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Viscosities of phase-separated glasses Bernheim, Philippe 1968

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V i s c o s i t i e s of Phase-separated Glasses by P h i l i p p e BERNHEIM A Thesis submitted i n p a r t i a l f u l f i l m e n t of the requirements f o r the degree of Master of Ap p l i e d Science i n the Department of METALLURGY We accept t h i s t h e s i s as conforming to the standards r e q u i r e d from candidates f o r the degree of Master of Ap p l i e d Science Members of the Department of Metallurgy The U n i v e r s i t y of B r i t i s h Columbia February, 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C olumbia, I ag r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department or by h.i.>s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f M e t a l l u r g y  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date A p r i l 30, 1968 ABSTRACT The e f f e c t s of heat treatments on v i s c o s i t y were analysed f o r three types of gl a s s e s . "Pyrex" glass could be approximated to a Newtonian l i q u i d i n the range of temperature 470 to 590°C.. The a c t i v a t i o n energy f o r viscous flow v a r i e d from 65 to 85 kcal/mole according to the previous thermal h i s t o r y . A l l glasses e x h i b i t e d phase s e p a r a t i o n to d i f f e r e n t degrees according to the heat treatments to which they had been submitted. Phase se p a r a t i o n and d i f f e r e n t degrees of d e v i t r i f i c a t i o n could account f o r the v a r i a t i o n i n v i s c o s i t y which have been encountered. Sev e r a l mathematical expressions were tested i n an attempt to c o r r e l a t e v i s c o s i t y change w i t h time. The best f i t was obtained w i t h a r e l a t i o n s h i p of the type n = (a + bt) where c v a r i e d i n the range 0.2 to 0.5. In the case of a ternary b o r o s i l i c a t e g l a s s , the value was found to be 0.5. This may suggest that phase s e p a r a t i o n i s a d i f f u s i o n c o n t r o l l e d process. I l l ACKNOWLEDGEMENT The author wishes to express h i s s i n c e r e g r a t i t u d e to Dr. A.CD. Chaklader f o r h i s advice and as s i s t a n c e throughout t h i s work. He a l s o wishes to express h i s thanks to a l l the members of the s t a f f who helped,, i n the r e a l i z a t i o n of the creep t e s t i n g equipment, and a l l those who a s s i s t e d i n any way i n t h i s i n v e s t i g a t i o n . F i n a n c i a l a s s i s t a n c e provided by the N a t i o n a l Research C o u n c i l under grant No. A - 2461 and by the Canada C o u n c i l as a research s c h o l a r s h i p to the author i s g r a t e f u l l y acknowledged. IV TABLE OF CONTENTS PAGE 1. INTRODUCTION 1 1.1 D e f i n i t i o n and importance of v i s c o s i t y f o r glass 1 f a b r i c a t i o n . 1.2 Measurement of the v i s c o s i t y of g l a s s . 2 1.3 Accuracy of the techniques. 3 1.4 Composition dependence of the v i s c o s i t y . 4 1.5 Thermal h i s t o r y dependence of the v i s c o s i t y . 6 1.6 Time dependence of the v i s c o s i t y . 6 1.7 Phase s e p a r a t i o n i n g l a s s . 7 1.8 Property changes of glass due to phase s e p a r a t i o n . 10 1.9 Obj e c t i v e of t h i s i n v e s t i g a t i o n . 11 2. EXPERIMENTAL 12 2.1 M a t e r i a l s e l e c t i o n . 12 2.11 The b o r o s i l i c a t e g l a s s e s . 12 2.12 The binary g l a s s . 14 2.13 The a l k a l i g l a s s e s . 19 2.2 Experimental procedure 19 2.21 Measure of the creep behaviour. 20 2.22 E l e c t r o n microscopy. 24 3. RESULTS 3.1 Measurement of the v i s c o s i t y . ' 24 3.11 C a l c u l a t i o n of the v i s c o s i t y from the creep 24 curves. 3.111 The formula. 26 3.112 P r e c i s i o n of the measurements. 26 3.113 Flow c h a r a c t e r i s t i c s of the g l a s s e s . 27 V TABLE OF CONTENTS (continued) PAGE 3.12 B o r o s i l i c a t e g l a s s e s . 27 3.121 "Pyrex" g l a s s . 27 3.1211 V a r i a t i o n of the temperature. 31 3.1212 E f f e c t of d i f f e r e n t heat t r e a t - 32 ments. 3.122 "Corning" g l a s s . 36 3.13 The binary g l a s s . 42 3.14 A l k a l i g l a s s e s . 42 3.141 A l k a l i g l a s s without a d d i t i v e s . 45 3.142, A l k a l i glass and quartz powder. 45 3.2 S t r u c t u r a l Study. 45 3.21 "Pyrex" g l a s s . 47 3.22 "Corning" g l a s s . 53 4. DISCUSSION 4.1 Theory of the p l a t e viscometer. 53 4.11 T h e o r e t i c a l treatment. 57 4.12 V a l i d i t y of the assumptions. 57 4.121 The specimen i s deformed under u n i a x i a l 59 compression. 4.122 H o r i z o n t a l planes remain h o r i z o n t a l during 59 the flow. 4.123 A given h o r i z o n t a l plane remains a plane 59 during the deformation. 4.2 V a r i a t i o n of the v i s c o s i t y w i t h time. 59 4.21 General c o n s i d e r a t i o n . 61 4.22 Expression f o r the v a r i a t i o n of the v i s c o s i t y w i t h 64 the time. 4.23 A c t i v a t i o n energy of the flow process. 76 VI TABLE OF CONTENTS (continued) PAGE 4.3 I n t e r p r e t a t i o n of the e l e c t r o n microscopy 76 4.31 "Corning" g l a s s . 77 4.32 The "Pyrex" g l a s s . 77 4.33 The other glasses 82 4.4 Changes of v i s c o s i t y due to phase se p a r a t i o n . 82 4.41 T h e o r e t i c a l c o n s i d e r a t i o n s . 82 4.411 E f f e c t s of the p h y s i c a l s t r u c t u r a l change. 82 4.412 E f f e c t s of-the chemical change. 88 CONCLUSION 93 SUGGESTION FOR FURTHER WORK 94 APPENDICES BIBLIOGRAPHY 95 119 LIST OF FIGURES VII FIGURE PAGE 1. Schematic phase diagram of binary glasses which w i l l 9 undergo phase se p a r a t i o n . 2. Schematic drawing of the hot pres s i n g assembly. 16 3. P a r t of the Na20-Si02 phase diagram. 18 4. Schematic diagram of the creep t e s t i n g equipment. 21 5. Photograph of the creep t e s t i n g equipment. 22 6. V a r i a t i o n of the v i s c o s i t y w i t h the pressure 28 7. Representative creep curves f o r tests at d i f f e r e n t 29 pressure. 8. V a r i a t i o n of the v i s c o s i t y w i t h the temperature: 33 glass as received. 9. V a r i a t i o n of the v i s c o s i t y w i t h the temperature: 34 annealed g l a s s . 10. Creep curves f o r tests at d i f f e r e n t . temperatures. 35 11. . V i s c o s i t y as a f u n c t i o n of the time of heat treatment 37 at 500°C. 12. V i s c o s i t y as a f u n c t i o n of the time of heat treatment 37 at 565°C. 13. V i s c o s i t y as a f u n c t i o n of the time of heat treatment 37 at 575°C. 14. V i s c o s i t y as a f u n c t i o n of the time of heat treatment 37 at 600°C. 15. V i s c o s i t y as a f u n c t i o n of the time of heat treatment 38 at 650°C. 16. V i s c o s i t y as a f u n c t i o n of the time of heat treatment 38 at 750°C. 17. V i s c o s i t y as a f u n c t i o n of the time of heat treatment 38 at 800°C. 18. V a r i a t i o n of.the r a t e of change of v i s c o s i t y versus 40 temperature of heat treatment. 19. V i s c o s i t y f o r d i f f e r e n t times of heat treatment versus 41 temperature of heat treatment. V I I I LIST OF FIGURES (continued) FIGURE PAGE 20. Creep curves of "Corning" glass specimens. 43 21. Binary g l a s s e s : v i s c o s i t y as a f u n c t i o n of the temperature. 44 22. A l k a l i g l a s s e s : v i s c o s i t y as a f u n c t i o n of the temperature: no heat treatment. 46 23. A l k a l i g l a s s e s : v i s c o s i t y as a f u n c t i o n of the of the temperature for d i f f e r e n t heat treatments. 46 24. A l k a l i g l a s s and quartz powder: v i s c o s i t y as a f u n c t i o n of the temperature: a f t e r d i f f e r e n t heat treatments. 46 25. D i f f e r e n t times of etching w i t h 2% HT d i l u t e d s o l u t i o n . 48 26. E f f e c t of d i f f e r e n t etch w i t h Au shadowing. 49 27. V a r i a t i o n of the s i z e of the second phase p a r t i c l e s w i t h the time of heat treatment. 51 28. V a r i a t i o n of the micros t r u e t u r e w i t h the temperature of heat treatment. 52 29. Change i n the shape of the p a r t i c l e s of the second phase. 54 30. Change of s t r u c t u r e observed w i t h "Corning" g l a s s . 56 31. Change of shape of the specimen during the creep t e s t . 60 32. <j>(h) as a f u n c t i o n of time f o r t e s t 3.1211-10-3.1211-17. 63 33. <)>(h) as a f u n c t i o n of time f o r t e s t s of i n c r e a s i n g d u r a t i o n . 66 34. D i f f e r e n t mathematical models of the v i s c o s i t y time r e l a t i o n s h i p f o r specimen' 3.122-00. 71 35. D i f f e r e n t mathematical models of the v i s c o s i t y time r e l a t i o n s h i p f o r specimen' 3.122-01. 71 36. D i f f e r e n t mathematical models of the v i s c o s i t y time r e l a t i o n s h i p f o r specimen 3.122-02. 71 37. D i f f e r e n t mathematical models of the v i s c o s i t y time r e l a t i o n s h i p f o r specimen'. 3.122-04. 71 38. D i f f e r e n t mathematical models of the v i s c o s i t y time r e l a t i o n s h i p f o r specimen 3.1212-309. 73 LIST OF FIGURES (continued)  FIGURE P A G E 39. D i f f e r e n t mathematical models of the v i s c o s i t y time 73 r e l a t i o n s h i p f o r specimen: 3.1212-310. 40. D i f f e r e n t mathematical models of the v i s c o s i t y time r e l a t i o n s h i p f o r specimen 3.1212-313. 73 41. V i s c o s i t y versus true time of heat treatment. 75 42. V a r i a t i o n of the s i z e of the p a r t i c l e s i n the micro-s t r u c t u r e of a specimen. 79 43. Islands of phase se p a r a t i o n . 80 44. D i f f e r e n t micros truetures encountered on specimen: 3.112-80. 115 45. D i f f e r e n t micros truetures encountered on specimen: 3.1212-407. 116 46. D i f f e r e n t m i c r o s t r u c t u r e s encountered on specimens: 46a,b, 3.1212-406, 46c,d, 3.1212-409, 46e,f, 3.1212-313 117 47. D i f f e r e n t aspects of phase separated g l a s s . 118 40. Growth r a t e and n u c l e a t i o n r a t e of lead t i t a n a t e . 81 49. Phase s e p a r a t i o n i n the b i n a r y g l a s s . 83 50. Phase s e p a r a t i o n i n the a l k a l i g l a s s . 84 51. C r y s t a l l i s a t i o n i n the b i n a r y g l a s s . 85 52. C r y s t a l l i s a t i o n i n the a l k a l i g l a s s . 86 53. R e l a t i v e v i s c o s i t y versus con c e n t r a t i o n of the suspended phase. 88 54. V a r i a t i o n of the v i s c o s i t y of a ternary glass w i t h the %2®2 content. 89 55. Log isokoms of the ternary system Na20-Si02~CaO. 89 56. V i s c o s i t y of a g l a s s of the Na 20-Nb 20-Si0 2 system at a r a t e of heating of 66°C/h. 91 57. V i s c o s i t y of a glass at d i f f e r e n t temperatures of heat treatment. 91 58. V a r i a t i o n of the v i s c o s i t y w i t h the time of heat treatment. 91 X LIST OF TABLES NUMBER PAGE 1. Chemical a n a l y s i s of the glasses used i n the present 13 work. 2. Parameters of the hot p r e s s i n g technique of the binary 17 glass specimens. 3. P r e s s u r e - v i s c o s i t y r e l a t i o n s h i p . 95 4. C o e f f i c i e n t of the l i n e a r r e l a t i o n s h i p between pressure 30 a n d . v i s c o s i t y . 5. Temperature v i s c o s i t y r e l a t i o n s h i p of "Pyrex" glass 97 teste d "as r e c e i v e d . " 6. Temperature v i s c o s i t y r e l a t i o n s h i p of "Pyrex" glass 98 "annealed" at 500°C. 7. Parameters of the v i s c o s i t y - t e m p e r a t u r e r e l a t i o n s h i p . 30 8. V i s c o s i t y - h e a t treatment r e l a t i o n s h i p . 99 9. Parameters of the v i s c o s i t y - h e a t treatment r e l a t i o n s h i p . 39 10. Parameters of the "Corning g l a s s " t e s t s . 101 11. Parameters of the bi n a r y glass t e s t s . 102 12. Parameters of the a l k a l i glass t e s t s . 103 13. Parameters of the a l k a l i glass and quartz powder t e s t s . 105 14. Parameters of the p a r a b o l i c approximation of <Kh). 62 15. Parameters of the p a r a b o l i c approximation of <)>(h) f o r 65 i n c r e a s i n g time of t e s t s . 16. Comparison of the d i f f e r e n t mathematical models f o r four 69 "Corning" glass specimens. 17. Comparison of the d i f f e r e n t mathematical modes f o r 3 72 "Pyrex" glass long l a s t i n g experiments. 18. Parameters of the time v i s c o s i t y r e l a t i o n s h i p f o r most 107 experiments. 1 1. INTRODUCTION 1.1 D e f i n i t i o n and Importance of V i s c o s i t y f o r glass F a b r i c a t i o n Glasses as a group of non-metallic m a t e r i a l s occupy a very l a r g e s e c t i o n of a l l i n o r g a n i c m a t e r i a l s being used f o r i n d u s t r i a l , t e c h n i c a l and s c i e n t i f i c purposes. Glass manufacture i s a la r g e s c a l e i n d u s t r y which has long been dependent on the s k i l l and d e x t e r i t y of i n d i v i d u a l craftsmen working under crude c o n d i t i o n s w i t h a l i t t l e understood m a t e r i a l . I t i s now dominated by continuous machine production which r e q u i r e s a p r e c i s e c o n t r o l of the flow c h a r a c t e r i s t i c s of g l a s s , i . e . v i s c o s i t y of the melt. Since g l a s s i s a supercooled l i q u i d , i t does not have a s p e c i f i c m elting p o i n t but softens g r a d u a l l y . This i s due to the continuous decrease of the v i s c o s i t y w i t h i n c r e a s i n g temperature. The knowledge of the v i s c o s i t y , n, of a glass i s of p r a c t i c a l importance at a l l stages of the manufacturing process since i t v a r i e s considerably. At room temperature 20 i t i s i n the range of 10 poises or more. Then the glass i s considered as " s o l i d " because i t i s i n a r i g i d s t a t e . As the temperature increases the v i s c o s i t y reaches the s t r a i n p o i n t (S.P.). This temperature i s 14. 6 defined as the temperature at which n = 10 ' poises and corresponds to the lower l i m i t of the annealing range. The annealing treatment used i n commercial p r a c t i c e s i s completed i n s i x t e e n hours at the S.P. 13.4 A v i s c o s i t y of 10 defines the annealing point A.P., and 11 12 a value of.10 to 10 poises i s the common range f o r the d i l a t o m e t r i c s o f t e n i n g p o i n t , where the viscous flow e x a c t l y counteracts the thermal expansion during measurements under given standard c o n d i t i o n s . When the 7 6 v i s c o s i t y drops to 10 " poises the f i b e r s o f t e n i n g p o i n t (F.S.P.) i s a t t a i n e d . Then a f i b e r of 0.5 to 1 mm i n diameter and 22.0 mm i n length 2 elongates under i t s own weight at a r a t e of 1 mm per minute, when heated at a r a t e of 5°C per minute under.standardized c o n d i t i o n s . A continuous decrease of the v i s c o s i t y leads to the glass working range. The upper end of t h i s range corresponds to manual work and blowing; the lower end i s used f o r the mechanical c a s t i n g of the g l a s s . The automatic machinery i n use i n i n d u s t r i a l processes re q u i r e s s u p p l i e s of glass having a more or l e s s constant v i s c o s i t y and al s o a given v i s c o s i t y - t e m p e r a t u r e r e l a t i o n -ship which corresponds to the r a t e of work f o r which the machine has been designed. v The degassing and the homogenization of the glass batches are 2 performed at v i s c o s i t i e s i n the order of 10 poises or l e s s . Then the f l u i d i t y of the glass i s that of a normal l i q u i d . 1.2 Measurement of the V i s c o s i t y of Glass The c o e f f i c i e n t of v i s c o s i t y of a f l u i d i s a measure of the i n t e r n a l f r i c t i o n i n the l i q u i d . Thus the measurement techniques r e q u i r e a knowledge of the forces or torques between the l i q u i d and some s o l i d s u r f a c e , or some v a r i a b l e r e l a t e d to these, such as the energy l o s s or the damping time. Since^ as mentioned p r e v i o u s l y , the v i s c o s i t y v a r i e s by more than twenty orders of magnitude, one cannot expect to cover the whole range of v i s c o s i t i e s w i t h a s i n g l e instrument. g For values i n the range below 10 poises standard methods of v i s c o s i t y measurement can be used; the glass i s then very c o r r o s i v e and the temperature very high so that s p e c i a l equipment made of platinum or carbon i s req u i r e d . Many authors have used apparatus based on Stoke's law, i n which the motion of a b a l l of heavy m a t e r i a l such as platinum, i s 3 followed i n the glassy medium. Nevertheless, the r o t a t i n g c y l i n d e r 3 viscometer i s considered to be the most accurate , e s p e c i a l l y f o r the 3 8 v i s c o s i t y range of 10 to 10 poises. I t i s used e i t h e r as a r o t a t i n g device by motion of the outer c y l i n d e r or as an o s c i l l a t i n g pendulum. 8 In the upper range of the v i s c o s i t y measurement (above 10 p o i s e s ) , g l a ss samples can be subjected to creep deformation. F i b r e s are twisted under t o r s i o n or s t r e t c h e d under a given load. More complicated methods, such as the e l o n g a t i o n of rings,have a l s o been used to c a l c u l a t e the v i s c o s i t y . Bending of specimens by 3 or 4 point l o a d i n g can be u t i l i z e d f o r the same purpose. F i n a l l y the p a r a l l e l p l a t e 4 viscometer, as i t has been used r e c e n t l y by other workers was employed" i n the present work. This i n v o l v e s the study of compressive creep of g l a s s c y l i n d e r s between two p a r a l l e l p l a t e s . For most of these methods, the deformation i n v o l v e d i n the experiment required d i f f i c u l t mathematical formulations of the viscous flow. E i t h e r a c a l i b r a t i o n of the apparatus, or r e s t r i c t i o n ' on the extent of the deformation a l l o w i n g s i m p l i f y i n g assumptions was used to overcome t h i s d i f f i c u l t y . This l i m i t a t i o n , however, i s not a p p l i c a b l e to the p l a t e viscometer. 1.3 Accuracy of the Techniques Measurements of the v i s c o s i t y of a glass w i t h reasonable accuracy are d i f f i c u l t . The problem encountered i s the c a l i b r a t i o n of the measuring device. Many co n t r o v e r s i e s e x i s t as to the p o s s i b l e v a r i a t i o n of the c a l i b r a t i o n f a c t o r of d i f f e r e n t types of r o t a t i n g viscometers. When deformation curves are used, the knowledge of the a c t u a l a p p l i e d s t r e s s i s c r i t i c a l . Furthermore the m a t e r i a l on which the experiment i s performed may vary. I t i s w e l l known that during 4 v i s c o s i t y measurements at a temperature above the l i q u i d u s of the g l a s s , d e v i t r i f i c a t i o n o f t e n occurs i n the glass-mix. Then the measurement gives the values of the v i s c o s i t y f o r a mixture of a l i q u i d (of unknown composition) and of s o l i d c r y s t a l l i n e p a r t i c l e s . The only way to avoid t h i s problem i s by r a p i d measurements. Since the l i q u i d u s temperature may be f a i r l y low and therefore corresponds to a high value of the v i s c o s i t y , t h i s source of e r r o r may not have been e a s i l y recognized or taken i n t o account i n the c a l c u l a t i o n of the v i s c o s i t y . In the determination of high v i s c o s i t i e s where creep curves are used, the " c a l c u l u s hypotheses" r e s t r i c t the v a l i d i t y of the method to very s m a l l deformations which are d i f f i c u l t to measure a c c u r a t e l y , and which can a l s o be a f f e c t e d by p a r a s i t i c phenomena. As i t i s explained l a t e r , the thermal h i s t o r y , the degree of s t a b i l i z a t i o n , r e l a x a t i o n phenomena and the phase-separation a l l occur i n the range of temperature of the measurement and can modify the glass s t r u c t u r e and th e r e f o r e i t s v i s c o s i t y . P r o t r a c t e d experiments which allow low s t r a i n rates and may p a r t l y e l i m i n a t e r e l a x a t i o n phenomena are nevertheless subject to some of the sources of e r r o r s mentioned above. 1.4 Composition Dependence of the V i s c o s i t y I n d u s t r i a l requirements have l e d to extensive work on the v i s c o s i t y - c o m p o s i t i o n r e l a t i o n s h i p i n glasses over a wide range of temperatures. Simpler systems, such as binary systems w i t h SiC^ as the main component and one of ^20"*, or or K^O^ as the second component, have been i n v e s t i g a t e d . As expected, at a given temperature the 5 v i s c o s i t y of the melts increases w i t h the SiO^ content. The change can be very important s i n c e a s l i g h t v a r i a t i o n of composition can produce a s e v e r a l - f o l d change i n the value of the v i s c o s i t y . The r e s u l t s of d i f f e r e n t authors are i n c o n f l i c t because of the use of d i f f e r e n t experimental techniques and p o s s i b l y due to the i n t r o d u c t i o n of i m p u r i t i e s during the measurement of the v i s c o s i t y . In a d d i t i o n , v i s c o s i t y has been determined on systems where one component i s replaced by another i n a binary glass without changing the c o n c e n t r a t i o n of the second component; e.g. i n the system Na20-SiCJ2, an e q u i v a l e n t amount of oxide such as CaO^, MgO or Al^O^ u s u a l l y replaces some of the Na20. Sometimes has been used to replace ^^^2 3 s i m i l a r way. In the f i r s t three cases an increase i n the v i s c o s i t y , and i n the l a s t case a decrease, i s observed. Thus l i n e s of equal v i s c o s i t y can be drawn on ternary diagrams. A s i m i l a r type of study on quaternary systems has been c a r r i e d out by v a r y i n g the r a t i o of two components keeping the other two constant. In commercial g l a s s e s , systematic replacement of SiC^ i n s m a l l amounts by other components had been e f f e c t e d i n order to o b t a i n c o r r e c t i o n charts of the temperature-viscosity r e l a t i o n s h i p . These charts are u s e f u l f o r p r a c t i c a l manufacturing processes. No c o n c l u s i o n can be drawn from these r e s u l t s , s i n c e a d d i t i o n a l work i s necessary to understand the deformation of the oxide-ion-network during the flow of the glassy m a t e r i a l and therefore to c o r r e l a t e the i n f l u e n c e of each c o n s t i t u e n t on the v i s c o s i t y of the g l a s s . The i m p u r i t i e s (other oxides or d i f f e r e n t ions) can act as a c a t a l y s t f o r c r y s t a l l i s a t i o n of the g l a s s . Thus the viscous p r o p e r t i e s of the glass may be a f f e c t e d by the i n t r o d u c t i o n of these i m p u r i t i e s . E i t h e r 6 c r y s t a l l i z a t i o n or phase s e p a r a t i o n , can change the v i s c o s i t y of two apparently i d e n t i c a l specimens by s e v e r a l orders of magnitude. 1.5 Thermal H i s t o r y Dependence of the V i s c o s i t y The d i f f e r e n t heat treatments undergone by a glass from the molten stage u n t i l the v i s c o s i t y measurement i s taken, can modify s i g n i f i c a n t l y (up to s e v e r a l orders of magnitude) the value of the measured v i s c o s i t y . Experiments conducted w i t h the same specimen provide r e s u l t s which change continuously. The v a r i a t i o n of some glass p r o p e r t i e s such as extension, index of r e f r a c t i o n , or e l e c t r i c a l c o n d u c t i v i t y as a f u n c t i o n of the temperature of the heat treatment, s u b s t a n t i a t e s t h i s change. When subjected to an i d e n t i c a l v a r i a t i o n of the con d i t i o n s of heat treatment the values of a l l . t h e s e parameters do not.vary i d e n t i c a l l y . The l a c k of r e p r o d u c i b i l i t y between experiments i s e s p e c i a l l y n o t i c e a b l e i n the tra n s -formation range of temperature. Considerations of the s t a b i l i z a t i o n of the gla s s have been made i n an attempt to account f o r these e f f e c t s . The high v i s c o s i t y encountered at temperatures below the t r a n s i t i o n p o i n t would not allow the motion of the glass " p a r t i c l e s " toward new s i t e s where an e q u i l i b r i u m s t a t e could be achieved w i t h i n a reasonable time. The technique of standard s t a b i l i z a t i o n f o r glass consists of a heat treatment f o r various lengths of time at the temperature at which the experiment i s to be performed. The time necessary to achieve some degree of s t a b i l i z a -t i o n v a r i e s w i t h the temperature i n the same manner as the v i s c o s i t y . Therefore, at a low temperature the s t a b i l i z a t i o n time exceed any reasonable p e r i o d and i s o f t e n such that s t a b i l i z a t i o n becomes impossible. 1.6 Time Dependence of the V i s c o s i t y Even on w e l l s t a b i l i z e d g l a s s e s , a constant value of the 7 v i s c o s i t y could not be obtained over a long p e r i o d . The v a r i a t i o n of the v i s c o s i t y w i t h time, which has been p r e v i o u s l y n o t i c e d by many researchers,was a t t r i b u t e d to poor s t a b i l i z a t i o n of the samples. This i s p a r t i c u l a r l y true i n the transformation range . P l o t s of v i s c o s i t y -time curves on a double l o g a r i t h m i c s c a l e e x h i b i t approximately l i n e a r • r ^ r. • - > • -1 9-10 shape i n the case of the b o r o s i l i c a t e glasses More recent works discovered new features on these curves''""'". For lead or b o r o s i l i c a t e glasses i n the high temperature region of the t r a n s i t i o n temperature range, the v i s c o s i t y - t i m e p l o t i s S-shaped. In the beginning,the r a t e of i n c r e a s e of the v i s c o s i t y grows f a s t e r w i t h time. For a given p e r i o d of time a constant r a t e i s then observed. E v e n t u a l l y the r a t e drops slowly to zero. The b o r o s i l i c a t e glass i n the lower temperature range e x h i b i t s a d i f f e r e n t behaviour, i n that no asymptotic value of the v i s c o s i t y i s a t t a i n e d . A survey of the l i t e r a t u r e leads to the c o n c l u s i o n that time i s an important parameter of the v i s c o s i t y . However, i n the annealing range s u f f i c i e n t c o n s i d e r a t i o n has not always been given to the delayed e l a s t i c e f f e c t and other r e l a x a t i o n phenomena, such as the change of density,which may give r i s e to s m a l l deformations. These p a r a s i t i c a l deformations of the specimen may completely obscure the deformation due only to the viscous flow and may lead to erroneous conclu-sions about the a c t u a l viscous behaviour, of the m a t e r i a l . 1.7 Phase Separation i n Glasses Often on c o o l i n g the components of a glass melt separate i n t o two or more d i s t i n c t intermixed l i q u i d s of d i f f e r e n t chemical composition. The n u c l e a t i o n and growth of one l i q u i d out of the other r e s u l t s from a very s i m i l a r process to the one by which c r y s t a l s nucleate and grow from a s o l u t i o n . In many glass systems, phase s e p a r a t i o n i n t o two l i q u i d phases occurs metastably below the l i q u i d u s temperature. Attempts to understand the o p a c i f i c a t i o n behaviour of Li20-B20^-Si02 ternary glasses which had o o 12 undergone heat treatment i n the range of 550 C to 880 C l e d to the discovery of l i q u i d i m m i s c i b i l i t y i n g l a s s e s . The e l e c t r o n microscopy of f r e s h - f r a c t u r e d surface of such g l a s s e s showed c l e a r l y immiscible regions where dr o p l e t s of one phase were surrounded by a matrix of another phase. The s i z e and d e n s i t y of these d r o p l e t s were c o r r e l a t e d w i t h the temperature and the time of the heat treatment. Recently the p o s s i b i l i t y of metastable l i q u i d i m m i s c i b i l i t y was discovered f o r d i f f e r e n t shapes of the l i q u i d u s l i n e on the phase 13 diagram . On the schematic phase diagram of a b i n a r y system as shown i n Figure l a , a mixture of the composition represented by arrow (1) might undergo spontaneous phase decomposition even during a quench. For a composition corresponding to that represented by arrow (2) on Figure l a or l b , s u b l i q u i d u s heat treatment i s required f o r the development of the phase separation. L i q u i d - i n - l i q u i d c o l l o i d a l i m m i s c i b i l i t y was found i n 14 quaternary systems, such as Li20-Ca0-Ti02"-Si02 . A systematic study of the p o s s i b l e r e gion of phase s e p a r a t i o n i n various systems has been c a r r i e d out by s e v e r a l workers. The phase s e p a r a t i o n phenomenon has a l s o been observed f o r S i 0 2 - L i 0 2 1 5 , and f o r S i 0 2 w i t h e i t h e r CaO, SrO, BaO or A ^ O ^ 6 . Ternary systems such as Si02-Al20.j-Y202 or Si02-Nb20,.-Y20.^'? and more 18 complex glasses e.g. commercial products such as "Pyrex" or "Vycor" have a l s o e x h i b i t e d the same phenomenon. An imprecise knowledge of the exact l i q u i d u s p o s i t i o n has hindered the i n v e s t i g a t i o n of phase s e p a r a t i o n i n gl a s s e s . 9 S p e c i a l ' Metastable 2 l i q u i d s b) Figure 1 : T h e o r e t i c a l phase diagrams of binary c o n s t i t u e n t s l e a d i n g to phase s e p a r a t i o n . Composition 1 2 Spontaneous phase separation ,even during quench . Phase s e p a r a t i o n r e q u i r e s a s u b l i q u i d u s heat t r e a t -ment . 10 The mechanism f o r the i n i t i a l development of the phase 18 s e p a r a t i o n has a l s o been i n v e s t i g a t e d . Charles proposed two i n i t i a l mechanisms to account f o r the f a c t that the phase se p a r a t i o n occurs below or above the s p i n o d a l . Each mechanism leads to d i f f e r e n t r e s u l t a n t character-i s t i c s t r u c t u r e s through d i f f e r e n t k i n e t i c s of the transformation. Below the s p i n o d a l , spinodal-composition-modulations form and increase i n amplitude to give a h i g h l y i r r e g u l a r two phase s t r u c t u r e . The k i n e t i c s of t h i s process, which are very f a s t , render u n l i k e l y the p o s s i b i l i t y of r e t a i n i n g a s i n g l e phase by quenching unless the v i s c o s i t y of the glass at the s p i n o d a l i s very high. 1.8 Property Changes of-Glass due to the Phase Separation A f t e r phase se p a r a t i o n , g l a s s i s no longer, indeed i f i t ever was, a homogeneous body, and the presence of the d i s t r i b u t e d heterogeneites modifies i t s p r o p e r t i e s . O p t i c a l l y the glass i s a l t e r e d by the s m a l l d r o p l e t s of the second phase. I t becomes opalescent or opaque or i t s index of r e f r a c t i o n changes. V a r i a t i o n s of the breaking strength of "Pyrex" rod according 14b to heat treatment i n the annealing range were observed . The f i n e s t r u c t u r e 14 revealed by e l e c t r o n microscopy of f r e s h - f r a c t u r e d surface of glass d i s -played a t a i l - l i k e wake l e f t by the f r a c t u r e f r o n t behind each d r o p l e t . The dispersed phase has a cohesive strength lower than the adhesive strength between the matrix and the d r o p l e t s i f the wake i s s i n g l e - t a i l e d . A double-t a i l e d wake i n d i c a t e d the opposite c o n c l u s i o n . A thorough study of the e l e c t r i c a l p r o p e r t i e s of the phase-11 18—20 separated glass has a l s o been c a r r i e d out r e c e n t l y D i e l e c t r i c measurements, explained i n terms of Maxwell's theory of inhomogeneous d i e l e c t r i c s , g i v e values f o r the l o s s f a c t o r and d i e l e c t r i c constant when the s p a t i a l d i s t r i b u t i o n i s known. The l a t t e r i s determined through e l e c t r o n microscopic observations. When the values of these constants are known, the e v o l u t i o n and d i s t r i b u t i o n of the new phase can be followed. The e l e c t r i c a l c o n d u c t i v i t y (d.c.) i s a s t r u c t u r e s e n s i t i v e property which gives more i n f o r m a t i o n of the d i s t r i b u t i o n of the most h i g h l y conducting phase. The a c t i v a t i o n energy f o r the conduction process i s an i n t r i s i c property of the most.highly conducting continuous phase. Considerable changes i n a l l these parameters a f t e r d i f f e r e n t heat treatments were observed. The recorded v a r i a t i o n was i n s a t i s f a c t o r y agreement w i t h the values deduced from e l e c t r o n - m i c r o s c o p i c observations i n the case of some commercial g l a s s e s . 1.9 Objectives of t h i s I n v e s t i g a t i o n The purpose of the present work i s to study the v i s c o s i t y of d i f f e r e n t commercial glasses i n order to o b t a i n a b e t t e r knowledge of t h e i r behaviour under v a r i a b l e c o n d i t i o n s of temperature, s t r a i n r a t e , composition and thermal h i s t o r y . Only the transformation range, w i t h v i s c o s i t i e s 12 15 v a r y i n g from 10 to 10 p o i s e s , (when the phase separation may be most s i g n i f i c a n t ) was considered. This range i s p a r t i c u l a r l y important as i t i s b e l i e v e d that heat treatments above the t r a n s i t i o n p o i n t could erase the previous thermal h i s t o r y of the glass and thus produce the standardized 1-2 m a t e r i a l required i n commercial p r a c t i c e s . This i s not true f o r glasses where d e v i t r i f i c a t i o n occurs i n the same range of temperature. i 12 2. EXPERIMENTAL 2.1 M a t e r i a l S e l e c t i o n Three types of g l a s s were used i n t h i s i n v e s t i g a t i o n and s i n c e they were a v a i l a b l e i n d i f f e r e n t forms the procedures f o r making the specimens v a r i e d . 2.11 The B o r o s i l i c a t e Glasses A b o r o s i l i c a t e glass of the "Pyrex" type was used f o r samples numbered 3.112-10 to 3.122-05. I t s composition i s given i n Table I . Batches of specimens were made from 9 mm rods a v a i l a b l e f o r glassware making. The rod was embedded i n p l a s t e r of P a r i s to form a slab of cross s e c t i o n 2 inches by 2 inches along the length of the rod. The glass rod and p l a s t e r of P a r i s s l a b was cut i n t o s l i c e s w i t h a diamond impregnated a l l o y wheel. This procedure produced ne g l i g e a b l e chipping of the rod. Furthermore, i t provided good f l a t specimen holders f o r a f i n a l p o l i s h i n g of the ends of. the specimens on a g r i n d i n g wheel. The r e s u l t i n g specimens were as p e r f e c t l y c y l i n d r i c a l as p o s s i b l e , w i t h both faces perpendicular to the a x i s of the c y l i n d e r . Their p a r a l l e l i s m was c o n t r o l l e d by measuring the height of the specimen on the three apices and the center of g r a v i t y of an e q u i l a t e r a l t r i a n g l e i n s c r i b a b l e on the f l a t faces. The v a r i a t i o n which -4 could be observed was g e n e r a l l y l e s s than 10 inches which corresponded to the accuracy of the micrometer used f o r the c o n t r o l . The heat treatment of these specimens was c a r r i e d out i n an argon atmosphere i n s i d e h o r i z o n t a l tube furnaces which were c o n t r o l l e d by "Wheelco" thermi s t o r s . For heat treatment i n the upper temperature range, the specimens were separated by molybdynum d i s k s to prevent them from s t i c k -i n g to one another, and a l s o placed i n s i d e a c l o s e f i t t i n g graphite mold i n order to. prevent the deformation due to viscous flow under t h e i r own weight. TABLE I THE CHEMICAL ANALYSIS. OF DIFFERENT GLASSES Type of Glass S i 0 2 F e2°3 A 1 2 0 3 T i 0 2 CaO MgO Na ?0 K2° L i 2 0 B2°5 "Pyrex" "Corning" A h a l i Glass 80.5 67.4 72.84 .09 2.2 .65 .02 11.88 .58 3.8 6.9 .2,95 0.4 .40 trac e 12.9 25.7 t r a c e Concentration of t h i s element i n % by weight 14 For the lower range of temperatures, no s p e c i a l care was taken and the specimens were heat tr e a t e d i n a i r . Another ternary b o r o s i l i c a t e g l a s s , the chemical a n a l y s i s of which i s given i n Table I , was s u p p l i e d by the "Corning Glass Works." The phase-separation of t h i s g l a s s has been stud i e d i n d e t a i l at Corning and other l a b o r a t o r i e s . Only ten c y l i n d r i c a l specimens, 9 mm i n diameter were received from t h i s company. 2.12 The Binary Glass A b i n a r y g l a s s of composition 70% Si02 - 30% Na^Q was made from a mixture of f i n e l y ground B r a z i l i a n quartz ( p a r t i c l e s i z e 55 microns) and Na2C0g anhydrous powder s u p p l i e d by the Ni c h o l s Chemical Company Li m i t e d (Montreal), the nominal p u r i t y of which was 99.0% Na2C0.j. The mixture was i n i t i a l l y melted i n a pure high d e n s i t y s i n t e r e d alumina c r u c i b l e which was subsequently replaced by a platinum c r u c i b l e to avoid contamination. Each batch was melted at about, 1200°C and he l d f o r s e v e r a l days at t h i s temperature. Then i t was poured i n t o water, f i n e l y ground and remelted at the same temperature. The c y c l e was repeated at l e a s t three times i n order to homogenize the glass and degas: < i t . The specimens were prepared from the ground powder by a hot p r e s s i n g technique to y i e l d c y l i n d e r s of 3/4 inches diameter and about 3/4 inches i n height. The hot p r e s s i n g was c a r r i e d out i n a graphite d i e w i t h a tungsten or molybdenum l i n e r and w i t h molybdenum plungers. The heating was provided by i n d u c t i o n w i t h a " P h i l i p s " i n d u c t i o n u n i t . A l l the steps of the hot p r e s s i n g were conducted i n vacuo. When they were completed, the specimen was quenched by f l u s h i n g the vacuum chamber w i t h argon. The compaction during the hot pre s s i n g was i n d i c a t e d by a d i a l 15 gauge attached to the end of the moving plunger. At f i r s t a f a i r l y f a s t motion of t h i s plunger, which l a s t e d about the same time as that needed to r a i s e the pressure on the specimen, was no t i c e d . In order to insure the com-p l e t i o n of the compaction, the specimen was l e f t under pressure f o r a period of time, r e f e r r e d to as the "hot pr e s s i n g time" taken from the i n s t a n t when no motion of the d i a l was perceptable. The assembly used f o r the hot pr e s s i n g procedure can be seen i n Figure 2. A t r i a l and e r r o r method was used i n order to determine the range of values of the d i f f e r e n t parameters of the hot p r e s s i n g , that i s , temperature, pressure and d u r a t i o n , which provided good specimens. Then, maintaining the temperature as low as p o s s i b l e , a systematic scanning of the ranges f o r the two other parameters defined the best c o n d i t i o n s f o r the process. Specimens f r e e of gas bubbles of a l l s i z e s could not be obtained at temperatures lower than the s o f t e n i n g p o i n t temperature. In order to overcome t h i s d i f f i c u l t y , a three step method was then used. The powder was f i r s t melted and degassed i n s i d e the d i e at the highest temperature achievable i n a reasonable period of time, without any s t r e s s a p p l i e d . Then i t was cooled down to the temperature at which the heat treatment, i f any, was c a r r i e d out. The l a s t step co n s i s t e d i n the a c t u a l hot p r e s s i n g at a much lower temperature i n order to shape the specimen. Table 2 gives the d e t a i l s of the f a b r i c a t i o n c o n d i t i o n s used f o r the p r e p a r a t i o n of the specimens. In Figure 3 the binary phase diagram of the g l a s s i s shown. This was used to i n t e r p r e t the s i g n i f i c a n c e of the d i f f e r e n t temperatures of heat treatment i n terms of the phases to be expected. The specimens obtained through t h i s procedure had a smooth shape 16 Figure 2 : Schematic drawing of the hot p r e s s i n g assembly 1 Specimen 2 Molybdenum ram 3 Molybdenum l i n e r 7 Graphite ram 9 Induc t i o n c o i l 4 Graphite sleeve 5 Graphite d i e 6 C o n t r o l thermocouple 8 I n s u l a t i n g glass tube 10 Vacuum chamber TABLE 2 PARAMETERS OF THE FABRICATION OF THE BINARY GLASS SPECIMENS Numb er Degassing treatment Heat Treatment Hot P r e s s i n g Temperature Duration Temperature Duration Temperature Duration Pressure °C minutes °C minutes PC minutes , p s i 3.13 -000 1150 80 690 8 1 1200 15 660 10 2 1150 15 660 10 2000 3 1200 15 No heat treatment 660 10 4 1020 28 600 10 5 1100 20 650 12 6 1100 32 670 12 3.13 -100 1100 40 840 4 650 8 1 1100 28 850 10 670 10 2 1050 28 840 10 660 12 2000 3 1005 30 840 10 700 20 4 1050 32 840 10 660 12 5 900 200 840 10 600 10 _ _ _ _ _ _ — — Figure 3 : P a r t i a l phase-diagram of N a 2 0 - S i 0 2 19 and were transparent. A brownish colour could be n o t i c e d i n areas which had been i n contact w i t h molybdenum, although no s i m i l a r c o l o r a t i o n was observed w i t h the tungsten. This may be explained by the d i f f u s i o n of some molybdenum ions i n t o the g l a s s . 2.13 The A l k a l i Glasses Glass beads of average diameter 400 microns were obtained from the "3M Company" St. P a u l Minnesota. Their chemical a n a l y s i s i s given i n Table 1. Two types of specimens were f a b r i c a t e d from the beads by the hot p r e s s i n g technique, but due to the d i f f e r e n c e i n the v i s c o s i t y - t e m p -erature r e l a t i o n s h i p between.the two g l a s s e s , the method had to be adapted to avoid the e x t r u s i o n of the specimen between the sleeve and the d i e . The same systematic v a r i a t i o n of the parameters of the f a b r i c a t i o n r e s u l t e d i n a simpler method of producing bubble f r e e specimens. This was achieved at a temperature of 650°C, i . e . , around the s o f t e n i n g p o i n t , by applying a pressure of 2,000 p s i over a p e r i o d of 5 minutes. O p t i c a l microscopy could b a r e l y detect the boundaries of the beads i n the specimens. A thinner l i n e r which e l i m i n a t e d the s c a l i n g of the specimen during the removal from the hot p r e s s i n g die could be used because the temperature was lower than f o r the binary g l a s s . T h i s r e s u l t e d i n l e s s c o r r o s i v e a c t i o n and reduced i n t e r a c t i o n between the metal of the sleeve and the g l a s s . A l l specimens numbered from 3.141-000 to 3.142-220 were pre-pared according to t h i s procedure. Specimens numbered from 3.141-000 to 3.141-303 were made of glass beads only. Specimens numbered from 3.142-000 to 3.142-220 contained f i n e B r a z i l l i a n quartz powder (55 microns) 5% by weight. The mixture of quartz powder and glass beads was wet-mixed w i t h 5% by weight 20 of water i n order to o b t a i n an homogeneous mixture and to prevent separation from each other during the handling. In s p i t e of the numerous precautions taken during the p r e p a r a t i o n , many specimens had to be discarded as t h e i r shape was d i s t o r t e d and no longer c y l i n d r i c a l . This was due to the bonding of the t i p s of the plungers to the specimens which f r e q u e n t l y broke during the removal of the specimen. The reshaping of the specimens was a very d e l i c a t e o p e r a t i o n , as dimensionally, a c l o s e tolerance was e s s e n t i a l f o r the accurate measure-ment of the v i s c o s i t y . 2.2 Experimental Procedure 2.21 Study of the Creep Behaviour Since the v i s c o s i t i e s to be determined were i n the range of 12 15 10 to 10 poises a p a r a l l e l p l a t e viscometer was used. The p r i n c i p l e of the measurements i s as f o l l o w s . The c y l i n d r i c a l specimen i s compressed between two p a r a l l e l p l a t e s under a constant load. The distance between the two p l a t e s of the viscometer i s recorded as a f u n c t i o n of time. Because of the c r i t i c a l dependence of the v i s c o s i t y on the temperature, the whole operation i s c a r r i e d out at a p r e c i s e l y c o n t r o l l e d temperature. For t h i s purpose a s p e c i a l equipment was b u i l t . A schematic diagram of the appara-tus i s shown i n Figure 4 and a photograph of i t can be seen i n Figure 5. The two p a r a l l e l p l a t e s were the t i p s of two 3/4 i n c h diameter molybdenum rods which were attached on s t a i n l e s s s t e e l water cooled rams. Only the upper ram was movable. A given load was a p p l i e d to the specimen by a p i s t o n i n s t a l l e d d i r e c t l y on the upper ram which was a c t i v a t e d by n i t r o g e n under pressure. S p e c i a l care was taken to reduce f r i c t i o n on the upper ram and to ensure u n i a x i a l alignment. The frame was water-cooled to avoid 21 Tip of the lower ram D i s p o s i t i o n of the thermocouples 1-2-3 C o n t r o l 4-5-6 Measure Figure 4 : Schematic diagram of the c r e e p - t e s t i n g apparatus Figure 5 : The c r e e p - t e s t i n g apparatus. 23 deformations which might r e s u l t from temperature changes during the experiments. The measurement of the displacement of the upper ram was obtained by the s i g n a l output from an electromagnetic transducer attached to the top of the ram. The c a l i b r a t i o n and r e s e t t i n g were accomplished by a micrometer screw mounted between the transducer and the ram. The t r a n s -ducer output was analysed by a " P h i l i p s PR 19300" d i r e c t reading measuring br i d g e . The output of the bridge.was fed i n t o a "Heathkit" servo recorder through a f i l t e r i n g device and.a p o t e n t i a l b i a s . The temperature was measured at s i x points equ a l l y spaced around the specimen by P t - P t 10% Rh thermocouples. These thermocouples were held a g a i n s t the specimen e l a s t i c a l l y . Three of them were used to check the temperature of the furnace w i t h a potentiometer. The added emf of the other three was fed i n t o a "Honeywell" d e v i a t i o n m u l t i p l i e r used as a monitoring system f o r the c o n t r o l of the furnace temperature. By t h i s arrangement the temperature could be hel d w i t h i n - .01°C. The heating was provided by a three-independent-element r e s i s t a n c e furnace . P r e l i m i n a r y experiments showed that i n s p i t e of the length of the heating zone w i t h respect to i t s diameter, i t was not p o s s i b l e to achieve a uniform temperature i n the specimen w i t h a s i n g l e element furnace due to the heat s i n k e f f e c t of the water cooled rams. To know and c o n t r o l the temperature d i s t r i b u t i o n i n s i d e the specimen, a s p e c i a l specimen was used i n which three s m a l l holes i n the center and c l o s e to i t s e x t r e m i t i e s had been d r i l l e d w i t h an u l t r a s o n i c - d r i l l e r . Three very t h i n chromel-alumel thermocouples were i n s e r t e d i n these holes and,by t h i s means,temperatures were recorded along the v e r t i c a l a x i s of the specimen. I t was observed that i n the case of the single-element furnace, gradients of temperature 24 up to 50°C could e x i s t i n s i d e the specimen under steady s t a t e c o n d i t i o n s . By the proper adjustment of the " V a r i a c " autotransformers which were attached to each of the three elements, a c l o s e c o n t r o l of the v e r t i c a l gradient of temperature could be maintained. F i n a l l y by t r i a l and e r r o r a temperature gradient of l e s s than 1°C (the s e n s i t i v i t y of the potentiometer used f o r the measurement) was achieved. The whole of the hot zone was enclosed i n an a i r - t i g h t container c o n s t a n t l y f l u s h e d w i t h argon.to prevent the o x i d a t i o n of the molybdenum p a r t s . In s p i t e of t h i s , considerable o x i d a t i o n of the molybdenum components occurred e s p e c i a l l y during the i n t r o d u c t i o n to and withdrawal from the hot furnace of the specimen. To i n s u r e the uniform q u a l i t y of the surface between which the specimen were compressed, the t i p s of the rams were replaced by pure high d e n i s t y s i n t e r e d alumina which broke f r e q u e n t l y . The operating procedure was as f o l l o w s . The furnace was maintained at the d e s i r e d temperature. The specimen was introduced through a removable door and loaded immediately under l i g h t pressure to ensure a u n i f h e a t i n g by the conduction heat t r a n s f e r through the e x t r e m i t i e s . The time re q u i r e d to reach the e q u i l i b r i u m temperature i n the specimen was found to be between ten and twenty minutes. This had been observed w i t h the c o n t r o l -specimen. During t h i s heating-up p e r i o d c a l i b r a t i o n of the creep-recording -4 equipment was c a r r i e d out. A dial-gauge w i t h an accuracy of 10 inches was used f o r t h i s purpose. The d e s i r e d pressure was then a p p l i e d and the deformation of the specimen was recorded. 2.22 E l e c t r o n Microscopy D i r e c t carbon r e p l i c a s of preshadowed f r e s h l y f r a c t u r e d 25 specimens were observed w i t h a " H i t a c h i " HU11A e l e c t r o n microscope. Chromium,gold or platinum was used f o r the shadowing. D i f f e r e n t etching techniques were a l s o used. The specimen was broken by ex p l o s i v e f r a c t u r e by squeezing i t i n a v i s e . The p r i n c i p a l e t ching agents were e i t h e r 2% HF d i l u t e s o l u t i o n used f o r d i f f e r e n t periods of time (one to f o r t y seconds), or one of 2N HC1, 2N H 2S0^, and 2% HF f o r two hours f o r "Pyrex" g l a s s , or d i s t i l l e d water f o r one second i n the case of the "Corning" g l a s s . For each specimen at l e a s t two d i f f e r e n t f r e s h l y f r a c t u r e d g l a s s surfaces were r e p l i c a t e d . From each r e p l i c a at l e a s t two d i s t i n c t pieces were mounted on copper g r i d s f o r observation. The r e p l i c a s were l i f t e d from the sample by f l o t a t i o n i n d i s t i l l e d water or i n some d i f f i c u l t cases using 2% HF s o l u t i o n . A systematic scanning of each g r i d was done to give s t a t i s t i c a l r e s u l t s of the observed f e a t u r e s . The m a g n i f i c a t i o n range v a r i e d from 5,000 to 80,000 times. 26 3. RESULTS 3.1 Measurement of the V i s c o s i t y 3.11 C a l c u l a t i o n of the V i s c o s i t y from the Creep Curves , 3.111 The Formula The theory of the p a r a l l e l p l a t e viscometer allows one to c o r r e l a t e , at any moment, the r a t e of creep of a c y l i n d r i c a l specimen compressed under a constant load to the v i s c o s i t y of the m a t e r i a l as f o l l o w s : V_ 1 m n 3V d h , l • V , u ' " dE (h2 + y-h-5) n i s the v i s c o s i t y c o e f f i c i e n t i n p o i s e s , F the t o t a l a p p l i e d l o a d , h the height of the specimen at.any time, t , V the volume of the specimen, and - -^7 the r a t e of creep, that i s , the slope of the creep curve. This r e s u l t has been e s t a b l i s h e d f o r Newtonian l i q u i d s , i . e . 23 l i q u i d s f o r which the v i s c o s i t y i s independent of the r a t e of shear 3.112 P r e c i s i o n of the Measurements The continuous r e c o r d i n g o f the creep curve could not be used d i r e c t l y f o r the measurement of the s l o p e , p r i m a r i l y because of the non-l i n e a r response of the transducer used f o r recording the change of height as a f u n c t i o n of time. Therefore the p l o t from the chart recorder -were transformed by the use of c a l i b r a t i o n curves. This was a discontinuous process which introduced some e r r o r i n the a c t u a l creep curve. This e r r o r was found to be l e s s than 5%. The measurements of the creep r a t e were c a r r i e d out on these r e p l o t t e d curves. Since t h i s measurement i n v o l v e d drawing a tangent to the curve, a new systematic e r r o r was introduced. However, i t was estimated to be i n the same order as the previous e r r o r . 27 Therefore, the c a l c u l a t e d values of the v i s c o s i t y obtained from these measurements i n h e r i t an e r r o r i n the range of 5% to 10%. Thus a l l the r e s u l t s given i n the t a b l e can be considered as accurate w i t h i n about 8%. 3.113 Flow C h a r a c t e r i s t i c s of the Glasses The flow c h a r a c t e r i s t i c s of the glasses used i n t h i s i n v e s t i g a -t i o n were f i r s t determined i n order to ensure that t h i s flow behaviour was Newtonian. This was done by studying the creep curves of the "Pyrex" glass as a f u n c t i o n of the s t r e s s a p p l i e d at 568°C which i s i n the tempera-ture range at which the subsequent measurements were made. The a p p l i e d load was v a r i e d from 3,900 p s i to 31,200 p s i . This allowed an 8-f o l d change i n the shear r a t e . In Table 3 (Appendix I) the values of the v i s c o s i t i e s a t d i f f e r e n t elapsed times and at the a p p l i e d pressure used i n the experiments are given. A p l o t of the values of the v i s c o s i t i e s obtained from the creep curves i s shown i n Figure.6. Some re p r e s e n t a t i v e creep curves are a l s o p l o t t e d i n Figure 7. The equation of the s t r a i g h t l i n e which gives the best l e a s t square f i t of the values of the v i s c o s i t y as a f u n c t i o n of the pressure can be w r i t t e n as f o l l o w s n = 1 0 1 4 x (A + B.P) i f n and P are measured i n poises and p s i r e s p e c t i v e l y . The values of A and B are given i n Table 4. For the maximum v a r i a t i o n of the pressure P, the c o n t r i b u t i o n to the value of the v i s c o s i t y due to the term B va r y i n g w i t h P i s 16% and 7% r e s p e c t i v e l y of the value of the constant term A, at 50 minutes and at 100 minutes. Therefore i t can be assumed that s i n c e the v a r i a t i o n i s approximately w i t h i n the p r e c i s i o n of the determination of A, the value of B can be set to zero. V i s c o s i t y 10 x poxses Figure 6 : V i s c o s i t y as a f u n c t i o n of the pressure Figure 7 : Creep curves f o r d i f f e r e n t pressures. 30 TABLE 4 COEFFICIENTS OF THE LINE OR RELATIONSHIP BETWEEN VISCOSITY AND PRESSURE at 50 mn at 100 mn C o e f f i c i e n t E r r o r P r e c i s i o n C o e f f i c i e n t E r r o r P r e c i s i o n A = .150 ± .006 4.0% A = .168 ± .007 4.2% B = (.856 ± .291)10" 6 34.0% B = (.548 ± .337)10 - 6 61.5% TABLE 7 PARAMETERS OF THE EQUATION LOG n = • A + B h at 50 mn at 100 mn Parameter Value E r r o r P r e c i s i o n % Parameter Value E r r o r P r e c i s i o n % Glass as received A -3.58 .51 14.4 A -3.45 .47 13.2 -4 B x 10 1.418 .042 2.9 -4 B x 10 1.421 .038 2.7 Q kcal/mole 65.4 1.9 2.9 Q kcal/mole 65.5 1.8 2.7 Glass a f t e r annealing A -9.27 .45 4.9 A -8.40 .60 7.2 -4 B x 10 1.904 .038 2.0 -4 B x 10 1.838 .050 2.7 Q kcal/mole 87.6 1.7 2.0 Q kcal/mole 84.4 2.3 2.7 31 The s c a t t e r i n the data i s a r e f l e c t i o n of the dependence of the v i s c o s i t y on the thermal h i s t o r y of the specimens and w i l l be discussed l a t e r on. However, the o v e r a l l r e l a t i o n s h i p as shown i n Figure 6 suggests that the glass behaves as a Newtonian l i q u i d . Equation 1 f o r the p a r a l l e l p l a t e viscometer can be used i n t h i s temperature range f o r the determination of the v i s c o s i t y . A l l subsequent.experiments were c a r r i e d out under a pressure of 16,500 p s i . 3.12 B o r o s i l i c a t e Glasses 3.121 "Pyrex" Glass 3.1211 V a r i a t i o n of the Temperature The temperature dependence of the v i s c o s i t y of the "Pyrex" glass was i n v e s t i g a t e d i n two sets of specimens. In the f i r s t s e t , specimens were used as received and without s u b j e c t i n g them to any heat treatment. In Table 5 (Appendix I) data f o r the values of the v i s c o s i t y versus temperature at a constant load are shown. In the same t a b l e , r e s u l t s from other experiments which were conducted i n the same con d i t i o n s of temperature and load have been included f o r comparison. These experiments, namely alignment of the ram a f t e r exchange of the t i p s and c a l i b r a t i o n of d i f f e r e n t p a r ts of the equipment, were necessary to i n s u r e the u n i f o r m i t y of the experimental c o n d i t i o n s . ( I n c i d e n t a l l y , other r e s u l t s obtained from the study of the glass d e n s i f i c a t i o n during the t e s t , were a l s o i n c l u d e d ) . A wider s c a t t e r i s n o t i c e a b l e f o r these values which correspond to the temperature of 568°C. I t can be accounted f o r by the v a r i a t i o n of the glass batches which were used f o r the f a b r i c a t i o n of the specimens. 32 The second set of specimens was "annealed" at 500°C f o r 44 hours i n a furnace w i t h an atmosphere of a i r . The r e s u l t s are given i n Table 6 (Appendix I) and are p l o t t e d i n Figure 9. In both Figures 8 and 9, the logarithm of the v i s c o s i t i e s i s p l o t t e d versus the i n v e r s e of the absolute temperature. The curves appear to be l i n e a r . The equation of the s t r a i g h t l i n e s i s as f o l l o w s : i , , B log n = A + -In order to determine an a c t i v a t i o n energy Q f o r the flow process the previous equation can be r e w r i t t e n : R Q = B loge. where R i s the gas constant. The values of the parameters A, B, and Q are given i n Table 7. The p r e c i s i o n of the data given i n t h i s t a b l e i s i d e n t i c a l to the accuracy of the determination of the parameters by the l e a s t square f i t method, assuming that the values of the v i s c o s i t i e s are exact. One n o t i c e s that t h i s p r e c i s i o n i s b e t t e r than the accuracy of the measurements of the v i s c o s i t i e s . Therefore, t h i s e r r o r could be neglected and Q could be given w i t h the same p r e c i s i o n as n, that i s 8%. Nevertheless two d i f f e r e n t values of Q were found f o r the two sets of experiments: Q-^  = 65 * 5 kcal/mole f o r the "as re c e i v e d " glass Q 2 = 8 7 - 7 kcal/mole f o r the "annealed" glass Some c h a r a c t e r i s t i c s of the creep curves are shown i n Figure 10. 3.1212 E f f e c t s of D i f f e r e n t Heat Treatments In order to study the e f f e c t of phase-separation on the v i s c o s i t y of the "Pyrex" g l a s s , heat treatments over the temperature range l o g n poise l o g n = f ( •=• ) IS 15 o 14 a . 13 V i s c o s i t y • at 50 mn O at 100 mn 10 3/ Figure 8 : V a r i a t i o n . o f the v i s c o s i t y w i t h the telaperature . Glass as received Figure 9 : V a r i a t i o n of the v i s c o s i t y w i t h the temperature . "Annealed " glass Figure 10 : Creep curves at d i f f e r e n t temperatures. 36 500 to 900°C,were c a r r i e d out. Below 500°C no change could be n o t i c e d . Above 700°C, problems of r e d u c t i o n of the oxides,or c r y s t a l l i s a t i o n of the glass i n contact w i t h the graphite mold have been encountered. Such problems l i m i t e d d r a s t i c a l l y the p e r i o d of time during which the specimen could be h e l d at the high temperatures without too many p a r a s i t i c t r a n s f o r -mations . Table 8 (Appendix I) summarizes the values of the v i s c o s i t i e s of the specimens subjected to d i f f e r e n t heat treatments. In Figures 11.to 17 the v i s c o s i t y values are p l o t t e d against the time of heat treatment i n a l o g - l o g p l o t . These p l o t s suggest a l i n e a r v a r i a t i o n of the logarithm of the v i s c o s i t y versus logarithm of the time. The equations of the s t r a i g h t l i n e s g i v i n g the best least-squares f i t are found i n Table 9, as l o g n = A + B l o g t , where n i s i n poises and t , the time of heat-treatments i n hours. In F i g u r e , 18 B i s p l o t t e d as a f u n c t i o n of the h e a t - t r e a t -ment temperature. There appears to be a maximum f o r the r a t e at which the v i s c o s i t y changes w i t h the time. This maximum i s a t t a i n e d at about, the temperature at which the measurements of the v i s c o s i t y had been c a r r i e d out. A b e t t e r account of the d i f f e r e n c e s i n the v a r i a t i o n s of the v i s c o s i t y i s shown i n Figure 19 where, f o r a given time of heat t r e a t -ment, the v i s c o s i t y i s p l o t t e d against the temperature of the heat treatment. 3.122 "Corning" Glass Only a few specimens were obtained from the "Corning" research l a b o r a t o r i e s . 37 Viscosity Poist • I0 1 4 10 ipo hour Time F i g u r e II : 5 0 0 s C F i g u r e 12 : 565 °C F i g u r e 13 : 5 7 5 ° C Viscos i ty poise a s * ' Y * ' * s 1 •P Time 1 0 0 hour Figure 14 6 0 0 » C 5 0 A O v i s c o s i t y value l e a s t square f i t and p r e c i s i o n at minutes 1 0 0 V i s c o s i t y as a f u n c t i o n of the time of heat treatment at d i f f e r e n t temperatures, 38 Mo" Viscosity Poise 14 h o ' 19 IO t _. IOO , hour Time Figure 15 : 650 °C Figure 16 : 7 5 0 ° C A O • . o ' 4 Viscosity poise ±==\ A • K ) , S .1 I hour IP T i m . F i g u r e 17 : 8 0 0 " C v i s c o s i t y value 50 least.square f i t and p r e c i s i o n at minutes. 100 V i s c o s i t y , as a function of the time of heat treatment a t ' " d i f f e r e n t temperatures. 39 TABLE 9 PARAMETERS OF THE EQUATION LOG n = A + B LOG t FOR DIFFERENT HEAT TREATMENTS Temperature At 50 mn At 100 mn °c A AA B AB A AA B AB 500 13.535 .111 -.059 .061 13.552 .103 -.056 .056 565 11.927 .094 1.32 .07 12.106 .161 1.19 .13 575 13.187 .087 .448 .048 13.229 .093 .519 . 110 600 13.440 .036 .411 .022 13.512 .037 .450 .025 650 13.424 .028 .107 .025 13.509 .093 .122 .083 750 13.195 .363 .0228 .0325 13.265 .040 .0095 .0365 800 13.321 .018 -.0097 .018 13.378 .018 .021 .018 . 40 Figure 18 V a r i a t i o n of the r a t e of change of the v i s c o s i t y w i t h the temperature of heat treatment . Figure 19 : V i s c o s i t y as a f u n c t i o n of the temperature of heat treatment a f t e r d i f f e r e n t times of heat treatment . 42 The v i s c o s i t y of these specimens was determined a f t e r they had been subjected to d i f f e r e n t heat-treatments. The unusual shape of the creep curves i s shown i n Figure 20. Table 10 (Appendix I) gives the d e t a i l of the c o n d i t i o n s of the experiments and the v i s c o s i t i e s obtained from them. 3.13 The Binary Glass Two sets of experiments were c a r r i e d out w i t h the binary g l a s s . In the f i r s t case, no s p e c i a l heat treatment was a p p l i e d . The glass was degassed at about 1,200°C, cooled down to 660°C, and hot pressed f o r f i v e minutes at t h i s temperature. On the second set of specimens, an a d d i t i o n a l heat treatment at 850°C was used. The l o c a t i o n s of the d i f f e r e n t temperatures with respect to the expected p h y s i c a l s t r u c t u r e of the binary system are shown 23 i n the phase diagram of Na20-Si02 as given i n Figure 3. The values of the v i s c o s i t y f o r both sets of data are p l o t t e d against the r e c i p r o c a l of the absolute temperature i n Figure 21 and are given i n Table I I (Appendix I ) . The l a r g e r s c a t t e r of the r e s u l t s could be explained from some v a r i a t i o n of the composition of the g l a s s , l a c k of constancy of the heat treatments and presence of minute gas bubbles which appeared during the f a b r i c a t i o n of the specimen. A d i f f e r e n c e i n c o l o r has a l s o been no t i c e d from one specimen to the other. Because of the d i f f i c u l t i e s encountered w i t h t h i s type of g l a s s , i n v e s t i g a t i o n s were discontinued. 43 44 . — : : , l o g n ] ® I -13.0 o o • A & S'8 A •12.5 10 3 x 1/T ' 1.25 1.27 Temperature 1.29 _ I I 1 1 I a) No heat treatment A 30 O v i s c o s i t y value at 60 minutes • ; 9j l o g n o A 13.0 O A • O ° A 2 A 12.5 1.25 Temperature 10 3 x 1/T 1.30] — l 1 1 1 J b) Heat t r e a t e d Figure 21 V i s c o s i t y of the binary the temperature . glass,as a f u n c t i o n of 45 3.14 A l k a l i Glasses 3.141 A l k a l i Glass without a d d i t i v e s Specimens made of hot pressed glass beads were tested at d i f f e r e n t temperatures. Before any t e s t was performed the batches of the specimens were subjected to various heat treatments. The d e t a i l s about these heat treatments and the corresponding values of the v i s c o s i t y are recorded i n Table' 12 (Appendix I ) . In Figures 22 and 23> a p l o t of the logarithm of the v i s c o s i t y versus the r e c i p r o c a l of the absolute tempera-ture can be found. A very l a r g e s c a t t e r of the r e s u l t s i s n o t i c e a b l e . I t could be accounted f o r by the l a c k of homogeneity of the m a t e r i a l and the i r r e g u l a r d i v i t r i f i c a t i o n of the gl a s s e s . To c o n t r o l the d e v i t r i f i c a t i o n of the specimens and ensure homogenization, experiments were conducted w i t h an a d d i t i o n of.quartz 1 c r y s t a l s to the bead mixture. 3.142 A l k a l i Glass + Quartz Powder Two batches of specimens were made w i t h a mixture of glass beads and quartz powder. The f i r s t s e r i e s was tested s t r a i g h t a f t e r f a b r i c a t i o n . The second s e r i e s was "annealed" f o r 20 minutes at 500°C. In Table 13 (Appendix I ) , experimental d e t a i l s of the t e s t s are given. In Figure 24 the logarithm of the v i s c o s i t y versus the r e c i p r o c a l of the absolute temperature curves are p l o t t e d . A l a r g e s c a t t e r s t i l l e x i s t s on the p l o t s . 46 log n Viscosity A • 13 .5 A A - 13 . 0 i , A * * A A A A A A A 1.15 I0 3« l / T > > Temperature .1.20 , No heal treatment Figure 2 2 Alkali g lass log n V i s c o s i t y 1-13.5 0 0. 0 o 13.0 O o * o A aA. A 0 o A L12_ \ 0 » | / T Temperature _1M5 , i i > _ A O • heat treated for 7 minutes at O 15 5 5 0 F i g u r e 2 3 Alkali glass log n V i s c o s i t y A -13.5 A A -13 .0 8ft a o A * i A ^ A A A « A a s fab r ica ted 0 O annea led 2 0 hours a t 5 0 0 ° C 1.15 1.20 I0*xl/T "fertpeiuture Figure 2 4 Alkali glass + quartz powder Viscosity value :^SS^ at mn A l k a l i glass : v i s c o s i t y as a f u n c t i o n of the temperature f o r d i f f e r e n t heat treatments and a d d i t i o n s . 47 3.2 S t r u c t u r a l Study 3.21 "Pyrex" Glass In an attempt to c o r r e l a t e the v a r i a t i o n of the measured v i s c o s i t i e s w i t h some i n t e r n a l change of s t r u c t u r e of the specimen, an e l e c t r o n microscopic study on a set of specimens was c a r r i e d out. The 4 f i n a l v i s c o s i t i e s of t h i s s et had values i n the range 0.2 to 6.0 (10 p o i s e s ) . D i f f e r e n t shadowing and etching techniques were used. The most s t r i k i n g f e a t u r e which i s n o t i c e a b l e , i s a constant grainy appearance of the micros t r u e t u r e s . To determine whether t h i s grainy s t r u c t u r e was due to some spurious e f f e c t o r i g i n a t i n g i n the method of p r e p a r a t i o n of the specimen, d i f f e r e n t techniques were used on the same specimen. In Figures 25 a, b, c and d, mic r o s t r u c t u r e s of the specimen which had been etched f o r 1, 5, 10 and 30 seconds r e s p e c t i v e l y , i n d i l u t e d 2% h y d r o f l u o r i c a c i d are represented. The d i f f e r e n t fragments were shadowed, a f t e r e t c h i n g w i t h platinum a t an angle of 30 degrees and carbon r e p l i c a s were made. In the backgroud of a l l the m i c r o s t r u c t u r e s , p a r t i c l e s of about 100 Sngstrom i n s i z e are d i s c e r n a b l e . A few l a r g e r p a r t i c l e s i n the range of 1,000 to 2,000 8 are a l s o seen i n these F i g u r e s . The specimen had been heat treated at 600°C f o r 19 hours and then creep tested f o r 100 minutes. In Figures 26 a and b, gold was used f o r shadowing because of i t s p e c u l i a r growth p r o p e r t i e s on glass s u b s t r a t e . Figure 26a represents an unetched specimen w h i l e Figure 26b shows the same specimen a f t e r an etch of 1 second i n d i s t i l l e d water. 4 8 Figure 25 : M i c r o s t r u c t u r e s of the same specimen etched f o r d i f f e r e n t times i n d i l u t e d 2% HF s o l u t i o n . a) No etching . b) 1 second i n water . Figure 26 : V a r i a t i o n of appearance of the m i c r o s t r u c t u r e due to the etching technique . 50 On both photographs p a r t i c l e s of about 200 to 400 Angstroms can be seen on a background of very f i n e p a r t i c l e s (about 50 8). The corresponding specimen was heat tr e a t e d f o r 24 hours at 600°C and subjected to creep deformation f o r 70 minutes. With d i f f e r e n t heat treatments of the g l a s s e s , various shapes, s i z e s and d e n s i t i e s of p a r t i c l e s have been encountered. In some cases the p a r t i c l e s were approximately s p h e r i c a l i n shape; t h e i r s i z e increased w i t h the i n c r e a s i n g period of heat treatment, and at the same time, a corresponding decrease of t h e i r density was observed. In Figure 27a,a uniform background of p a r t i c l e s of about 120 X i n diameter i s evident. This micros true t u r e i s e x h i b i t e d by a specimen which had undergone a creep t e s t of only 150 minutes. In Figure 27b, the micros true t u r e of a,specimen, heat,treated f o r 4 hours at 600°C and then creep tested f o r 100 minutes i s represented. The s i z e of the p a r t i c l e s v a r i e d from 500 to 1,000 £ w h i l e t h e i r d e n s i t i e s decreased w i t h the corresponding increase i n s i z e . A heat treatment of 19 hours at the same temperature followed by a 100 minutes v i s c o s i t y measurement t e s t , revealed a s t r u c t u r e of l a r g e r p a r t i c l e s which had coalesced. Their diameter ranged.from 800 to 1,500 £ as shown i n Figure 27c. E v e n t u a l l y , a f t e r 24 hours at 600°C and a t e s t of 110 minutes, p a r t i c l e s between 2,000 and 3,000 £ covered the whole specimen, Figure 27d. For a given period of heat treatment at d i f f e r e n t temperatures, various m i c r o s t r u c t u r e s were observed. In Figure 28, the d i f f e r e n t s t r u c t u r e s which were encountered a f t e r a heat treatment l a s t i n g 24 hours at i n c r e a s i n g temperatures are shown. There was a general tendency f o r the p a r t i c l e s to increase i n diameter w i t h i n c r e a s i n g temperature up to about 575°C. This was followed by a decrease i n the s i z e of the p a r t i c l e s which composed the background,with an infrequent occurrence of l a r g e r p a r t i c l e s . Figure 27 : V a r i a t i o n of the s i z e of the p a r t i c l e s w i t h the time of heat treatment . a) At 500°C b) At 565°C Figure 28 : V a r i a t i o n of the s i z e of the p a r t i c l e s w i t h the temperature of heat treatment f o r 24 hours . 53 The change of the p a r t i c l e shape was a l s o n o t i c e a b l e i n the high temperature range. Under some c o n d i t i o n s , a transformation from a s p h e r i c a l shape to a more i r r e g u l a r angular one has been observed. In Figures 29a, b, near p e r f e c t spheres and odd shaped p a r t i c l e s c o e x i s t . The specimens which represent t h i s m i c r o s t r u c t u r e were heat tr e a t e d f o r 2 and 4 hours at 750°C r e s p e c t i v e l y and then tes t e d f o r 70 minutes. A longer heat treatment (22 hours at the same temperature) or a heat treatment f o r the same length of time but at higher temperature (4 hours at 800°C) l e d to the e l i m i n a t i o n of smooth s p h e r i c a l p a r t i c l e s . These are replaced by rough-surfaced i r r e g u l a r ones w i t h o c c a s i o n a l square c r y s t a l s . Some of these c r y s t a l s have been found to achieve a s i z e up to s e v e r a l microns across as shown i n Figure 29d. 3.22 "Corning" Glass An obvious change of the s t r u c t u r e of these glasses could be observed,without any m a g n i f i c a t i o n j a f t e r the creep t e s t or a f t e r any heat treatment. The gl a s s which was almost transparent but w h i t i s h due to g r i n d i n g , came out y e l l o w i s h i n transmitted l i g h t and b l u e i s h i n r e f l e c t e d l i g h t . Because of the high s e n s i t i v i t y of t h i s m a t e r i a l to any heat treatment, the d i f f e r e n t specimens were c l a s s i f i e d according to t h e i r f i n a l v i s c o s i t y , r a t h e r than according to t h e i r heat treatments. In Figure 30, the d i f f e r e n t s t r u c t u r e s observed f o r decreasing v i s c o s i t y are shown. The highest value of the v i s c o s i t y corresponds to a s t r u c t u r e i n which only very f i n e p a r t i c l e s form the background. Their diameters v a r i e d from 70 to 120 X. When the v i s c o s i t y dropped from 0.73 54 c) 22 hours at 750 C. d) 4 hours at 800°C. Figure 29 : Transformation of the shape of the p a r t i c l e s 55 to 0.58 (10 p o i s e s ) , the dimensions of the p a r t i c l e s grew and ranged from 100 to 200 £ as seen i n Figure 30b. For a f u r t h e r decrease of the 14 v i s c o s i t y down to 0.42 (10 p o i s e s ) , the range of the s i z e s of the p a r t i c l e s was s h i f t e d towards higher values and lay between 150 and 300 £ as shown i n the m i c r o s t r u c t u r e represented by Figure 30c. E v e n t u a l l y , 14 f o r the sm a l l e s t achieved measurement of the f i n a l v i s c o s i t y , 0.28 x 10 po i s e s , a new type of s t r u c t u r e appeared. Large smooth c i r c u l a r p a r t i c l e s , i n the range of 400 to 600 £,emerged from a background of grainy s m a l l p a r t i c l e s of 80 £ (Figure 30d). The l a r g e r p a r t i c l e s seemed to have coalesced. a) 0.73 x 10 p o i s e s . b) 0.58 x 10 p o i s e s . 57 4. DISCUSSION 4.1 Theory of the P l a t e Viscometer 4.11 T h e o r e t i c a l Treatment The purpose of a theory i s to express the p h y s i c a l law which r e l a t e s the measured v a r i a b l e s w i t h the p r o p e r t i e s of the tested m a t e r i a l . In the case of the p l a t e viscometer, i t i s des i r e d to f i n d an equation r e l a t i n g the p l a t e s e p a r a t i o n at any time to the v i s c o s i t y of the specimen. I t i s w e l l known that the behaviour of p l a s t i c m a t e r i a l s under s t r e s s i s very complex because both e l a s t i c and viscous deformations take place simultaneously. No s i n g l e theory i s able to take i n t o account the a c t u a l viscous flow which i s encountered i n p r a c t i c e . 24 The f i r s t attempt i n t h i s f i e l d used the f o l l o w i n g s i m p l i f y i n g assumptions about the flow behaviour. I f the c e n t r a l p o i n t of the c y l i n d e r undergoing viscous flow i s considered as a f i x e d o r i g i n , the gradi e n t of v e l o c i t y w i t h respect.to t h i s o r i g i n , i s l i n e a r along each a x i s w i t h respect to i t s e l f . In t h i s case, the f o r c e , a p p l i e d on the specimen, d i s s i p a t e s energy i n the volume of the specimen according to the 3 2 d 1 r e l a t i o n s h i p : E = — n V a' (2) where E i s the energy (F , n the v i s c o s i t y and a the c o e f f i c i e n t of the gradient of v e l o c i t i e s . This leads to the r e l a t i o n s h i p : _ _ . „ _ _ _ . (3) 6nV h h 0 ^ where h Q i s the height of the specimen at time t = 0 and h the height at time t . This r e s u l t i s v a l i d only f o r very s m a l l deformations. 25 A more elaborate theory o r i g i n a t e d from the general equation of motion of a Newtonian f l u i d of v i s c o s i t y n. 58 P IT + pv . Vv = - Vp + riV 2v + ^ n VxV . v (4) o t J Considerations of c o n t i n u i t y and symmetry, along w i t h the r e s t r i c t i o n to very slow motion of every point, permitted a r e d u c t i o n to equation (5). P ^ = - Vp + r,V2v (5) Equation (5) was then solved i n c y l i n d r i c a l co-ordinates i n t r o d u c i n g the boundary c o n d i t i o n s . Assumptions were made that no s l i p p a g e occurred at the p l a t e s and that the steady s t a t e flow c o n d i t i o n was achieved. Therefore t h i s r e l a t i o n s h i p i s not a p p l i c a b l e f o r the i n i t i a l stage of the creep deformation. A f u r t h e r s i m p l i f i c a t i o n was subsequently introduced which i s j u s t i f i e d when the radius of the c y l i n d e r R i s at l e a s t ten times l a r g e r than the height h. Then a p a r a b o l i c v a r i a t i o n of the v e l o c i t y of the f l u i d i s observed. I t was a l s o assumed that a given, h o r i z o n t a l plane remained h o r i z o n t a l during the deformation. Under these c o n d i t i o n s equation (5) can.be i n t e g r a t e d i n t o equation (6). L± m rk _ I,, J L ( 6 ) 3^V " ( h * " hj> 8 tr 26 F i n a l l y , a theory has been derived on the b a s i s of the . 27 c a l c u l a t i o n of the apparent Young's modulus f o r rubber c y l i n d e r s . This produced the r e l a t i o n s h i p given i n equation (1) where the r e s t r i c t i o n of R >10h and the l i m i t a t i o n to a s m a l l amount of deformation can be eliminated, The d e t a i l s of t h i s d e r i v a t i o n are given i n Appendix ( I I I ) . 3nv h h Q 8 tt >ir V*QJ K±J This r e l a t i o n s h i p : i s p a r t i c u l a r l y a p p l i c a b l e to a wide range of specimen thickness because, as the assumption became improbable, t h e i r consequences are r e l a t i v e l y unimportant. The corresponding term becomes n e g l i g i b l e w i t h respect.to the others. 59 4.12 V a l i d i t y of the Assumptions 4.121 The specimen i s deformed under u n i a x i a l compression Although, because of the design and t h e f a i r l y t i g h t f i t of the s l i d i n g elements, no motion of the rams could occur p e r p e n d i c u l a r l y to t h e i r a x i s , the a x i a l symmetry of each specimen was checked a f t e r the t e s t to ensure that no shear s t r e s s had been a p p l i e d on them during the measurement. 4.122 H o r i z o n t a l planes remain h o r i z o n t a l during the flow S p e c i a l care was taken to use specimens w i t h p a r a l l e l f a c e s , and faces perpendicular to the a x i s of the specimen. The p a r e l -l e l i s m was c o n t r o l l e d by measuring the thickness of the specimen on three e q u a l l y spaced p o i n t s around the perimeter. Specimens having a thickness -4 d i f f e r e n c e of more than 10 inc h (the accuracy of the micro-meter used f o r the c o n t r o l ) were discarded. S i m i l a r measurements were c a r r i e d out to check the p a r e l l e l i s m a f t e r the creep t e s t . Due to the wear of the t i p s of the rams, _3 v a r i a t i o n s of the thickness of the specimen up to 10 or more were some-times observed. In such cases, the r e s u l t s were considered to be d o u b t f u l . The p a r a l l e l i s m was e s p e c i a l l y c r i t i c a l i n the case of high v i s c o s i t y v a l u e s , where the whole deformation of the specimen could be of the same order as the t i l t i n g deformation. 4.123 "A given h o r i z o n t a l plane remains a plane during the deformation." As already mentioned, t h i s leads to specimens, the a x i a l s e c t i o n of which c o n s i s t s of two segments of a parabola. On the photographs of Figure 31, the a x i a l s e c t i o n of some deformed specimens can be seen; they show the p a r a b o l i c deformation. b) A f t e r deformation : the s c a l i n g of the edge i s due to the adhesion to the t i p of the ram. Figure 31 : P a r a b o l i c deformation of the specimen 61 4.2 V a r i a t i o n of the V i s c o s i t y w i t h Time 4.21 General Considerations In Appendix I I I , i t has been e s t a b l i s h e d that the time, the v i s c o s i t y and the height of the specimen can be r e l a t e d together by a formula as: • ^ T T dt = <j> (h) ( i ) Jo 3 n V where F i s the load a p p l i e d on the specimen,n the v i s c o s i t y at time t and V the volume of the specimen; the right.hand s i d e i s a f u n c t i o n o f the height of the specimen. In the casewherethe v i s c o s i t y i s a constant w i t h respect F to the time, the expression can be reduced to: <jj (h) = -^-^ t ( j ) . That i s , a l i n e a r r e l a t i o n s h i p e x i s t s between <j> (h) and t . To determine to what e x t e n t s was a f u n c t i o n of the time, the v a r i a t i o n of. <|>(h) w i t h the time has been compared with a l i n e a r v a r i a t i o n . To evaluate the d e v i a t i o n of <j> (h) from a s t r a i g h t l i n e , - <j> (h) has been developed i n t o a polynomial of second degree w i t h respect to t , as : 2 <J> (h) = a + bt + c t . The values of the unknown parameters a, b,and c were determined by the l e a s t square f i t method which was programmed i n a 7044 IBM computer. A l l c a l c u l a t i o n s i n v o l v e d i n t h i s work were done by the same computer. In Table 14^ the values of the parameters a, b and c are given and i n Figure 32, the corresponding curves are p l o t t e d . The r a t i o r = c/b and the product r t ^ are c a l c u l a t e d at the same time, t f i s the t o t a l time of each experiment. r gives an e s t i m a t i o n of the r e l a t i v e importance of the p a r a b o l i c term in.((>(h) at the beginning of the experiment, w h i l e r t f gives the same i n f o r m a t i o n f o r the end of the experiment; thus, they give a measure of the departure of <f> (h) from a l i n e a r expression. In t h i s c a l c u l a t i o n , TABLE 14 PARAMETERS OF THE PARABOLIC APPROXIMATION OF <j> (h) FOR SOME EXPERIMENTS Number A x 10 X A x DA/A B x 10 B y y DA/B C x 10 Z' C 7. DC/C C/B T f T f *C/B 3.1211 -10 -0.501 -02 -3.955 0.139 00 0.049 -0.205 -02 -0.253 -0.0148 11.46 -0.170 11 0.601 -01 0.182 0.118 00 0.028 -0.257 -02 -0.087 -0.0219 13.10 -0.287 12 0.221 -01 0.369 0.769 -01 0.040 -0.219 -02 -0.117 -0.0284 10.38 -0.295 13 0.176 -01 0.196 0.341 -01 0.029 -0.437 -03 -0.141 -0.0128 14.12 -0.181 14 0.165 -01 0.241 0.226 -01 0.064 -0.581 -03 -0.201 -0.0257 10.46 -0.269 15 0.160 -01 0.206 0.121 -01 0.083 -0.293 -03 -0.232 -0.0242 12.94 -0.313 16 0.578 -02 0.444 0.813 -02 0.109 -0.292 -03 -0.233 -0.0358 12.50 -0.448 Figure 32 : <j>(h) as a f u n c t i o n of time f o r d i f f e r e n t t e s t s . 64 an attempt was made to take i n t o account the t r a n s i e n t phenomena such as delayed e l a s t i c i t y which might i n t e r f e r e w i t h the viscous flow. D i f f e r e n t weights have been given to the d i f f e r e n t measurements of h so that the f i r s t t h i r t y minutes of the experiment were not taken i n t o account to the same extent as the l a t e r p a r t of the experiment. The value of t h i r t y minutes 32 had been suggested by the r e l a x a t i o n time of t h i s type of g l a s s f o r t h i s range of temperature. In order to e l i m i n a t e e n t i r e l y the " t r a n s i e n t phenomenon," the same c a l c u l a t i o n was c a r r i e d out n e g l e c t i n g completely the f i r s t f i f t y minutes of the creep. S i m i l a r r e s u l t s were found where the importance of the parameter c i s not n e g l i g i b l e f o r long periods of time. F i n a l l y , because of the exponential v a r i a t i o n of the r e l a x a t i o n time w i t h the temperature, an i d e n t i c a l procedure was used w i t h creep curves of v a r y i n g length of time to determine whether a t r a n s i e n t e f f e c t would a l t e r the r e s u l t s f o r even longer periods of t e s t i n g than had been considered so f a r . In Table 15!, the values of a, b and c f o r the d i f f e r e n t curves are reported. The p l o t s of $ (h) versus time i s given i n Figure 33^ . Even a f t e r a long p e r i o d of time,the p l o t of (j>(h) i s not a s t r a i g h t . l i n e , that i s , the slope of the l o g - l o g p l o t i s not equal to 1. Thus, i t i s reasonable to assume that a v a r i a t i o n of the v i s c o s i t y w i t h the time has been detected. In a d d i t i o n , the v a r i a t i o n of the v i s c o s i t y w i t h the d u r a t i o n of the heat treatment at 565 and 575°C as reported p r e v i o u s l y , showed a maximum i n t h i s range of temperature which i s a l s o the range at which most of the creep t e s t s were c a r r i e d out. Thus, a time dependence of the v i s c o s i t y was expected. 4.22 Expressions f o r the V a r i a t i o n of the V i s c o s i t y w i t h the Time At any moment, the v i s c o s i t y of the specimen can be obtained from the slope of the tangent to the curve <j>(h) versus time by the r e l a t i o n s h i p : TABLE 15 PARAMETERS OF THE PARABOLIC APPROXIMATION FOR TESTS OF VARIOUS DURATION Number A x 10 X DA/A B x 10 y DA/B C x 1 0 Z DC/C C/B T * C/B A X B v C z f f 3.1212 303 0.895" -02 0.076 0.161 -02 0.028 -0.365: -05 -0.159 -0.0023 75,70 -0.171 414 0.298: -01 0.043 0.261 -02 0.050 -0.212 -04 -0.134 -0.0081 43.30 -0.352 309 0.174 -01 0.067 0.622: -02 0.023 -0.425 -04 -0.085 -0.0068 36.00 -0.246 404 0.463: -01 0.128 0.173 -01 0.054 -0.467 -04 -0.655 -0.0027 30.00 -0.081 308 0.236 -01 0.075 0.553 -02 0.047 -0.752: -04 -0.109 -0.0136 29.10 -0.395 310 0.612 -02 0.145 0.461 -02 0.029 -0.6.05 -05 -0.698 -0.0013 28.80 -0.038 311 0.652 -02 0.118 0.232. -02 0.052 -0.264: -04 -0.155 -0.0114 27.20 -0.310 601 -0.785 -02 -0.481 0.336' -01 0.020 -0.390: -03 -0.068 -0.0116 23.50 -0.273 204 0.203 -01 0.147 0.159: -01 0.055 -0.222 -03 -0.230 -0.0139 17.00 -0.237 412 0.661 -02 0.160 0.507 -02 0.062 -0.113 -03 -0.161 -0.0223 17.00 -0.379 413 0.123 -01 0.130 0.521 -02 0.074 -0.125 -03 -0.160 -0.0241 17.00 -0.409 305 0.903 -02 0.187 0.133 -01 0.040 -0.175 -03 -0.188 -0.0132 15.90 -0.209 411 0.952: -02 0.170 0.760 -02 0.055 -0.209 -03 -0.112 -0.0275 15.60 -0.430 3.1212 -415 0.361 --02 0.456 0.507: -02 0.113 -0.198 ,-03 -0.226 -0.0390 11.20 -0.437 3.112 \ 41 -0.115 \ -00 -0.158 0.821 -01 0.090 -0.346: -02 -0.195 -0.0422 9.00 -0.380 3.1212 307 0.543 -02 0.428 0.656 -02 0.164 -0.296 -03 -0.384 -0.0451 8.10 -0.366 409 0.521 -02 0.470 0.105 -01 0.123 -0.480 -03 -0.327 -0.0456 7.00 -0.319 408 -0.595 -02 -0.210 0.112 -01 0.056 -0.383 -03 -0.187 -0.0343 7.00 -0.240 | ! i 66 1000 4 Figure 3 3 : D i f f e r e n t appearence. of the <t>(h) p l o t s f o r various durations of experiment . 67 " d<j>(h) 3V dt To get a reasonable v a r i a t i o n of n, a f a i r l y accurate knowledge of the shape of <fi(h) i s necessary. An approximation i n the values of the r e q u i r e d slopes gave a p r e l i m i n a r y i d e a of the shape of n = f ( t ) . The f o l l o w i n g procedure was used. At a given p o i n t A of the p l o t of <(>(h), the slope of the tangent i s the angular c o e f f i c i e n t of the secant which j o i n s i t s two nearest neighbours. This procedure i s in a c c u r a t e f o r small, values of curvature of the p l o t . A damping e f f e c t of the e r r o r s could be obtained by the use of a secant j o i n i n g p o ints f u r t h e r apart from A. P l o t s of long l a s t i n g e x p e r i -ments were analysed by t h i s improved method. The other attempt c o n s i s t e d i n the use of the f i r s t computed approximation of <fi(h). ' By d i f f e r e n t i a t i o n of the polynomial which approximates <(> (h), n i s given as f o l l o w s : _ L 1 n = ' 3V b + 2ct (8) The shortcoming of t h i s method i s that i t provides a p l o t of the v i s c o s i t y with a curvature which has the opposite s i g n of the curvature of the p l o t obtained by the previous method. Some other functions have been t r i e d . The shape of some <j>(h) curves and the expected d i f f i c u l t i e s brought about by a delayed e l a s t i c e f f e c t , l e d to the attempt-of an exponential decay of <|>(h) w i t h the time which gives as f o l l o w s : n = n M { 1 - a exp ( - |) } (9) Because of the l a c k of p r e c i s i o n of the f i r s t approximation, i t was not p o s s i b l e to l i n e a r i z e the system of equations which gives the parameters n^, a and x. Thus the recurrence c a l c u l a t i o n became divergent and could not provide the values of the parameters. 68 To account f o r the l i n e a r v a r i a t i o n of the v i s c o s i t y which had 9-10 been found by e a r l i e r workers , a l o g a r i t h m i c f i t of <f>(h) versus t was attempted, to c a l c u l a t e the values of A and B i n the r e l a t i o n s h i p : n = A + Bt . (10) The f i t t i n g f u n c t i o n could not be determined with enough accuracy, suggest-ing that a c l o s e r f i t t i n g equation might be found. Considerations of mechanisms such as d i f f u s i o n , which might c o n t r o l the change of v i s c o s i t y , l e d to a t e n t a t i v e p a r a b o l i c v a r i a t i o n of the v i s c o s i t y w i t h time: n = (a + ht)h (11) A much b e t t e r f i t of the data was obtained. T h i s suggested that an improvement could be made by i n t r o d u c i n g a t h i r d parameter, the exponent of the f u n c t i o n , o b t a i n i n g the f o l l o w i n g r e l a t i o n s h i p : n = (a + b t ) C (12) A good f i t f o r the data r e s u l t e d from t h i s new equation. In Table 16, the parameters i n these four d i f f e r e n t mathematical models are compared using the data of the "Corning" glass experiments. The l a s t model: n = (a + bt) appears to give the most coherent accuracy f o r the determined parameters. In Figures 34 - 37 , the p l o t s of the d i f f e r e n t models are compared to the p l o t of the "measured" values of the v i s c o s i t y . The former r e f e r s to the data obtained from the measurement of the slope of the tangent of the <f(h)-versus-time curve. In Table 17' and i n Figures 38 to 40 , a s i m i l a r com-p a r i s o n was made w i t h the longest l a s t i n g creep t e s t s . I n Table 18 (Appendix II ) , the values of a , b and c f o r most experiments are reported w i t h t h e i r corresponding a c c u r a c i e s . From these t a b l e s , i t can be n o t i c e d that the exponent i s not completely independent of the other v a r i a b l e s , but no systematic c o r r e l a t i o n could be e s t a b l i s h e d between c and some other experimental parameters... TABLE 16 COMPARISON OF THE DIFFERENT MATHEMATICAL MODELS FOR "CORNING" GLASS SPECIMENS Function A AA/A B AB/B C AC/C n = 10** x C/'(A - Bt) n - 107; x (A + Bt) n = 1077 x Cv'A + Bt n = 10 x (a + Bt) 1.00 .480 -.0249 -.00596 Specimen No. .077 .118 3.69 .146 3.122 - 00 .0666 .551 1.00 .0226 .159 .010 .011 .050 .179 .145 .514 .000 .018 n = C x 1 0 1 4 / ( A - Bt) n = (A + Bt) . 1013 n = C x I O W A + Bt n = i o l 4 (A + Bt)C 1.00 .717 -.448 -.0212 Specimen No. .072 .144 .213 .037 3.122 - 01 .0728 .894 1.00 .0479 .150 .014 .012 .059 .259 .226 .4896 .000 .015 n= C x IO 1 4/(A - Bt) n= (A + B t ) 1 0 l 3 n= c x IO14/A + Bt n = 1 0 l 4 ( A + B t ) c 1.00 .529 .271 .00685 Specimen No. .079 .079 .456 .126 3.122 - 02 .0676 .567 1.00 .0279 .157 .007 .014 .043 .186 .152 .537 .000 .000 .014 TABLE 16 continued Function A AA/A B AB/B C AC/C n = l o | 4 x C/(A - Bt) n = 1 0 ^ x (A + Bt) n = 10 7 x C/A + Bt n = 10 x (A + Bt) Sp 1.00 .626 -.463 -.00242 ecimen No. .046 .245 .144 .587 3.122 - 04 .0460 .849 1.00 .0378 .103 .018 .007 .033 .335 .221 .442 .000 .011 71 H. Viscosi ty 14 10 x p o i s e s O n f rom the s l o p e — n • IO**x.l79/( I -.066 t) — H 1 I0 IS(.478 + 55I t) n = KDVl45.rt-.025 — n ' l o ' V o o s g + . o ^ e t ) 9 1 4 F i g u r e 3 4 : n ° 3 . 1 2 2 - 0 0 V iscos i ty 14 10 x poises O r| f rom the slope 14 — n - 10 x . 2 5 9 / t l - . 0 9 2 t ) •I. 13 — T =10 (.717 + 3 9 4 t) — - t ) =10 x . 6 2 6 \ T t - . 4 4 8 ? ° T i m e F i g u r e 3 5 : n ° 3 . 1 2 2 - 0 1 V iscos i t y 10 poises I-1. O n f rom the s lope — n = I0 ,4x . 3 3 5 A I . - . 0 4 6 D = I 0 l 3 ( . 6 2 6 + . 8 4 9 t ) / — n = I0l4x . 2 2 1 ^ 1 * 4 6 3 ) ^ — q = l d 4 ( - . 0 0 2 4 * 0 3 7 8 ^ -/ / 1 0 0 Time Figure 37 : n* 3.122-04 -1 j 1 Viscosity 1 lO 1 4 * po ises 1 1 O T t rom the s lope 1 1 I — n - . 1 8 6 / ( 1 - 0 6 7 1 ) / - - n = l o ' 3 ( 5 2 9 + . 5 6 7 t) ! / •—11= I 0 l 4 x , l 5 2 / - t + . 2 7 l ) / 14 437' — n= 10 ( . 0 0 6 8 * . 0 2 7 9 t ) / . / / ° - . 5 y. M Jy r '/ 0 50 mtait. 1 0 0 TkM Figure 3 6 : n* 3.122-02 Compari son between the d i f f e r e n t mathematical models f o r the v i s c o s i t y - t i m e r e l a t i o n s h i p i n the case of the "Corning" glass . TABLE 17 COMPARISON OF THE DIFFERENT MATHEMATICAL MODELS OF THE VARIATION OF n WITH £ FOR 3 LONG LASTING "PYREX" GLASS EXPERIMENTS Function A AA/A B AB/B C AC/C n = 1 0 ^ C/(A - Bt) n = ior;r (A + Bt) n = 10 (A + B t r Spec] 1.00 .444 .0693 .men No. 3 .023 .931 .186 .1212 - 309 .0137 1.46 .181 .085 .027 .024 1,02 .293 .000 .008 n = 10** C/(A - Bt) n = 10:7 (A + Bt) n = 10 (A + Bt) Spec] 1.00 .773 .840 .men No. 3 .028 .888 .188 .1212 - 310 .00262 1.99 1.74 .702 .041 .029 1.32 .104 .000 .012 n = 1 0 ^ C/(A - Bt) n = l o r ? (A + Bt) n = 1 0 ^ (A + B t r Spec: 1.00 .373 115. Lmen No. 3 .025 .425 .226 .1212 - 313 .00452 3.11 120. .159 .001 .051 3.73 .255 .000 .014 , 73 Viscos i ty 14 10 x po ises O n f r o m the s l o p e n = 10 x 1.02/(1-0(33 t ) 10 (.444+ 1.46 t) 14 .293 — n » 10 (069+.l8lt) 1 0 0 200 T ime »0ninute V i s c o s i t y 14 10 x poises O 1 f r o m the s l o p e — n ' I0 l 4 x 1 . 3 2 / ( I - . 0 0 2 6 1 ) — - n • I 0 I 3 ( . 7 7 3 + 1.991) 0 * K) ( .84 • 1.74 t ) 100 minute 2 0 0 Time F i g u r e 3 9 : n ° 3 . 1 2 1 2 - 3 1 0 F i g u r e 38 : n ° 3.1212-309 V iscos i t y 14 10 x p o i s e s ° --—^ -.5 o - . 4 O ° / o n f rom the s lope — n = I 0 , 4 x 3 . 7 3 / ( l . - . 0 0 4 5 0 - . 3 / — n 14 286 = 10 (115. + 120. t ) 3 0 0 ^ 6 0 0 T i m . 3 0 0 F i g u r e 4 0 : n» 3 . I 2 C - 3 I 3 Comparison between the d i f f e r e n t mathematical models f o r the v i s c o s i t y - t i m e r e l a t i o n s h i p i n the case of long l a s t i n g experiments. 74 However, i t appears that the s c a t t e r of the values of c decreases as the temperature of heat treatment of the specimen i n c r e a s e s , and that i t i s l e s s than that of the experiments where the specimens have not been subjected to any heat treatment before the t e s t . This would suggest the existence of some homogenization process l e a d i n g towards a more uniform s t r u c t u r e , thus erasin g the previous thermal h i s t o r y of the g l a s s . The value of c, then would b e . i n the v i c i n i t y of 0.3. In the case of the "Corning" glass the values of c a r e very c l o s e . 13 to 0.5. This would confirm the f i n d i n g s of other workers , namely, that the c o n t r o l l i n g mechanism of.phase s e p a r a t i o n i n the range of temperature of the transformation, i s a d i f f u s i o n process. But as no data are a v a i l a b l e to show the importance of the geometric d i s t r i b u t i o n of the second phase and the r e l a t i v e v i s c o s i t y of the two phases, which may modify the value of the o v e r a l l v i s c o s i t y of the g l a s s , the experimentally determined values of c may be f o r t u i t o u s . With respect to the v i s c o s i t y which i s c o r r e l a t e d to the phase-sep a r a t i o n , the temperature of heat treatment was found to be important, as a maximum was observed on.the p l o t of the v i s c o s i t y versus heat treatment temperature, Figure 18 and 19. Simple thermodynamical c o n s i d e r a t i o n s of the m i s c i b i l i t y gap,added to the v a r i a t i o n of the v i s c o s i t y w i t h the temperature, could e x p l a i n t h i s maximum. The v i s c o s i t y data of the specimens subjected to the heat t r e a t -ment at 575°C were f u r t h e r analysed. In Figure 41, the v i s c o s i t y of these specimens i s p l o t t e d against the r e a l time of heat treatment i n a l o g - l o g p l o t . The r e a l time r e f e r s to the time of heat treatment i n the heat treatment furnace, plus the time in v o l v e d i n the creep t e s t i n g . For a short period of heat treatment, i t appears that the values of the v i s c o s i t y are much c l o s e r to. the approximate s t r a i g h t l i n e than they were on the previous p l o t . Figure 41 : V i s c o s i t y versus true time of heat treatment. 76 This r e f l e c t s the r e l a t i v e importance of the time of creep on the o v e r a l l heat treatment of the specimen. On the other hand, when the creep t e s t was c a r r i e d out a f t e r a much longer p e r i o d of heat treatment and when the v i s c o s i t y at the beginning of the creep t e s t i s much higher, the change i n the values of the v i s c o s i t y during the creep t e s t was more r a p i d than f o r the same period of heat treatment when no s t r e s s was ap p l i e d on the specimen. The explanation f o r t h i s change i n the flow behaviour of the glass may be e i t h e r that the glass no longer acted as a Newtonian l i q u i d when a c e r t a i n degree of phase separation was achieved, or that the growth r a t e of the second phase under s t r e s s changed when a given s i z e of the d r o p l e t s of the second phase was reached. 4.23 A c t i v a t i o n Energy of the Flow Process In the case of the "Pyrex" glass the temperature c o e f f i c i e n t of the v i s c o s i t y can be represented by a r e l a t i o n s h i p as f o l l o w s : n = a eNR. T . This r e l a t i o n s h i p i s commonly accepted by most authors to r e l a t e the temperature dependence of the v i s c o s i t y . The narrow range' of temperature which had been considered here»resulted i n a constant value of the a c t i v a t i o n energy Q f o r the whole range. The value of 87 kcal/mole which was obtained f o r the "annealed" g l a s s , was i n good agreement w i t h the value of 32 80 kcal/mole found f o r s t a b i l i z e d b o r i c glasses i n the transformation range On the other hand, the glass which had been tested as re c e i v e d , had an a c t i v a t i o n energy of 65 kcal/mole, f o r the flow process. At the lower end of the temperature range the values of the v i s c o s i t y of t h i s g l a s s were lower than those of the heat tr e a t e d g l a s s . The presence of r e s i d u a l s t r e s s e s i n s i d e the glass could account f o r t h i s e f f e c t . But the higher value of the a c t i v a t i o n energy of the flow process suggested a higher energy b a r r i e r which nay be due to some ordering e f f e c t w i t h i n the glass s t r u c t u r e r e s u l t i n g from some i n i t i a l stage of phase s e p a r a t i o n , although the l a t t e r was not detected . 77 4.3 I n t e r p r e t a t i o n of the E l e c t r o n Microscopy 4.31 "Corning" Glass D i f f e r e n t degrees of phase s e p a r a t i o n were e a s i l y detected w i t h specimens made out of "Corning" g l a s s . The formation and the growth of dr o p l e t s of a new l i q u i d phase were observed during d i f f e r e n t heat treatments. But a q u a n t i t a t i v e c o r r e l a t i o n between the v a r i a t i o n of the v i s c o s i t y and the development of the second phase.was not p o s s i b l e without extensive i n v e s t i g a t i o n s i n v o l v i n g the q u a n t i t a t i v e determination of the amount of the new phase present.at any time. Only the s t r u c t u r e which corresponded to the end of a given t e s t could be observed. No i n f o r m a t i o n whatsoever on the previous s t a t e s of e v o l u t i o n of the phenomenon could be deduced from t h i s observation i n order to discuss the d i f f e r e n t shapes of p l o t s of the v a r i a t i o n of the v i s c o s i t y w i t h time. The previous data on phase s e p a r a t i o n of t h i s g l a s s could not be used as the phase s e p a r a t i o n i n the present case was p a r t i a l l y c a r r i e d out w h i l e the specimen was under-going the creep t e s t ; that i s , w h i l e under s t r e s s . For, i f the phase separation i s assumed to be a d i f f u s i o n c o n t r o l l e d process, the e f f e c t of the applied load used during the t e s t may be s i g n i f i c a n t enough to change the n u c l e a t i o n and growth c h a r a c t e r i s t i c s of the second phase as evidenced i n Figure 30 . ' 4.32 The "Pyrex" Glass Extensive evidence of phase se p a r a t i o n can be seen i n t h i s g l a s s . The formation, growth and the eventual d i s s o l u t i o n of a second phase w i t h d i f f e r e n t heat treatments are e a s i l y n o t i c e a b l e . This second phase has d i f f e r e n t mechanical p r o p e r t i e s from the bulk as suggested by the appearance of f r e s h l y - f r a c t u r e d surfaces which were etched before r e p l i c a t i o n . 78 The v i s c o s i t y of the second phase i s expected to be d i f f e r e n t from the bulk. Therefore i t s s p a c i a l d i s t r i b u t i o n and the s i z e s of the d r o p l e t s are to be taken i n t o c o n s i d e r a t i o n to account f o r the observed change of the o v e r a l l v i s c o s i t y . In a d d i t i o n , although the mi c r o s t r u c t u r e s shown i n Figures 25 to29', are q u i t e r e s p r e s e n t a t i v e of a given specimen, other s t r u c t u r a l c h a r a c t e r i s t i c s i n i s o l a t e d places have a l s o been observed i n the same specimen. The d i s t r i b u t i o n of the second phase was not always uniform i n the bulk of the whole specimen. In Figure 42, d i f f e r e n t areas of the same specimen can be seen. The d i s t r i b u t i o n of the d r o p l e t s of the second phase v a r i e s widely from cases where almost no d r o p l e t s are observed (Figure 42a) to cases where the d r o p l e t s are q u i t e dense (Figure 42d), w i t h a l l p o s s i b l e intermediate s t a t e s . Several i s l a n d s of phase-separated g l a s s e x i s t amongst zones where no phase s e p a r a t i o n has occurred. From a t h e o r e t i c a l l y 18 computed model proposed f o r the e a r l y stage of phase separation,Charles considered the formation of t h i s type of s t r u c t u r e . I n most cases,these i s l a n d s are of a c i r c u l a r type as shown i n Figure 43 . D i f f e r e n t appearances of phase se p a r a t i o n are shown i n Figures 44 to47.(Appendix I V ) . For the same specimen, wide v a r i a t i o n s of the over-a l l s t r u c t u r e were observed. The s i z e , shape, d i s t r i b u t i o n and density of the apparent features a l s o v a r i e d g r e a t l y . In the s e c t i o n d e a l i n g w i t h the r e s u l t s , a continuous t r a n s i t i o n i n the shape of the p a r t i c l e s from a pure s p h e r i c a l shape.to angular c r y s t a l s has been reported. The f a c t that many authors considered phase se p a r a t i o n as an e a r l y stage of c r y s t a l l i s a t i o n concurs w i t h t h i s observation. This remark a p p l i e s e s p e c i a l l y to the form-a t i o n of glass-ceramics by converting the pure glassy s t a t e to the completely c r y s t a l l i s e d ones. An analogy between the shape of the two curves given i n Figure 48. and Figure 19 favors t h i s explanation. a) 80 Figure 43 : Islands of phase-separation. 81 g( u/mm) 900 Figure 48 : Growth r a t e and r e l a t i v e n u c l e a t i o n r a t e of PbT i O ^ c r y s t a l s from glass as a f u n c t i o n of temperature. 82 The former respresents the v i s c o s i t y - t e m p e r a t u r e of heat treatment r e l a t i o n -ship at a given time. The l a t t e r i s the p l o t of the n u c l e a t i o n and growth 29 r a t e of lead t i t a n a t e c r y s t a l s from a glass . The growth rates of c r y s t a l s from a g l a s s and the corresponding v i s c o s i t i e s have been e m p i r i c a l l y c o r r e l a t e d . I n the case of the c r y s t a l l i z a t i o n of s i l i c a t e s t h i s r e l a t i o n -30 ship can be w r i t t e n as f o l l o w s V = -1 + k 2 l o g n (13) n where V i s the growth r a t e , and are two e m p i r i c a l constants and n i s the v i s c o s i t y . Since phase s e p a r a t i o n can be considered as an e a r l y stage of c r y s t a l l i s a t i o n , a s i m i l a r r e l a t i o n s h i p may e x i s t between the v i s c o s i t y and the growth r a t e of the new phase. Unfo r t u n a t e l y , the s t r u c t u r a l study d i d not r e v e a l s u f f i c i e n t i n f o r m a t i o n on the growth of the second phase to a l l o w an e v a l u a t i o n of these c o e f f i c i e n t s i n the present case. 4.33 The other Glasses E l e c t r o n micrographs of the b i n a r y glass and the a l k a l i glasses showed the presence of phase s e p a r a t i o n i n a l l the specimens. The degree of s e p a r a t i o n v a r i e d from one specimen to the other, and the s i z e and,density of the new phase s i m i l a r l y v a r i e d widely as shown i n Figures 49 and 50 . On the other hand a f a i r l y high degree of c r y s t a l l i n i t y was encountered i n the same specimens as i t can be seen i n Figures 51 and 52 . 4.4 Changes of V i s c o s i t y due to Phase Separation 4.41 T h e o r e t i c a l Considerations 4.411 E f f e c t s of the P h y s i c a l S t r u c t u r a l Changes As p r e v i o u s l y mentioned, a phase-separated glass c o n s i s t s of a hetero-geneous system of at l e a s t one immiscible phase f i n e l y dispersed i n the form 84 c) d) Figure 50 : Phase separation i n the a l k a l i glass . a and b Micros true t u r e of a glass as f a b r i c a t e d , c and d Micros true t u r e of a glass heat tr e a t e d f o r 10 mn at 500°C. Figure 51 : D i f f e r e n t degree of c r y s t a l l i n i t y i n the binary g l a s s . Figure 52 : D i f f e r e n t degrees of c r y s t a l l i n i t y i n the a l k a l i g l a s s . 87 of d r o p l e t s , the s i z e of which may vary from one hundred to s e v e r a l thousand Angstroms i n diameter. This c l a s s i f i e d the mixture at the lower l i m i t of 31 32 emulsions and i n the f i e l d of c o l l o i d s Considerations on the Reynold's number R of the flow of a glass undergoing creep lead to the f o l l o w i n g remarks. In the present case R i s of _ 2 2 U d 5 the order of 2 x 10 . R = —— , the shear r a t e U = 2 x 10 cm/s, . V -4 the distance between the p a r t i c l e s d = 10 cm and the dynamic c o e f f i c i e n t 13 2 of v i s c o s i t y v = 10 cm/s.) This value i s very low and the type of flow should be s i m i l a r to the flow of a f l u i d of lower v i s c o s i t y at a very slow v e l o c i t y between c l o s e l y packed o b s t a c l e s . The surface tension and the p a r t i c l e -p a r t i c l e i n t e r a c t i o n have not been taken i n t o account i n t h i s comparison. Both of these however, should have a major e f f e c t on the a c t u a l flow. Although a p s e u d o - p l a s t i c behaviour should occur f o r t h i s type of f l u i d , the flow can s t i l l be Newtonian because of the very low v e l o c i t y i n v o l v e d i n t h i s study. 33 34 The v i s c o s i t y of emulsions or c o l l o i d s i s very s e n s i t i v e to parameters of the suspended phase such as volume co n c e n t r a t i o n , s i z e , s i z e d i s t r i b u t i o n of the d r o p l e t s and r e l a t i v e v i s c o s i t y . Although i n the case of c o l l o i d s no s i n g l e equation can completely represent the volume-concentration-v i s c o s i t y r e l a t i o n s h i p , exponential expressions of the type l o g n = logn + kc where k i s approximately constant, are g e n e r a l l y adequate to r e l a t e the experimental behaviour. At higher concentrations of the second phase, coalescence and b r i d g i n g of the d r o p l e t s might occur and give a sudden change i n the v i s c o s i t y of the mixture. I t would be s i m i l a r to the i n v e r s i o n of emulsions. Then the p l o t of r e l a t i v e v i s c o s i t y of the s o l u t i o n ( v i s c o s i t y of the m i x t u r e / v i s c o s i t y of the s o l v e n t ) , as a f u n c t i o n of the c o n c e n t r a t i o n , f o l l o w s a curve as shown i n Figure 53. 88 R e l a t i v e v i s c o s i t y n v , In v e r s i o n 0 100 Concentration Figure*.53 : V i s c o s i t y versus concentration of the suspended phase f o r an emulsion . From the arguments above, , i t f o l l o w s that v i s c o s i t y values may vary by s e v e r a l orders of magnitude due only, to p h y s i c a l s t r u c t u r a l changes w i t h . phase s e p a r a t i o n i n g l a s s . 4.412 E f f e c t s of the Chemical.Change When the second phase p r e c i p i t a t e s out.from the glassy m a t r i x , the c o n c e n t r a t i o n of the d i f f e r e n t components of the matrix should change ac c o r d i n g l y . Although i t i s not p o s s i b l e to f o r e s e e the exact changes of composition of the phase separated g l a s s , the importance of the v a r i a t i o n s of the v i s c o s i t y due to the composition f l u c t u a t i o n s can be estimated from a v a i l a b l e d a t a , i n the l i t e r a t u r e . 35 For a , b o r o s i l i c a t e .glass having the formula, Na 20 + x B^O^ + (4 - x) S i 0 2 , the change,of the v i s c o s i t y w i t h the conc e n t r a t i o n of B 20,j, at 575°C, i s g i v e n . i n Figure 54. A.rapid change of two orders of magnitude occurs i n the v i s c o s i t y values when the conc e n t r a t i o n of l^O^ increases from 0 to 10%. 36 o For the binary g l a s s N a 2 0 - S i 0 2 , at 575 C, the logarithm of the v i s c o s i t y increases from 9.4 to . 10.3 when the concentration of S i 0 2 v a r i e s from 74.05 to 79.84%. • 89 Figure 55 : Log isokoms i n poises i n the system Na„0-CaO-SiO at 575°C. -13 V i s c o s i t y logo poise -12 -II / 0 IP 2.0 % B 2 0 3 3.0 4 0 Figure 54 : V a r i a t i o n of the v i s c o s i t y w i t h the composition i n a g l a s s of formula Na.O + x B o0„ + ( 4- x ) SiO_ _ - , r . 0 „ I L J i. at 575 C. 90 For the a l k a l i g l a s s , some in f o r m a t i o n i s provided by the l o g -3 7 — 39 o isokoms of the system Na20-Ca0-Si02 at 575 C given i n Figure 55. Although the presence of other i m p u r i t i e s are not being considered, small v a r i a t i o n s of the composition of the main c o n s t i t u e n t s are followed by important changes of v i s c o s i t y . Compositional changes alone can modify the v i s c o s i t y values by s e v e r a l orders of magnitude. 4.42 Other Experimental Observations In support of the above consi d e r a t i o n s experimental evidence of s i m i l a r phenomena are discussed below. The great v a r i a t i o n of the observed s t r u c t u r e s , as w e l l as the great s c a t t e r of the v i s c o s i t y measurements suggest a probable phase se p a r a t i o n phenomenon which i s followed by the c r y s t a l l i s a t i o n of the 40 type observed by other workers. For glasses of the Na20-Nb20^-Si02 system, d r a s t i c changes i n the s t r u c t u r e are detected f o r d i f f e r e n t rates of heat treatment as w e l l as f o r s l i g h t v a r i a t i o n s of the f i n a l temperature of the heat treatment. A few degrees of temperature d i f f e r e n c e may lead to important changes i n the s t r u c t u r a l c h a r a c t e r i s t i c s . At the same time the v i s c o s i t y - temperature r e l a t i o n s h i p e x h i b i t s an unusual shape w i t h a minimum a f t e r which the v i s c o s i t y increases sharply w i t h the temperature. The corresponding p l o t s are shown i n Figure 56 f o r the glass of the system Na20-Nb20^-Si02 and i n Figure 57 f o r a glass the composition of which i s 41 • unknown . The same author a l s o found a curve showing a minimum f o r the p l o t of the v i s c o s i t y as a f u n c t i o n of the heat treatment. This minimum corresponds to a very short p e r i o d of heat treatment which has not been i n v e s t i g a t e d i n the present case (Figure 58). One can suggest that a s i m i l a r V-shaped curve e x i s t s f o r the p l o t of the v i s c o s i t y versus temperature 91 Figure 5.6 : V i s c o s i t y measurements at a heating r a t e of 66 C/h f o r d i f f e r e n t glasses 1. Figure 5 7 : E f f e c t of the tempera-ture on the v i s c o s i t y of the o r i -g i n a l g l a ss melt and of the c r i s -t a l l i s a t i o n products . o i n d u c t i o n p e r i o d A 3000s a f t e r the s t a r t of c r i s -t a l l i s a t i o n • 7000s V i s c o s i t y p o i s e l o g n i 1 2 ^ .. - 3 •' Time 10 x seconds 1 0 -T • 2 3 4 ' = ' ' — 5 6 7 8 1 .. i — 1 1 Figure 5 8 : V a r i a t i o n s of v i s c o s i t y w i t h time during heat treatment of s u b m i c r o c r y s t l l i z i n g g l a ss . o 754°C A 784°C 92 i n the glasses s t u d i e d here. The s c a t t e r i n the data would then r e s u l t from the f a c t that i n the course.of the heat treatments or i n the p r e p a r a t i o n of the specimen, some c r i t i c a l temperature corresponding to the sharp change of the v i s c o s i t y - t e m p e r a t u r e of heat treatment r e l a t i o n s h i p was l o c a l l y exceeded. This i s q u i t e l i k e l y because of the d i f f i c u l t i e s t hat have been, encountered i n c o n t r o l l i n g the temperature of the high frequency furnace. 93 CONCLUSION V a r i a t i o n of v i s c o s i t y i n . t h e t r a n s f o r m a t i o n temperature range w i t h respect t o thermal h i s t o r y , were i n v e s t i g a t e d f o r d i f f e r e n t g l a s s e s . A p a r a l l e l - p l a t e viscosmeter which allowed few r e s t r i c t i o n s on the c o n d i t i o n s of the viscous f l o w , was used f o r the v i s c o s i t y measurements. "Pyrex" glass could-be approximated to a Newtonian l i q u i d i n the range of temperature +^70 t o 590°C f o r the flow c o n d i t i o n s of the experiments. Then, the a c t i v a t i o n energy of the viscous flow v a r i e d from 6 5 to 8 5 kcal/mole according to the previous t h e r m a l , h i s t o r y . A l l glasses e x h i b i t e d a v a r i a t i o n of v i s c o s i t y w i t h time. S e v e r a l mathematical expressions were t e s t e d i n an attempt t o express t h i s v a r i a t i o n . The best f i t of the experimental r e s u l t s , was obtained w i t h a r e l a t i o n s h i p of the type .n = ( a + bt)° where a, and b are parameters f o r which no p h y s i c a l i n t e r p r e t a t i o n was.found. The exponent c v a r i e d i n the range .2 to .5 and appeared t o depend on the temperature of the heat treatment. M i c r o s t r u c t u r e s of the d i f f e r e n t glasses revealed phase sep a r a t i o n t o d i f f e r e n t degrees according t o the thermal h i s t o r y . In some cases d i f f e r e n t amounts of d e v i t r i f i c a t i o n were, n o t i c e d . Although no q u a n t i t a t i v e c o r r e l a t i o n could be n o t i c e d , phase separation could be an e x p l a n a t i o n f o r the v a r i a t i o n -of :the v i s c o s i t y w i t h heat treatment. On the other hand, i n the case of a t e r n a r y b o r o s i l i c a t e g l a s s , where phase separation was q u i t e e x t e n s i v e , the value of c = 0.5 i n the v i s c o s i t y -t i m e - r e l a t i o n s h i p suggested•that d i f f u s i o n may be the r a t e c o n t r o l l i n g process of phase separation. 94 SUGGESTIONS FOR FURTHER WORK • A p r e c i s e c o r r e l a t i o n between the v i s c o s i t y and the s t r u c t u r a l changes can be.obtained by extensive work w i t h "Corning" type glasses which e x h i b i t an important and e a s i l y c o n t r o l l a b l e phase s e p a r a t i o n . Some complementary i n f o r m a t i o n can be obtained from e l e c t r i c a l measurements c a r r i e d out.at the same time on the glass which undergoes phase separation. For the other.types of b o r o s i l i c a t e glasses which have been used i n the present work, measurements of the v i s c o s i t y at va r i o u s tempera-tures may determine t h e . c r i t i c a l . t e m p e r a t u r e where the v i s c o s i t y and heat treatment r e l a t i o n s h i p showed a minimum. APPENDIX I TABLE 3 PRESSURE VARIATION No. Pressure Duration Viscosity i o 1 4 poises in psi in mn at 50 mn at 100 mn or at t max -10' 3900 70 .154 .196 1 71 .153 .173 20 >ioo .143 .154 1 7800 >ioo .156 .156 2 >100 .187 .205 3 73 .156 .156 30 74 .153 .167 1 11700 71 .157 .166 2 70 .143 .153 40 72 .251 .279 1 90 .224 .274 2 70 .215 .274 3 70 .177 .193 4 85 .228 .228 5 15600 >100 .228 .286 6 80 .147 .184 7 80 .148 .174 8 >100 .164 .168 9 77 .174 .192 9b 70 .173 .185 , 50 85 .146 .162 1 84 .170 .198 2 19500 73 .164 .168 3 70 .170 .170 4 80 .163 .171 TABLE 3 continued No. Pressure Duration V i s c o s i t y i n 1 4 10 poises i n p s i i n mn. a t 50 mn at 100 mn or at t max -60 100 .182 .205 1 84 .153 .158 2 3 23400 100 76 .191 .185 .196 .185 4 100 .191 .191 5 0 .202 .202 70 100 .156 .156 1 73 .164 .164 2 3 27300 72 70 .185 .170 .183 .184 80 50 .271 1 74 .158 .170 2 31200 77 .177 .189 3 90 .170 .208 4 75 .190 .197 APPENDIX I (Continued) TABLE 5 VISCOSITY TEMPERATURE RELATIONSHIP Pyrex glass "as rec e i v e d " No. Temperature V i s c o s i t i e s i n 10 poises °C at 50 mn at 100 mn 3.121 -10 474 2.63 3.21 1 473 2.62 2.87 2 506 3.60 4.62 3 504 3.35 4.15 4 526 2.81 5 524 1.92 2.26 6 543 .823 .898 7 544 .691 .840 8 585 0754 0942 Con t r o l t e s t s : 3.112 -56 567 .148 .184 9 567 .174 .192 9B 567 .173 .185 0 568 .251 .279 1 568 .224 .274 2 568 .215 .274 3 568 1 , .177 .228 4 569 .228 .193 5 568 .228 .284 7 568 .148 .174 8 568 .164 .169 APPENDIX I (Continued) TABLE 6 TEMPERATURE-VISCOSITY RELATIONSHIP OF "PYREX" GLASS ANNEALED 44 HOURS AT 500°C No. Temperature 14 V i s c o s i t y i n 10 poises °C at 50 mn at 100 mn 3.121 -20 591 .0541 .0770 1 588 .0778 .0991 2 578 .126 .150 3 568 .241 .255 4 558 .446 .469 5 548 .846 .955 6 538 1.45 2.12 7 528 3.44 APPENDIX I (Continued) TABLE 8 EFFECT OF THE HEAT TREATMENT ON THE VISCOSITY No. Heat treatment V i s c o s i t y Temperature Time at 50 mn at 100 mn °C hours i o 1 4 poises 3.1212 -101 500 3 .231 .247 2 4.71 .361 .361 3 24.21 .287 .318 4 62.41 .395 .395 5 109.3 .332 .358 6 133.3 .275 .275 7 289. .195 .198 8 289. .182 .205 201 565 11. .190 .209 2 11. .211 .249 3 23.8 .569 .578 4 23.8 .538 ,555 301 575 2.16 .224 .250 2 4.33 .221 .298 3 14.9 .824 .736 4 15.7 .690 .970 5 19. .537 .537 6 26.5 .630 .630 1 26,5 .547 .852 8 176. 1.95 1.9S 9 176. 1,30 1,73 18 199 1.03 1.36 1 199, 1,36 1.36 3,1212 =312 575 340 2,71 3,86 3 1 340 2,62 3.93 3.1212 -401 600 ,5 .178 .203 2 .5 .191 ,217 3 1, .282 .368 4 1. .356 .381 5 3.89 .466 .773 6 3.89 .443 .582 7 19. .819 .905 1 8 19. .895 .959 ! TABLE 8 continued 100 No. Heat treatment Temperature Time °C hours V i s c o s i t y at 50 mn at 100 mn 10 poises 9 10 1 2 3 4 5 6 501 2 3 4 5 6 7 .1212 601 2 3 4 I 6 7 8 9 10 1 2 •701 2 3 4 5 6 7 8 9 10 11 -801 2 3 650 750 800 900 24.4 24.4 71.5 75.5 212. 213. 236. 236. .5 .5 9.9 9.9 9.9 48. 48. 2,05 4*15 4,15 21.7 21.7 21.7 47.8 47.8 47.8 .033 ,083 ,083 .166 2.71 2.71 4.6 4.6 22.8 22.6 22.6 1. 1. 1. 1.34 1.03 1.37 2.23 1.69 1.79 2.68 3.11 .257 .219 ,352 .397 .350 .406 .393 .146 ,143 ,177 .142 ,148 .209 .201 .193 .160 .150 .146 .217 .191 .213 .207 .217 .234 .160 .184 .179 .246 .233 .178 .196 .209 1.63 1.197 2.01 2.92 2.80 3.66 3.29 4.58 .300 .239 .398 .432 ,385 .446 .411 .152 ,153 .227 ,215 ,184 ,214 .210 .231 .202 .162 .158 ,217 .191 .221 .251 .250 .279 .212 .223 .237 .263 .287 .190 .216 .222 APPENDIX I (Continued) 101 TABLE 10 PARAMETERS OF THE "CORNING" GLASS TESTS No. Heat Treatment V i s c o s i t y temperature time at 50 mn at 100 mn °C hours 10 poises 3.122 -00 .348 .442 1 560 3. .484 .675 2 560 3. .503 .687 3 600 3. .174 4 690 3. .493 .658 5 750 3.1 .109 .211 APPENDIX I (Continued) TABLE 11 CONDITIONS OF THE TESTS WITH THE BINARY GLASS No. F a b r i c a t i o n Test Heat treatment Temperature V i s c o s i t y Time time temperature at 30 mn at 60 mn at 90 mn mn. mn °C °c 10 poises • mn 3.13 000 15 minutes 510 .0551 .0375 80 1 at 1200°C 513 .0403 .0381 70 2 f o r degassing, 515 .0367 .0376 -• 70 3 then heat 518 .144 .139 70 4 treatment , i f 521 .0465 .0539 80 5 any then 5 326 .0596 .0623 70 6 min at 660°C 527 .0397 .0412 .0413 100 under 2000 3.13 p s i . -100 180 850 506 .0502 .0620 .0739 90 1 10 850 515 .0551 .0526 70 2 10 850 517 .0275 .0295 62 3 10 850 518 .108 .148 .157 130 4 10 850 518 .0516 .0564 70 5 10 850 518 . .0738 .0878 1 i 80 o TABLE 12 APPENDIX I(Continued) EXPERIMENTAL CONDITIONS OF THE TEST WITH THE ALKALI-GLASS.. No. Heat treatment temperature time °C mn Temperature °C Test 14 V i s c o s i t y 10 poises at 30 mn at 60 ran at 90. mn ...... Time mn 3.141 -000 558 .303 .582 70 1 568 .196 .258 70 2 571 .115 .132 80 3 572 .0513 .0596 80 4 575 .0583 . .0703 • CO 5 575 .0753 .0930 70 6 576 .0802 .129 70 7 576 .137 .154 .185 100 8 576 .149 .156 70 9 578 .0765 .104 70 10 578 .0612 .0694 70 11 595 .1014 .101 .100 ?o 100 5i 50 2 0 582 .0739 .0870 70 1 583 .201 .241 70 2 | 583 .200 .237 80 3 585 .105 .115 70 4 585 .0748 .0841 70 5 592 .0877 .0855 7" 6 606 .0502 .0620 .739 90 7 550 20 610 .0550 .0375 TABLE 12 continued No. Heat treatment temperature time °C mn Temperature °C Test 14 V i s c o s i t y 10 poises at 30 mn at 60 mn. a t 90 mn Time mn 3.141 -200 555°C 1' 577 .161 . 146 70 . 1 • . 582 .0669 .638 70 2 588 .129 .127 70 3 592 .0551 .0555 70 .: 4 " 555°C If 602 : .0820 .105 \:. 70 3 0 0 - 65o°c 7 , 573 . .128 .142." ."; 70 1 576 .355 .353 70 2 578 .151 .149 70 . 3 : 65 0°G 7 583 .125 .129 70 400 65.0°C 20 559 .119 .159 .322 90 1 586 .135 .132 .130 90 2 588 .236 .234 60 3 589 .135 .252 70 4 598 .138 .198 .212 150 5 650 C 20. 599 .155 .152 80 500 670°C 27 576 .142 .194 70 1 1 577 .0904 .109 70 2 670 C 27 579 .650 .648 70 o APPENDIX I (Continued) TABLE 13 105 EXPERIMENTAL CONDITIONS OF THE ALKALI GLASS AND QUARTZ POWDER TESTS No. Temperature V i s c o s i t y Time at 30 mn at 60 mn at 90 mn °C 10 poises mn 3.142 -000 533 .262 .280 .389 110 1 545 .411 .408 .406 100 2 546 .148 .176 .274 140 3 550 .425 .424 .422 92 4 553 .133 .192 .227 100 5 554 .0429 .0525 .0683 120 6 558 .0329 50 7 559 .0213 .0227 60 8 559 .112 .252 .646 130 9 559 .115 .113 80 10 561 .109 .139 60 1 561 .092 .127 .130 120 2 561 .128 .152 80 3 576 .0932 .244 .333 120 4 576 .128 .258 80 5 578 .0709 .105 280 6 587 .0737 .105 80 7 590 .278 .316 21 8 591 .451 .447 .443 100 9 591 .197 .274 .293 100 20 591 .101 .0985 60 1 592 .0697 .0766 60 2 598 .123 .123 60 3 610 .0272 .0356 .0557 140 4 612 .228 .229 .306 100 3.142 -100 589 .276 .273 .283 160 1 594 .0944 .155 .172 120 2 595 .129 .131 70 3 595 .153 .145 .142 180 4 596 .0841 .0990 70 5 598 .122 .149 60 6 598 .0958 .113 60 I 106 APPENDIX I I The tables which f o l l o w c o n t a i n a l l the in f o r m a t i o n about the parameters of the mathematical model of the v i s c o s i t y - t i m e r e l a t i o n s h i p given by: n - (a + b t ) C 1 0 1 4 i n poises The computed values of the c o e f f i c i e n t s a, b and c are.given w i t h t h e i r computed r e l a t i v e e r r o r Aa/a, Ab/b and Ac/c r e s p e c t i v e l y . The r a t i o a/b i s provided as a measure of the r e l a t i v e importance of both terms. t ^ i s the t o t a l ; time of the experiment i n minutes/10. o i s a measure of the q u a l i t y of the f i t and i s defined as. f o l l o w s : ' n » M O Q - »c(h> J. where and fy, are r e s p e c t i v e l y , the measured and the computed values of <J>(h) at a given time t,and n i s the number of values of t f o r which the c a l c u l a t i o n has been done. TABLE 18 Number a x 1 0 X a x Aa/a b x ] b .o y Ab/b C x i c 0 Z A c/c a/b fcf 0 y 3.112 -10 0.130 -02 1.243 0.148 -02 0.426 0.376 00 0.065 0.8810 7.00 0.006 1 0.318 -03 2.021 0.699 -03 0.593 0.320 00 0.080 0.4557 ..7.10 0.006 1 0.805 -23 3.127 0.115 -22 1.751 0.364 -01 0.040 0.7024' 7.00 0.006 2 0.625 -03 0.707 0.129 -22 0.169 0.340 00 0.035 0.4851 16.10 0.013 3 0.798 -22 6.886 0.476 -22 3.431 0.374 -01 0.109 1.6818 7.30 0.007 30 -0.218 -10 2.885 0.849 -10 1.420 0.858 -01 0.077 ' -0.2568 7.30 0.026 1 0.344 -08 0.628 G.131 -07 0.227 0,111 00 0.018 0.2623 7.10 0.004 2 -0.125 -04 0.472 0.452 -04 0.186 0.232 00 0.027 -0.2770 7.00 0.007 40 -0.216 -06 1.115 0.370 -05 0.100 0.155 00 0.014 -0.0583 7.16 0.002 2 -0.197 -02 0.279 0.243 -02 0.126 0.328 00 0.026 -0.8099 8.00 0.004 3 -0.109 -04 0.277 0.377 -04 0.080 0.195 00 0.012 -0.2883 7.00 0.001 4 -0.319 -02 0.245 0.277 -02 0.122 0.229 00 0.025 -1.1528 8.50 0.007 5 -0.575 -03 0.107 0.108 -02 0.030 0.277 00 0.007 -0.5343 15.00 0.003 6 -0.143 -02 0.738 0.295 -02 0.297 0.420 00 0.085 -0.4856 8.00 0.024 7 0.651 -03 0.268 0.104 -02 0.058 0.371 00 0.023 0.6280 8.00 0.004 8 0.113 -24 3.425 0.597 -25 0.855 0.318 -01 0.074 1.8880 10.50 0.009 9 0.576 -06 3.283 0.307 -04 0.143 0.194 00 0.022 0.0188 7170 0.005 9b 0.420 -24 5.614 0.657 -24 2.236 0.330 -01 0.089 0.6399 6.50 0.006 50 -0.165 -04 0.137 0.464 -04 0.048 0.218 00 0.009 -0.3558 8.50 0.003 1 -0.241 -03 0.424 -.570 -03 0.142 0.296 00 0.036 -0.4233 15.10 0.021 2 0.324 -05 0.656 0.302 -04 0.101 0.193 00 0.017 0.1074 7.30 0.004 3 -0.635 -03 2.948 0.309 -02 0.675 0.388 00 0.193 -0.2053 7.00 0.038 4 0.269 18 6.354 0.615 18 3.380 0.441 -01 0.279 0.4372 7.10 0.028 60 -0.305 -03 0.200 0.107 -02 0.038 0.330 00 0.014 -0.2838 11.50 0.006 1 0.933 -08 0.825 0.379 -07 0.151 0.125 00 0.032 0.2460 8.40 0.010 2 0.140 -34 7.800 0.546 -34 2.484 0.236 -01 0.097 0.2570 9.00 0.031 3 0.178 -04 2.267 0.173 -03 0.252 0.238 00 0.064 0.1028 7.60 0.013 4 -0.173 -05 2.469 0.723 -04 0.057 0.220 00 0.017 -0.0239 12.80 ! 0„009 TABLE 18 continued Number a x 1 0 X a x Aa/a b x 10 y b y Ab/b C x i c o z A.c/c a/b fcf a 5 0.180 02 0.322 0.274 01 0.509 0.448 00 0.190 6.5618 10.80 0.003 6 0.186 -04 0.928 0.106 -03 0.163 0.216 00 0.035 0.1747 7.10 0.008 70 0.733 -08 0.913 0.441 -07 0.118 0.122 00 0.029 0.1661 9.20 0.008 1 0.120 -16 1.178 0.311 -16 0.328 0.492 -01 0.030 0.3854 7.20 0.008 2 0.104 -11 0.999 0.148 -11. 0.287 0.656 -01 0.035 0;7050 7.20 0.008 3 0.542 -07 1.460 0.111 -06 0.383 0.121 00 0.072 0.4891 7.00 0.014 80 -0.993 -04 2.578 0.326 -02 0.131 0.317 00 0.034 -0;0304 5.16 0.004 1 -0.522 -04 0.275 0.148 -03 0.073 0.249 00 0.024 -0;3526 7.30 0.008 2 0.103 -13 1.721 0.820 -13 0.339 0.600 -01 6;036 0.1252 7.65 0.008 3 0.120 -05 0.421 0.356 -05 0.067 0,156 00 0.022 0.3369 9.05 0.006 4 0.422 -37 4.738 0.358 -37 . 0.967 0.198 -01 0.070 1.1775 7.50 0.012 3.1211 -10 0.000 -38 5.229 0.000 -38 1.491 0.200 -01 0.063 0.3255 11.46 0.027 1 0.311 -06 2.344 0.139 -04 0.031 0.266 00 0.015 0.0223 13.10 0.008 2 -0.129 -03 0.120 0.371 -03 0.032 0.323 00 0.0.0 -0.3487 10.38 0.008 3 -0.101 -05 2.844 0.118 0.042 0.197 00 0.007 -0.0086 14.12 0.004 4 0.241 -02 0.208 0.973 -02 0.049 0.289 00 0.007 0.2475 10.46 0.002 5 -0.433 -03 0,113 00 0.133 0.409 00 0.021 -0.0038 12.94 • 0.009 6 0.7431 00 1.742 0.615 00 0.615 0.380 00 0.078 1.2077 12.50 0.013 3.1211 -00 0.324 -03 2.840 0.415 -02 3.610 0.196 00 0.178 0.0781 7.10 0.073 1 0.287 03 5.340 0.474 03 3.118 0.130 00 0.107 0.6061 7.20 0.017 6 0.673 -01 0.716 0.128 00 0.269 0.291 00 0.028 0.5254 7.30 0.005 i 7 0.153 -01 0.442 0.606 -01 0.118 0.305 00 0.014 0.2525 10.70 0.004 i 1 8 0.756 -07 0.287 0.195 -06 0.040 0.184 00 0.018 0.3883 9.30 0.005 TABLE 18 continued Number a x 10 X a x Aa/a b x 10 y b y Ab/b C x 1 0 Z c z A-c/c a/b o -3.1212 -101 2 3 4 5 6 7 8 -201 2 3 4 -301 2 4 9 5 6 7 11 10 13 -401 2 3 4 0.275 0,321 -0=266 0.847 0.368 0.225 0.636 -0.233 0.345 0.653 0.761 0.435 0.864 0.426 0.528 -0.364 0.890 0.223 0.219 0.841 0.693 0.116 -0.542 ! -0.166 | -0.393 ! 0.192 -03 -03 -02 -04 -04 -04 -05 -05 -02 -02 -04 -04 -05 -05 -01 -01 -03 -02 -02 00 -01 02 -03 -04 | -02 !-03 0.339 0. 215 0.218 0.682 6.319 1.553 3.809 0.569 1.116 0.510 0.707 0.298 0.646 1.197 1.990 0.595 0.312 0.405 2.603 0.583 0.298 0.485 0.096 0.316 0.223 i 0.343 0.303 0.184 0.609 0.750 0.286 0.997 0.685 0.381 0.354 0.831 0.137 0.258 0.385 0,122 0.505 0.277 0.338 0.139 0.248 0.174 0.181 0.120 0.545 0.366 j 0.390 1 0.920 -02 -02 -02 -03 -04 -03 -04 -04 -01 -02 -03 -03 -04 -03 -01 00 -02 -01 -01 01 00 02 -03 -04 -02 -03 0.032 0.067 0.088 0.092 3.130 0.106 0.515 0.060 0.150 0.171 0.212 0.059 0.173 0.102 0.815 1 0.0-74 0.099 0.118 0.428 ' 0'. 263 0.066 0.164 0.049 [ 0.096 0.108 0.053 0.370 0.259 0.370 0.220 0.112 0.256 0.207 0.198 0.338. 0.242 0.229; 0.231 0.175 0.174 0.246 0.515 : 0.171 0.221 0.277 0.104 0.293 0.255 0.270 0.190 0. 300 \ 0.215 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 OOi 00 00 00 00 00 00 00 00 0.008 0.009 0.019 ; o.oi4 0.219 0,015 0.084 0.009 0.023 0.024 0.041 0.010 s 0.022 0.011 0.089 0.014 0.009 0.012 0.050 0.012 0.009 ! 0.015 i 0.009 1 0.017 ! 0.019 i o.oio 0.0908 0=1743 -0.4371 0.1130 1.2882 0.0226 0.0929 -0.0612 0.0973 0.7852 . 0.5564 0.1686 0.2244 0.0350 1.0459 -0.1312 0.2635 0.1608 0.0882 0.4831 0.3823 0.9651 -0.9959 -0.4552 -1.0083 '• 0.2086 9.20 13.32 11.00 16,70 11.40 11.60 10.48 12.08 12.20 17.00 8.00 9.00 9.44 10.60 12.10 29.10 14.00 15.90 10.20 28.80 36.00 75.70 7.00 11.00 11.40 30.00 0.002 0.008 0.009 0.008 0.067 0.013 0.023 0.005 0o009 0.010 0.007 0.008 0.006 0.004 0.026 0.009 0.004 0.013 0 013 0.013 0.010 0.023 0.001 0.004 0.006 0.013 TABLE 18 continued Number a x 1 0 X a x Aa/a b x 10 y b y A b/b C x 1 0 Z c z A c/c a/b a 5 0.385 -02 1.700 0.269 -01 0.353 0.324 00 0.048 0.1428 8.00 0.008 6 0.382 -02 0.402 0.100 -01 0.116 0.268 00 0.016 0.3817 11.50 0.006 7 0.511 -01 0.457 0.955 -01 0.138 0.374 00 0.02Q 0.5354 10.00 0.004 8 -0.607 -01 1.583 0.108 00 0.957 0.186 00 0.062 -0.5641 7.00 0.008 9 0.142 00 2.366 0.210 00 1.008 0.261 00 0.090 0.6769 7.00 0.012 13 0.280 00 0.932 0.637 00 0.216 0.474 00 0.032 0.4389 17.00 0.010 3.1212 -501 -0.375 -08 0.497 0.294 -07 0.089 0.233 00 0.015 -0.1274 13.40 0.007 2 -0.292 -03 0.699 0.179 -02 0.176 0.317 00 0.034 -0.1636 7.00 0.005 3 0.116 -02 0.412 0.446 -02 0.095 0.283 00 0.016 0.2598 11.40 0.004 4 0.556 -02 0.192 0.595 -02 0.055 0.294 00 0.011 0.9345 18.00 0.005 5 0.956 -04 2.740 0.441 -02 0.092 0.285 00 0.016 0.0217 13.80 0.009 6 0.169 -03 3.469 0.204 -02 0.527 0.231 00 0.066 0.0826- 9.60 0.014 7 0.350 -03 1.039 0.226 -02 0.241 0.229 00 0.029 0.1548 9.20 0.008 -601 -0.144 -04 0.590 0.729 -04 0.214 0.238 00 0.057 -0.1981 9.60 0.009 2 -0.434 -05 3.059 0.359 -04 0.290 0.226 00 0.076 -0.1211 8.30 0.009 3 0.219 -04 3.751 0.545 -03 0.100 0.279 00 0.028 0.0401 12.50 0.006 4 -0.629 -03 0.159 0.428 -03 0.082 0.253 00 0.013 -1.4696 12.00 0.008 5 -0.162 -03 0.423 0.249 -03 0.101 0.288 00 0.035 -0.6521 14.90 0.013 6 0.107 -03 0.796 0.217 -03 0.144 0.294 00 0.053 0.4947 12.00 0.016 7 -0.120 -03 0.154 0.242 -03 0.067 0.279 00 0.014 -0.4973 8.10 0.006 8 -0.289 -03 0.672 0.977 -03 0.110 0.300 00 0.031 -0.2957 13.00 0.009 9 0.482 -05 0.903 0.275 -04 0.131 0.184 00 0.024 0.1754 9.00 0.004 10 -0.906 -03 0.298 0.126 -02 0.097 0.311 00 0.026 -0.7173 12.10 0.008 6 0.297 -04 8.463 0.994 -03 0.283 0.346 00 0.083 0.0299 9.00 0.018 2 -0.527 -04 0.803 0.116 -03 0.168 0.240 00 0.048 -0.4562 13.90 0.018 TABLE 18 continued Numb er a x 1 0 X Aa/a b x 10 y Ab/b C x 1 0 Z Ac/c a/b V 0 ' a V" h Q ct U y 3.1212 -701 0.107 -04 0.645 0.122 -03 0.067 0.219 00 0.012 0.0875 12.88 0.006 2 0.199 -05 1.023 0.386 -05 0.324 0.154 00 0.046 0.5149 11.86 0.014 3 0.112 -05 0.481 0.190 -05 0.143 0.138 00 0.021 0.5881 14.32 0.010 4 -0.126 -04 1.361 0.699 -05 0.720 0.165 00 0.076 -1.8065 23.50 0.226 5 -0.506 -03 0.359 0.154 -02 0.114 0.330 00 0.026 -0.3284 11.00 0.010 6 -0.196 -02 0.648 0.480 -02 0.253 0.394 00 0.059 -0.4082 9.00 0.019 10 -0.222 -02 0.166 0.236 -02 0.068 0.328 00 0.016 -0.9426 14.00 0.006 7 -0.346 -02 0.066 0.265 -02 0.030 0.393 00 0.009 -1.3059 10.30 0.003 8 -0.750 -03. 0.220 0.210 -02 0.061 0.378 00 0.018 -0.3580 12.20 0.009 9 -0.765 -03 0.277 0.131 -02 0.078 0.329 00 0.022 -0.5839 11.80 0.005 -801 -0.127 -02 0.091 0.221 -02 0.032 0.370 00 0.008 -0.5717 7.00 0.001 2 -0.124 -03 0.345 0.255 -03 0.116 0.239 00 0.021 -0.4836 9.00 0.003 3 -0.801 -03 0.273 0.157 -02 0.083 0.332. 00 0.021 -0.5096 9.00 0.004 3.122 -00 -0.596 -02 0.279 0.227 -01 0.066 0.514 00 0.018 -0.2629 13.00 0.007 5 -0.806 -02 0.167 0.116 -01 0.049 0.713 00 0.038 -0.6931 15.30 0.016 4 -0.242 -02 0.687 0.378 -01 0.053 0.442 00 0.011 -0.0640 16.60 0.005 2 -0.685 -02 0.233 0.280 -01 0.051 0.537 00 0.014 -0.2449 13.40 0.005 1 -0.212 -01 0.189 0.479 -01 0.077 0.490 00 0.015 -0.4425 11.44 0.006 APPENDIX I I I 112 THEORY OF-THE PLATE VISCOMETER The i n i t i a l c a l c u l a t i o n s and hypotheses only apply f o r the deformation under load of rubber c y l i n d e r s between two p a r a l l e l p l a t e s . I f i t i s assumed that the c y l i n d e r . i s bounded to the p l a t e s and thus there i s no s l i p p a g e a t the p l a t e s , the t o t a l deformation i s derived as f o l l o w s : h The t o t a l displacement of any element of the c y l i n d e r can be eemiderad to. a r i s e from the s u p e r p o s i t i o n of two iimple,displacements? l - s pure homogeneous deformation defined by the displacement of one,plat© towards the other, 2 - the subsequent,displacements necessary to.cause p o i n t s i n the planes of the bounded surfaces to.be r e s t o r e d t o . t h e i r o r i g i n a l p o s i t i o n . Since rubber i s incompressive, the f o r c e F^ which must be ap p l i e d to maintain the f i r s t deformation, that i s , a compressive s t r a i n . e i n the v e r t i c a l d i r e c t i o n , i s given by equation ( a ) . F = . E a' e. (a) where a i s the radius of the d i s c and E the Young's modulus of the m a t e r i a l . In order to c a l c u l a t e the corresponding system of forces which has to be ap p l i e d to the bounded s u r f a c e , i n order to account.for the second d i s p l a c e -ment, one may assume t h a t h o r i z o n t a l . p l a n e s remain planes during the deformation. Therefore v e r t i c a l d i a m e t r a l s e c t i o n s of the c y l i n d e r take 28 up p a r a b o l i c forms , so that the volume contained w i t h i n them i s unchanged. a r dr 113 " t r P+dP \ i H i i / For a c y l i n d e r , o f diameter r, we have the r o l l o w i n g r a l a t i o n s h i p : 2 2 4 ?rr h = Tir (h - d) + k - j tt r h When the compressive displacement d, i s small.and the radius r, much greater than the r a d i a l displacement k, then k i s given by equation (b) (b) In an elementary crown of width dx, the displacement k may be maintained by an.excessive h y d r o s t a t i c pressure dP, a c t i n g on the curved face.. C l a s s i c a l e l a s t i c i t y theory gives dP as i n equation ( c ) . 8 E k 1 3 d k = -7-. r — 4 h dP - - 3 h' dr (c). I n t e g r a t i n g (c) between r =.r and.r = a w i t h P = 0 on the outside s u r f a c e , one gets equation (d). P = Ed ( a 2 - r 2 ) (d) This pressure a c t i n g on the p l a t e s creates a corresponding f o r c e F 2 obtained by i n t e g r a t i n g P2 if dr between r = 0 and r = a. That gives equation (e) 4 F 2 * E d f (e) To apply these r e s u l t s to a viscous l i q u i d c y l i n d e r ^ i t s u f f i c e s to s u b s t i t u t e the v i s c o s i t y to the r i g i d i t y modulus and the r a t e of approach of the p l a t e s w i t h respect to one another,to the displacement. One then obtains equations ( f ) and (g) 114 F 2 - " 3 " v ' ITT 5 ft  <s> C a l l i n g F = F^ + the v i s c o s i t y and the ra t e of creep can be r e l a t e d by equation (k) F 1 (1) 3 V ( _ d h 1 V , ^ dt h 2frh 5 J On i n t e g r a t i n g between the l i m i t s t = t and t = 0, one gets equation ( i ) ,1 3 ^ V d t = * ( h ) ( 1 ) w i t h • ( h ) = i - i ( ± 4 - £ 4> (k) o o When the v i s c o s i t y does not depend upon the time the r i g h t hand si d e of F t the equation becomes: = <f> (h) ( j ) 115 APPENDIX IV 116. APPENDIX IV (Continued) 117 Figure 46 : D i f f e r e n t m i c r o s t r u c t u r e s encountered on a same specimen heat t r e a t e d 24 h at 600 C. APPENDIX IV (Continued ) 1 1 8 a ) b ) LIST OF REFERENCES 119 1. L i t t l e t o n J.T., Ind. Eng. Chem., 25, 748, (1933). 2. Obrey G.N., I. Am. Ceram. Cos., 17, 315, (1934). 3. L i l l i e H.R., Phys. Rev., 36, 347, (1930). 4. G i l a r d P. and de Bast, J . 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