, flexure strength decreases with increase of volume fraction of crystalline phase. It has been shown by Phillips72, Watanabe73, MCMillan7l, and Freiman and Heinch48, that flexure strength of certain glasses increases with increase of volume fraction of crystalline phases. McMillan found 168 Fig. 64. Plot of (A) Uniaxial and biaxial strength of a soda-borosilicate glass containing alumina spheres 60p in diameter (Hasselman and Fulrath45). (B) Uniaxial strength of a soda-borosilicate glass containing spherical pores 600 in diameter (Hasselman and Fulrath45). (C) Uniaxial strength of sodium-borosilicate glass containing spherical pores (Bertolotti.and Fulrathyo). (A) Strength (psi x I03) I I I I I I4 _ I I I I A II Y2 IO“— ‘1 It 1 ' 3 II _. - I: - 5 8 g n '0 — .2; Es In 6 -— ‘i .— .‘s’ .5 ixf E . Uniaxial _ 5 4 —- \ I - \ D Bioxiol \\ i 2 — \‘ - O l l I l l 0 DJ 0.2 0.3 0.4 0.5 I I I 1 0 Volume fraction Alzos 0 DJ 0.2 0.3 0.4 0.5 Volume fraction Spherical pores STRENGTN 00' PSI) 02 O norm”, s-IOu s-IOII (alcohol mend) IO'ZOp IO'SCp u-«p ro-IOSu 05d“! DIOOODQI I ' l I 5 I0 20 30 4O VOLUME PERCENT POROSITY 169 that LiZO-ZnO-SiOZ-PZO5 type glass showed an increase of strength from 1800 to 3500 kg/cmZ, as a result of heat treatment at 600°C for 1 hour. The differential contraction of crystalline and glass phases causes a favorable system of microstresses to deve10p in the glass-ceramics. This system results in the increase of flexure strength. Freiman and Hench, working with a series of glass-ceramics developed by heat- treating a glass of molecular percentage composition (L120)33(8102)67, showed that mechanical strength increases with increasing volume fraction of crystalline phase. The volume fraction of crystals ranged from 0.5 to 0.95, the crystal sizes varied from 4.2 to 61pm. The Griffith flaw size for the glassy phase was 85pm. The strengthening effect is greatest for small-size crystals, because fracture is controlled by the initiation of flaws in the glass phase rather than in the lithium disilicate spherulites. Although the microstructure of partially crystallized selenium is very similar to that of (L120)33(8102)67 system, the flexure strength of partially crystallized selenium decreases with increase of volume frac- tion of spherulites. The spherulite of selenium is the weakest phase in the partially crystallized selenium specimens. Consequently, fracture invariably originates in the spherulites, and the flexure strength of partially crystallized selenium is always lower than that of amorphous selenium. V. CONCLUSIONS In an elemental glass-former selenium, the spherulites that are pro- duced by heat-treating the glassy selenium weaken the glass-crystal composite. As a consequence, partially crystallized selenium is mechanically weaker than amorphous selenium. The fracture nucleates in the spherulitic region and prefers to pro- pagate through the crystallized regions, indicating that certain regions of spherulite are weaker than the glassy matrix. The peri- pheral regions of spherulites appear to be mechanically weaker than the core. The fracture originates from microcracks that develop in the peripheral regions of the spherulites. Increase in volume fraction for a given size of spherulites, as well as increase of size for a given volume fraction of spherulites, decreases the mechanical strength of selenium. Increased volume fraction of small size (IO-200m) spherulites drastically decreases the flexural strength of the partially crystallized selenium. Increased volume fraction of large-size spherulites (20-50um) does not decrease the flexure strength of the composite as drastically as increased volume fractions of small-size crystals. In amorphous selenium, fracture propagates with no preferred direc- tion, as seen in oxide glasses. The fracture in partially crystal- lized selenium as indicated by fractographs shows the four distinct 170 8. 171 regions of fracture (origin, mirror, coarse, and hackle) as seen in glass-ceramics and glass-crystal composites. Fractographs reveal that the mirror area of fracture is small for higher-strength materials. With decrease of flexure strength, the mirror area increases in size. Above a certain volume fraction of spherulites (V82> 15%), the fracture becomes mixed, and regions of fracture are not clearly identifiable. Elastic modulus of partially crystallized selenium increases with increase of volume fraction of crystallites. For small size of spherulites (ds 5 20pm), the increase of modulus of elasticity is drastic. For larger-size crystals (d8 fESOum), the increase of elastic modulus with increase of volume fraction of crystallites is not as drastic as that for the small-size crystals. The Maxwell- Eucken relationship best predicts the experimentally-determined elastic modulus of crystallized selenium for the 50.8um-size crystals. The scatter of flexure-strength data is greatest for amorphous selenium. With increasing volume fraction of crystallites, the scatter is reduced. The following properties of amorphous and partially crystallized selenium have been determined during the course of this work: (a) Amorphous Selenium Flexure strength (a): most probable value 450kg/cm2. Elastic modulus (E): 0.7 - l x lOSkg/cm?. Fracture surface energy (v): 1.6289 ergs/cm?. 10. 11. 12. 172 (b) Partially Crystallized Selenium Flexure strength (p): 250-400 kg/cm? Elastic modulus (E): 14 - 24 x 105 kg/cmz. Fracture surface energy (v): 1.4974 ergs/cmz. The mechanical behavior of the partially crystallized selenium can be explained completely with the existing theories for glass-ceramic and glass-crystal composites. The fracture surface energy of partially crystallized selenium does not depend on the volume fraction or the size of spherulites. The fracture surface energy of amorphous selenium is approximately the same as that of partially crystallized selenium. - 2 = 2 (very — 1.4974 ergs/cm and Yamo 1.6289 ergs/cm ) The ratio of calculated Griffith crack size to crystal diameter in partially crystallized selenium varies from 1.10 to 3.00 approxi- mately. This ratio is a) 1.10 for composites having large-size crystals (dsjg 50.8um), b) 1.71 for composites having medium-size crystals (d.8 f 50.8um), and c) 3.00 for composites having small-size crystals ((18 £315.0um). The size of flaws present in amorphous selenium specimens is extremely small compared with the size of flaws in oxide glasses (Sum in amor- phous selenium versus 50pm in oxide glasses). LIST OF REFERENCES 10. 11. 12. 13. 14. 15. 16. 17. LIST OF REFERENCES R. S. Caldwell and H. Y. Fan, Phys. Rev., 114 664 (1959). G. Briegleb, Z. Physik. Chem., A 144 321 (1929), referred as in W. C. LaCourse Ph.D. Thesis, RPI, New York. H. Krebs, Fundamentals of Inorganic Crystal Chemistry (McGraw-Hill, London 1968), referred as in W. C. LaCourse Ph.D. Thesis, RPI, New York. H. Krebs, Z. Ffir Naturforschg., 12b 795 (1957), referred as in W. C. LaCourse Ph.D. Thesis, RPI, New York. 0. Foss, Acta Chem. Scand., Z 1221 (1953), referred as in W. C. LaCourse Ph.D. Thesis, RPI, New York. F. Tunistra, Ph.D. Thesis, Tech. Hogesch. Delft, Netherlands, (1967), referred as in W. C. LaCourse Ph.D. Thesis, RPI, New York. L. A. Niman, V. D. Neff, R. E. Cantley and R. D. Butler, J. Mo1. Spectry., 22 105 (1967), referred as in W. C. LaCourse Ph.D. Thesis, RPI, New York. E. Mooser and W. B. Pearson, J. Phys. Chem. Solids, 1_65 (1958), referred as in W. C. LaCourse Ph.D. Thesis, RPI, New York. R. E. Marsh, L. Pauling and J. D. McCullough, Acta Cryst., referred as in W. C. LaCourse Ph.D. Thesis, RPI, New York. R. D. Burbank, Acta Cryst., 4 140 (1951). A. Eisenberg and A. Tobolsky, J. Polymer Science, 46 19 (1960). R. C. Keezer and M. W. Baily, Material Research Bull., g_185 (1967). B. Fitton and C. A. Griffiths, J. Appl. Phys., 32 No. 8, 3663 (1968). K. H. Meyer and C. Ferry, Helv. Chim. Acta., 18, 570 (1935), referred as in W. C. LaCourse Ph.D. Thesis, RPI, New York. A. Eisenberg and A. V. Tobolsky, J. Polymer Science, 61 483 (1962). M. Polanyi, Z. Phys., Z_323 (1921), referred as in Ref. 34. J. Frankel, Z. Phys., §Z_572 (1926), referred as in Ref. 34. 173 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 174 E. Orowan, Trans. Instn. Engrs. Shipb. Scot., 82 165 (1946). C. E. Inglis, Trans. Instn. Nav. Archit., London, §§_219 (1913). A. A. Griffith, Phil. Trans., A221 163 (1921). A. A. Griffith, Proc. Int. Congr. Appl. Mech. (Delft) (1924). R. A. Sack, Proc. Phys. Soc., Lond., _§ 729 (1946). H. A. Elliott, Proc. Phys. Soc., Lond., §2_208 (1947). W. Weibull, Ingen.Vetensk.Akad. Hand1., 151 153 (1949), as referred in Ref. 34. J. C. Fisher and J. H. Holloman, Trans. Am. Inst. Met., 1 l 546 (1947). R. J. Charles and J. C. Fisher, Non-Crystalline Solids, p. 491 (Wiley, New York 1960). E. J. Saibel, J. Chem. Phys., 15 760 (1947). P. Gibbs and I. B. Cutler, J. Am. Ceram. Soc., 34 200 (1951). N. W. Taylor, J. Appl. Phys., 1§_943 (1947). E. Poncelet, Trans. Am. Soc. Math., 495_201 (1948). S. M. Cox, J. Soc. Glass Tech., 32 127 (1948). J. C. Fisher, J. Appl. Phys., 12 1062 (1948). W. Weibull, Ingen. Vetensk.Akad., Proc. 151 No. 153 (1939): as referred in Ref. 34. W. D. Kingery, Introduction to Ceramics, p. 599 Table 17.1 (Wiley, New York 1967). P. W. McMillan, Keynote Lectures, The Glass Phase in Glass-Ceramics, presented in the conference on the role of vitreous phases in technical materials, in Leeds 1972. Z. Hashin, J. Appl. Mech., 29|l| 143 (1962). Z. Hashin and S. Shtrikman, J. Mech. Phys. Solids, 11|2| 127 (1963). F. F. Y. Wang, Matl. Sci. & Engg., 1_109 (1971). R. C. Rossi, J. Am. Ceram. Soc., §1_433 (1968). B. Paul, Trans. AIME, 218 36 (1960). D.P.H. Hasselman and R. M. Fulrath, J. Am. Ceram. Soc., 48 No. 4 218 (1965). 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 175 D. P. H. Hasselman and R. M. Fulrath, J. Am. Ceram. Soc., 4§_No. 10 548 (1965). P. W. McMillan, S. V. Phillips and G. J. Partridge, J. Matls. Sci., .1 269 (1966). H. L. McCollister and M. A. Conrad, "Fracture of Glass-Ceramics" presented at Annual Meeting of American Ceramic Society, (1966), as referred in Ref. 35. D. P. H. Hasselman and R. M. Fulrath, J. Am. Ceram. Soc., 59_No. 8 399 (1967). Y. Usumi and S. Sakka, J. Am. Ceram. Soc., 59 369 (1967). D. P. H. Hasselman and R. M. Fulrath, Ceramic Microstructures, p. 343 (Wiley, New York 1965). S. W. Freiman and L. L. Hench, J. Am. Ceram. Soc., 25 86 (1972). E. B. Shand, J. Am. Ceram. Soc., 31 No. 2 52 (1954). E. B. Shand, J. Am. Ceram. Soc., 31 No. 12 559 (1954). A. Smekal, J. Soc. Glass Tech. Trans., 29 432 (1936). J. B. Murgatroyed, J. Soc. Glass Tech., 26 22 155 (1942). J. M. Barson, J. Am. Ceram. Soc., §l_No. 2 75 (1968). E. Golz, Z. Phys., 129 773 (1943), referred as in Ref. 50. E. B. Shand, J. Am. Ceram. Soc., 42 No. 10 474 (1959). E. B. Shand, J. Am. Ceram. Soc., 44 No. 9 451 (1961). Helmut Wallner, Ceram. Abstr., 19(6) 137 (1940). A. Smekal, Ergeb. Exakt. Naturw., 15 106 (1936), referred as in Ref. 55. N. Terao, J. Phys. Soc. Japan, 8 545-549 (1953). W. C. Levengood, J. Appl. Phys., 29(5) 820 (1958) and Ceram. Abstr., 2301 (1958). L. Orr, Shand's Footnote ref. 55 p. 474. R. W. Davidge and G. Tappin, J. Matls. Sci., 3 165 (1968). W. J. Frey and J. D. Mackenzie, J. Matls. Sci., 2’124-130 (1967). D. B. Binns, Science of Ceramics, p. 315 [Academic Press (1962)]. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 176 J. Selsing, J. Am. Ceram. Soc., 44 419 (1961). W. D. Kingery, Introduction to Ceramics, p. 500 (Wiley, New York 1967). ibid., p. 472, Table 14.1. Y. Usumi and S. Sakka, J. Am. Ceram. Soc., 53 286 (1970). E. Orowan, J. Welding, 34 157 (1955). R. L. Bertolotti and R. M. Fulrath, J. Am. Ceram. Soc., 59 No. 11 558 (1967). P. W. McMillan, Glass-Ceramics, p. 134 [Academic Press, London and New York (1964)]. S. V. Phillips, unpublished data Nelson Research Lab. (1962), referred as in Ref. 71. M. Watanabe, R. V. Caporali, and R. E. Mould, "Symposium on nucleation and crystallization of glasses," Am. Ceram. Soc., 23-28 (1962), referred as in Ref. 71. G. R. Irwin, Trans. ASM, 40A 147 (1948). D. H. Winne and B. M. Wundt, Trans. ASME, §Q_1643 (1958). J. E. Srawley and W. F. Brown, ASTM (1964) and NASA TM X-52030 (1964). B. Cross and J. E. Srawley, NASA REPORT, Tn-2603 (1965). J. M. Corum, USAEC REPORT ORNL-4030 (1966). APPENDIX APPENDIX A METHODS FOR DETERMINING FRACTURE SURFACE ENERGY OF GLASS AND GLASS-CRYSTAL COMPOSITES The effective surface energy is defined in Section 1.2.2 as the work done to create unit area of new fracture face, not taking into account the fine-scale surface irregularities of the fracture face. For a given material, the effective surface energy (y) is not necessarily the same at all stages in the fracture process. Davidge and Tappin62 define YI as effective surface energy pertaining to the initiation of fracture, and YF as work of fracture, the value averaged over the whole fracture process. v1 is the value of y appearing in the Griffith equation, and is related to the strain-energy release rate at the instant of fracture by (GU/GA) 2 v1 where A is the area of new fracture face. (bU/bA) may be obtained by theoretical or experimental methods. The theoretical method involves determination of (bU/bA) from the mathematically-computed stress dis- tribution around the notch for the particular specimen geometry. The experimental method derives (DU/DA) solely from experimental load- deflection curves. 177 178 Theoretical Method Let y be the effective surface energy determined by this method, G when the notch depth (c:) is small compared with the beam depth (d). YG is given by Irwin74 and Orowan69 as yG = -(bU/0A) = (1 - 02)n onc/ZE [28] where v is Poisson's ratio, E is Young's modulus, and CF is the fracture stress. Plane-strain conditions are assumed in deriving this equation. Equation 28 is essentially the original Griffith's equation. Winne and Wundt75 have presented corrections to the Equation 28 for the case when old 2 0.1. Srawley and Brown76 have reviewed these mathematical treatments and represented the expressions in the form 9(1 - v2) PF2£2f(c/d) Y = [29] G 8EB2(d - c)3 where f(c/d) is a dimensionless parameter. At small arguments, (c/d)'; nc(d - c)3/d4, and Equation 29 reduces to Equation 28. Figure 65 shows f(c/d) as a function of its argument, as calculated from results of Gross and Srawley77 by Corum78. Figure 65 strictly applies only to beams deformed in four-point bending, but provided that £23 8b approximately, corrections due to the shear stresses present during three-point bending are small [<2101], as discussed by Srawley and 76 Brown . The effective surface energy YG can be determined using Equation 28 or the following equation: 179 Fig. 65. Plot of function f(c/d) versus c/d, as calculated from results of Gross and Srawley by Corum (Davidge and Tappin62). Fig. 66. The general form of stiffness (K) versus crack area (A) plot (Davidge and Tappin62). STIFFNESS k GI 02 O 3 04 05 06 . _ . - _ ,-_-_-.-__I CRACK AREA A - 2bc 180 2 2 yG - (1 - 0 )PF F/E [30] where F = 912f(c/d)/8b2(d-c)3. Values of F can be substituted in the above equation for different values of f(c/d). Experimental Method Let Yc represent the effective surface energy determined by the experimental method. This value should be equal to YG' The load- deflection curve in Figure 13 is given by P = K6, so that the stored energy at the instant of fracture is V = P 6 /2 or K6 2/2. F F ’ F Now, Yc equals -(bU/bA), and fracture occurs at a fixed deflec- tinn. Then, Yc = -(bU/0K)6 (GK/0A)6 - But (GU/GK) = 6F2/2 and thus vc = '5F2(bK/0A)/2 . [31] The specimen stiffness K is experimentally determined and plotted as a function of the initial crack area, A = 2bc. For each notch depth, (bK/bA) is obtained from the slope of the curve at the appropriate value of A, as in Figure 66. Substitution of these values of (OK/DA) in Equation 31 with the experimental values for 6 thus gives a series of F Yc values for each notch depth. Davidge and Tappin have determined the surface energy at the instant of fracture (VI) for four brittle materials, namely alumina, poly-methylmethacrylate, glass, and graphite. The theoretical and 181 experimental methods for the determination of Y1 show good agreement. There is no variation of either YG or ye as a function of c/d, and both YG or Yc show a scatter of up to'i 20% from the mean value. The values of YG and Yc for a particular material are within 10 to 20% of each other. Davidge and Tappin conclude that YI values are highly dependent on the precise testing procedure, in particular on the initial distribution of crack sources. The values of Y determined, I are strictly relevant only to specimens prepared and tested by the above method. "IIIIIIIIIIIIIIIIIII