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Electrical resistivity of hot-pressed compacts 1970

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ELECTRICAL RESISTIVITY OF HOT-PRESSED COMPACTS by THIAGARAJAN RAMANAN B.Sc. (Physics), U n i v e r s i t y of Madras, India, 1965. B.E. (Metallurgy), I.I.Sc., Bangalore, India, 1968. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of METALLURGY We accept t h i s thesis as conforming to the standard required from candidates f o r the degree of Master of Applied Science Members of the Department of Metallurgy THE UNIVERSITY OF BRITISH COLUMBIA May, 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g 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 s t u d y . I f u r t h e r a g r e e t h a 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 t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r 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 . Depar tment o f Metallurgy The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date June 9, 1970 i ABSTRACT An attempt has been made to study the change i n the e l e c t r i c a l r e s i s t i v i t y of a powder compact during the i n i t i a l stages of hot-pressing. Theoretical models have been formulated on the basis of p l a s t i c deformation of spheres i n a compact. The r e s i s t i v i t y change during d e n s i f i c a t i o n has been derived for various packing arrangements. For small deformation of spheres, the f i n a l equation i s and the more generalized equation for larger deformation i s ^ = a ( D 2 / 3 e 2 / 3 R 2 _ , ) c where a and a are the c o n d u c t i v i t i e s of a compact of spheres having m c a r e l a t i v e density D, and at the t h e o r e t i c a l density (D = 1), r e s p e c t i v e l y . D q i s the i n i t i a l r e l a t i v e density of the compact before deformation. a i s a constant depending on geometry and R i s the radius of spheres at any stage of deformation i n a r b i t r a r y u n i t s . The derived r e l a t i o n s h i p was tested by: (a) measuring the e l e c t r i c a l r e s i s t i v i t y as a function of density during hot-pressing of compacts of glass spheres, (b) measuring the e l e c t r i c a l r e s i s t i v i t y of d i f f e r e n t compacts of n i c k e l spheres at room temperature, and (c) comparing previous r e s i s t i v i t y data with the t h e o r e t i c a l equation. ACKNOWLEDGEMENTS The author i s g r a t e f u l for the advice and encouragement given by his research d i r e c t o r , Dr. A. C. D. Chaklader. Thanks are also extended to other f a c u l t y members and fellow graduate students for many h e l p f u l discussions. Special thanks are extended to Dr. R. B l a i r and Asst. Prof. R. G. Butters for th e i r help and advice. F i n a n c i a l assistance from both Defence Research Board of Canada and National Research Council i s g r a t e f u l l y acknowledged. i i i TABLE OF CONTENTS PAGE I. INTRODUCTION 1 1.1 PREVIOUS RESISTIVITY MEASUREMENTS 2 1.2 QUANTITATIVE APPROACH 5 .1.3 OBJECTIVES OF THE PRESENT WORK 12 I I . THEORETICAL DEVELOPMENTS 14 11.1 GEOMETRIC RELATION AND CURRENT NETWORK 14 11.2 THEORETICAL MODELS 16 11.3 GEOMETRIC RELATIONSHIPS 17 a) Simple Cubic Packing 17 b) Orthorhombic Packing 19 c) - Rhombohedral Packing 21 d) i ) B.C.C. Packing 23 i i ) : Tetrakaidecahedron Packing 25 11.4 DEDUCED RELATIONSHIPS 27 a) When R = R Q 27 b) When R ± R Q 28 11.5 PRESENT VS. PREVIOUS CONDUCTIVITY EQUATIONS . . . . 32 II I . EXPERIMENTAL VERIFICATION OF THEORY . . . . • 34 I I I . l EQUIPMENT 34 I I I . 2 PROCEDURE 37 111.3 MEASUREMENT ON NON-POROUS GLASS 38 111.4 PROCEDURE FOR NICKEL SPHERES 38 i v TABLE OF CONTENTS (continued) PAGE IV. RESULTS AND DISCUSSION 43 IV. 1 CONDUCTIVITY VS. DENSITY FOR GLASS 43 IV.2 CONDUCTIVITY VS. RELATIVE DENSITY FOR NICKEL . . . 43 IV. 3 TEST OF THEORETICAL MODELS 43 IV.4 ELECTRICAL CONDUCTIVITY OF GLASS 47 IV.5 RELATIVE CONDUCTIVITY VS. RELATIVE DENSITY . . . . 52 IV.6 ELECTRICAL CONDUCTIVITY OF THE GREEN COMPACT . . . 52 IV.7 VERIFICATION OF THE THEORETICAL MODELS WITH PREVIOUS RESISTIVITY DATA 56 IV. 8 PACKING GEOMETRY INSIDE THE DIE 59 IV. 9 DEFORMATION GEOMETRY INSIDE THE DIE 62 IV.10 EFFECTS OF OTHER PARAMETERS ON CONDUCTIVITY MEASUREMENTS 63 a) Surface E f f e c t 63 b) Size E f f e c t 63 V. SUMMARY AND CONCLUSIONS 66 VI. SUGGESTIONS FOR FUTURE WORK 67 APPENDICES 68 BIBLIOGRAPHY 90 V LIST OF FIGURES PAGE 1. E f f e c t of compacting pressure on the e l e c t r i c a l r e s i s t i v i t y of coarse (75-100IJ) carbonyl n i c k e l powder compacts during s i n t e r i n g (After Hausner' ) 3 2. E l e c t r i c a l r e s i s t i v i t y vs. density of coarse and f i n e (<44p) carbonyl n i c k e l powder compacts heated i n hydrogen from 600°C to 1100°C (After Hausner 8) 3 3. E f f e c t of atmosphere on change of e l e c t r i c a l r e s i s t i v i t y i n r a i s i n g temperature of copper compacts (After Kimura ). . 4 4. Porous bronze; v a r i a t i o n of e l e c t r i c a l conductivity (20°C) with density. A-̂ , A2, etc. r e f e r to porous bronze specimens ( (After Grootenhuis 1 0) 4 e l e c t r i c a l conductivity at 20°C (After 6 e l e c t r i c a l conductivity at 20°C (After 6 7. Geometric d i s t r i b u t i o n of phases ( p a r a l l e l slabs) and di r e c t i o n s of current-flow 8 8. Conductivity curves for p a r a l l e l and series c i r c u i t s of a two-phase system. Maxwell's equation for s p h e r i c a l i n c l u s i o n i s also included. ( o ^ = 10a^ assumed) 8 9. Change i n conductivity with r e l a t i v e density for pressed copper powder (After H u t t i g ^ ) 10 10. Measured and calculated e l e c t r i c a l c o n d u c t i v i t i e s of high- purity sintered copper specimens (After K l a r 2 ) 10 11. Geometry of deformation of two spheres i n contact and equivalent e l e c t r i c a l network 15 12. Spheres i n two-dimensional cubic array and equivalent e l e c t r i q a l network 15 13. Geometric r e l a t i o n s h i p of simple cubic model and current- path (x - x) 18 14. Geometric r e l a t i o n s h i p of orthorhombic model and current- path (x,- x) 20 5. Porous copper: Grootenhuis 1 0) 6. Porous n i c k e l : Grootenhuis 1 0 ) LIST OF FIGURES (continued). NO. PAGE 15. Geometric r e l a t i o n s h i p of rhombohedral model and current-paths. 22 16. Geometric r e l a t i o n s h i p of b.c.c. model and current-paths. . . 24 17. Geometric r e l a t i o n s h i p of tetrakaidecahedron model and the a d d i t i o n a l current-path 26 18. Theoretical r e l a t i o n s h i p of R vs. a f o r the models (After Kakar 2 6) 29 19. T h e o r e t i c a l r e l a t i o n s h i p of r e l a t i v e conductivity vs. Relative density f o r the proposed models 31 20. Comparison of the present and previous t h e o r e t i c a l equations. 33 21. Photograph of a) glass spheres,0.42 mm average d i a . , b) glass spheres, 0.70 mm average d i a . , and c) n i c k e l spheres, 0.65 mm average dia 35 22. Schematic diagram of the equipment used for r e s i s t i v i t y measurements of glass spheres 36 23. Schematic diagram of the die used for hot-pressing n i c k e l spheres 40 24. Photograph of the v i s e used to measure the e l e c t r i c a l conductivity of n i c k e l compacts. . . 41 25. Conductivity vs. Relative density f o r glass compacts at d i f f e r e n t temperatures 44 26. Conductivity vs. Relative density f o r n i c k e l compacts at room temperature 46 r > D \ 2 / 3 i 27 Log a m vs. log — J - 1 for glass compacts 48 (_ o 28. Log a vs. — for the non-porous glass at d i f f e r e n t times . . . 50 29. Log a (calculated) vs. — for non-porous glass and 1 log a m vs. - f o r porous glass (0.70 r e l a t i v e density). . . 51 30. Relative conductivity vs. r e l a t i v e density for glass at 550°C 53 v i i LIST OF FIGURES (continued) NO PAGE 31. Relative conductivity vs. r e l a t i v e density for glass at 600°C 54 32. Relative conductivity vs. r e l a t i v e density for glass at 650°c 55 33. Relative conductivity vs. r e l a t i v e density for n i c k e l at room temperature 57 34. Corrected values of r e l a t i v e conductivity vs. r e l a t i v e density for n i c k e l compacts 58 35. Results of previous investigations compared with proposed models . 60 36. E f f e c t of container-size on the e f f i c i e n c y of packing one-size spheres (After McGeary-^) 61 (35) 37. Basic systems of s p h e r i c a l packings (After Morgan ) . . . 71 38. Theoretical r e l a t i o n s h i p of D vs. a/^ for the proposed models (After Kakar 2 6) 73 39. Geometric re l a t i o n s h i p s for the u n i t - c e l l s i n d i f f e r e n t orientations with respect to current path 83 v i i i LIST OF TABLES NO. PAGE I. E l e c t r i c a l conductivity equations f or two-phase composites 13 I I . Geometric constants for the proposed models 32 I I I . Basic methods of packing and t h e i r construction . . . . 69 CHAPTER 1 I. INTRODUCTION S i n t e r i n g ^ i s a complex process by which d e n s i f i c a t i o n of a powder compact takes place at a temperature below the melting (2) point of the (bulk of the) material. Hot-pressing i s a s i n t e r i n g process, i n the presence of an applied pressure. Hot-pressing methods promise products of greater density at much lower temperatures and for shorter times than conventional s i n t e r i n g processes. The degree of d e n s i f i c a t i o n during and a f t e r s i n t e r i n g or hot-pressing can be determined i n several ways, some of which are l i s t e d below: 1) measurement of density changes, 2) measurement of strength, 3) microscopic examination, 4) e l e c t r i c a l measurement, 5) thermal conductivity determination, 6) sound v e l o c i t y determination, 7) X-ray d i f f r a c t i o n a n alysis, etc. Density determination i s the most widely used technique. For small objects, this does not pose any problem, but f o r large pieces such as those fabricated by hot-or c o l d - r o l l i n g , hot-or cold-extrusions, density measurement i s not easy and frequently involves destruction of the objects. Measurements of strength and thermal c h a r a c t e r i s t i c s are d i f f i c u l t and are of low precision,when porous bodies of very low strength are to be measured. Microscopic observations show very l i t t l e change during early stages of s i n t e r i n g . Measurement of sound v e l o c i t y i s 2 f a i r l y c o m p l i c a t e d as a r e X-ray d i f f r a c t i o n measurements. Compared w i t h a l l t h e s e methods, measurements o f e l e c t r i c a l p r o p e r t i e s , s u c h as e l e c t r i c a l r e s i s t i v i t y and t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t a n c e , have many a d v a n t a g e s . They can be made w i t h f a i r l y s i m p l e equipment. Because of the much g r e a t e r change i n r e s i s t i v i t y t han i n d e n s i t y , they a r e e x c e l l e n t i n d i c a t o r s i n e a r l y s t a g e s o f s i n t e r i n g when o t h e r methods can be a p p l i e d o n l y w i t h a low degree o f p r e c i s i o n o r w i t h g r e a t d i f f i c u l t y . I . l . PREVIOUS ELECTRICAL RESISTIVITY MEASUREMENTS (3) The e a r l i e s t work i n t h i s f i e l d was t h a t o f T r z e b i a t o w s k i who r e p o r t e d a r a p i d d e c r e a s e i n e l e c t r i c a l r e s i s t i v i t y o f s i n t e r e d copper and g o l d compacts w i t h i n c r e a s e i n t e m p e r a t u r e and d e n s i t y . T h i s work was f o l l o w e d by t h a t o f Iwase and O g a w a ^ \ Myers ̂ \ H u t t i g ^ \ A d l a s s n i g and F o g l a r ^ ^ e t c . . H a u s n e r ^ s t u d i e d the b e h a v i o u r o f copper and n i c k e l powders, h o t - p r e s s e d i n hydrogen a t 5 t o 80 t . s . i . i n t h e te m p e r a t u r e range 600°C t o 1000°C. He p l o t t e d t h e e l e c t r i c a l r e s i s t i v i t y as a f u n c t i o n of d e n s i t y and c o m p a c t i n g p r e s s u r e a t v a r i o u s t e m p e r a t u r e s . ( F i g u r e s 1 and 2 ) . He n o t e d a d i f f e r e n c e i n the e l e c t r i c a l r e s i s t i v i t y a t a g i v e n d e n s i t y f o r powders o f d i f f e r e n t s i z e s , b o t h i n t h e as-compacted s t a g e and h o t - p r e s s e d c o n d i t i o n . The e l e c t r i c a l r e s i s t i v i t y o f the g r e e n compact was found t o be a f u n c t i o n o f the p a r t i c l e s i z e . (9) Kimura and H i s a m a t s u c a r r i e d o u t s i n t e r i n g s t u d i e s on copper and n i c k e l powders i n hydrogen, a r g o n and vacuum. They p l o t t e d the e l e c t r i c a l r e s i s t i v i t y as a f u n c t i o n o f t e m p e r a t u r e i n d i f f e r e n t atmospheres., ( F i g u r e 3 ) , and n o t e d d i f f e r e n t s t e p s i n the r e s i s t i v i t y 3 100 VO I 90 1 \ O Fine n i c k e l o *—( X CJ 80 i i i i i i TV " Coarse n i c k e l Figure 2. oh m—  ci  70 i i i i • j E l e c t r i c a l r e s i s t i v i t y 60 i i i vs. density of coarse i i ! and fine (<44(J) • H > 50 i i | carbonyl n i c k e l powder • H 4-1 - A* [ compacts heated i n cn • H i hydrogen from 600 to _ cn cu 40 1 1 [ 1100°C (After Hausner ). u | El ec tr  30 20 10 1 \ T i i 6.1 6.2 6.3 Density, g/cc 3 § .3 Q • 1 in Hydrogen in Uxua in Argon \ \ w \ \ \ \ \ s V N \ ..i Figure 3. 200 300 Temperature. °C too 500 E f f e c t of atmosphere on change of e l e c t r i c a l r e s i s t i v i t y i n r a i s i n g temperature of copper compacts. (After Kimura ). Density (g /cm 3 ) 4 5 6 7 8 9 I I I Solid Bronze Porosity (percent) gure 4. Porous bronze : v a r i a t i o n of e l e c t r i c a l conductivity (20°C) with density. A^,A2 etc refer to porous' bronze specimens. (After Grootenhuis 5 curves,in agreement with Myers' w o r k ^ \ In hydrogen, two steps of rapid decrease were observed. In argon and vacuum, only one step was observed. The f i r s t step i n hydrogen at low temperature could be due to the reduction of the surface oxide layer or removal of adsorbed gases. The subsequent step at higher temperature ( i n a l l atmospheres) i s due to s i n t e r i n g and bonding of metal powders. Grootenhuis et a l ^ ^ studied the e l e c t r i c a l r e s i s t i v i t y (3 7 of sintered bronze (Figure 4). They replotted the works of others * ' 8,1] to 15) ^ copper and n i c k e l powders (Figures 5 & 6), and claimed that i n a l l cases the experimental data conformed to the s t r a i g h t l i n e , drawn from the point for s o l i d metal to cut the x-axis at a porosity of 47.6%. This pqorpsity corresponds to the maximum porosity, which can be attained on packing equal sized spheres i n simple cubic array. Consequently, zero conductivity was assumed for a simple cubic packing of spheres, and the increase i n conductivity with density was a t t r i b u t e d to increase i n contact area between p a r t i c l e s , and increasing i n t e r p a r t i c l e bonding. The rather large scatter i n the r e s u l t was caused by d i f f i c u l t i e s i n obtaining the data from various figures i n the publications, and the difference i n q u a l i t y of the specimens used by the i n v e s t i g a t o r s . 1.2. QUANTITATIVE APPROACH A l l these studies (so far l i s t e d ) have been q u a l i t a t i v e i n nature. A rigorous mathematical approach to p r e d i c t accurately the conductivity of porous compacts from the known conductivity values of the s o l i d materials was not a v a i l a b l e . This i s p r i m a r i l y due to the fact that the packing geometry of the random shaped powders i s very Density (g/cm J) 5 6 7 8 0 6 | 0-5 . 0 4 0 3 c 0 2 0 1 • Adlassig iFoglar o Goetzel x Hausner O Hensel, Larsen 4 Swaiy & Kieffer 4 Hotop • Sauerwald & Kubik + Trzebiatowski 5 5 5 43 8 32 5 21 2 Porosity (per cent) Figure 5. Porous copper: e l e c t r i c a l conductivity at 20°g (After Grootenhuis ). gure 6. Porous n i c k e l : e l e c t r i c a l conductivity at 20°C (After Grootenhuis ). 7 complicated and i t i s very d i f f i c u l t to p r e d i c t the r e s i s t i v i t y of such a network. However, the studies c a r r i e d out on the thermal and e l e c t r i c a l c o n d u c t i v i t i e s of two phase systems,in terms of the volume fra c t i o n s of the two phases shave thrown much l i g h t i n this f i e l d . The equations derived for two phase systems^in many cases, allow one to clos e l y estimate the e l e c t r i c a l conductivity of porous sintered materials s by assuming one of the phases to be the pore phase. A s i m p l i f i e d approach i s to consider the material as having a regular o r i e n t a t i o n and a structure ssuch as the p a r a l l e l slabs $shown i n Figure .7. . If the current flow i s p a r a l l e l to the plane of the slabs, they are equivalent to a p a r a l l e l e l e c t r i c a l c i r c u i t . The t o t a l conductivity of the material o m i s given by °m = Vl°l + V 2 ° 2 CD where and V 2 are the volume f r a c t i o n s (equal to c r o s s - s e c t i o n a l area) and a j and a 2 are the co n d u c t i v i t i e s of each component. ± ' e * = (1 - V 2) !L + V 2 a 2 P2 If a2 > y o i ' f ° r example^component 1 being a i r , m̂ = V 2 (2) If the slabs are arranged normal to current flow, they are equivalent to a serie s e l e c t r i c a l network and crm ax o2 8 Figure 7. Geometric d i s t r i b u t i o n of phases ( p a r a l l e l slabs) and d i r e c t i o n s of current-flow. 9 or a a l g 2 (3) m and a m V l°2 + V l °2 " V 2 + <1-V2>f2 °1 If a 2 >> o^, fm q l / g 2 (4) a 2 (1-V 2) In Figure 8 , equations (1) and (3) are plotted for = lOo^. (22) Huttig has shown that a l l conductivity data on sintered porous materials should f a l l w i t h i n the region bounded by the two curves, given by equations'(2) and (4). These two equations define the upper and the lower bounds for conductivity data (Figure 9). The above equations are i d e a l i s e d . In p r a c t i c e , i t i s e s s e n t i a l to use equations derived for random s p h e r i c a l i n c l u s i o n s i n a continuous matrix phase,or s p h e r i c a l p a r t i c l e s i n a continuous minor phase. Relationships applicable to random mixtures have been derived by various authors from Maxwell's equation f o r a continuous matrix phase o^, with s p h e r i c a l dispersed phase The conductivity (16) of the mixture a i s given by m ' VjCo"! - a 2) ( 5 ) a m + '2a2 CT ^ + 2 a 2 0 0.2 0.4 0.6 0.8 1.0 Relative Density Figure 10. Measured and calculated e l e c t r i c a l c o n d u c t i v i t i e s high-purity sintered copper specimens (After Klar 11 and are the volume f r a c t i o n s of the dispersed and matrix phases, When a^ y > °\> f ° r example, phase 1 being s p h e r i c a l pores, a = a_ m 2 1 + Vj/2 2(1 - V ) = a — (6) 2 + Vl Equation (5) i s included i n Figure 8 and i s found to s a t i s f y some of the experimental r e s u l t s . Similar equations derived by Juretschke et a l ^ Doebke^^, T o r k a r a n d G r e k i l a and T i e n ^ ^ \ a l l s t a r t i n g from (16) Maxwell's r e l a t i o n , are shown i n Table I. These equations are converted for applying to porous compacts and are also included i n the table. A l l these equations are found to hold good only f o r c e r t a i n sets of data, and for the f i n a l stages of s i n t e r i n g and hot-pressing. For (21) instance, Klar and Michael tested the equations on sintered copper powder and found good agreement with Maxwell's equation (equation (6))> but only at higher densities (> 80% bulk density), as shown i n Figure 10. / o o \ ( / \ Mal'ko et a l and Litvinenko et a l have considered another set of equations f o r the e l e c t r i c a l conductivity of porous metal compacts, which are l i s t e d below: (2 - 3V X) a = o m 2 „ (a) 1 - V l °m = °2 7 ~ ~ — (b) 12 3 0.9 - V a m 2.1 + V* (c) 2 They were d e v e l o p e d u s i n g O d e l e v s k i i ' (22) f o r m u l a e f o r s t a t i s t i c a l m i x t u r e s and m a t r i x systems. These e q u a t i o n s a r e s i m i l a r t o t h o s e l i s t e d i n T a b l e I . E q u a t i o n (a) was found t o f i t t h e i r e x p e r i m e n t a l r e s u l t s s a t i s f a c t o r i l y a t h i g h e r d e n s i t i e s . 1.3. OBJECTIVES OF THE PRESENT WORK A l i t e r a t u r e s u r v e y r e v e a l s t h a t t h e r e i s no s a t i s f a c t o r y e q u a t i o n t o p r e d i c t the e l e c t r i c a l c o n d u c t i v i t y o f a powder compact d u r i n g the i n i t i a l s t a g e s of s i n t e r i n g o r h o t - p r e s s i n g . I n v e s t i g a t i o n s c a r r i e d o u t so f a r d u r i n g the e a r l y s t a g e s o f d e n s i f i c a t i o n m e r e l y p o i n t o u t , t h a t the e x p e r i m e n t a l d a t a s c a t t e r around a l i n e a r r a t e o f i n c r e a s e o f e l e c t r i c a l c o n d u c t i v i t y w i t h i n c r e a s i n g d e n s i t y o f h o t - p r e s s e d compacts. Towards the end of h o t - p r e s s i n g , however, M a x w e l l ' s and o t h e r s i m i l a r e q u a t i o n s have s u c c e s s f u l l y p r e d i c t e d t h e v a r i a t i o n o f the r e l a t i v e e l e c t r i c a l c o n d u c t i v i t y —— w i t h the r e l a t i v e d e n s i t y ( o r volume f r a c t i o n V„; d e n s i t y o f phase 1 - a i r - c a n be c o n s i d e r e d t o be n e g l i g i b l e ) . a) t o d e r i v e an e q u a t i o n t o p r e d i c t the c o n d u c t i v i t y o f a porous compact as a f u n c t i o n o f i t s r e l a t i v e d e n s i t y i n the range 0.6 t o 0.75. Compacts of r e l a t i v e d e n s i t y 0.6 t o 0.75 c o n s t i t u t e the e a r l y s t a g e s of d e n s i f i c a t i o n d u r i n g s i n t e r i n g and h o t - p r e s s i n g , and b) .to t e s t the d e r i v e d e q u a t i o n w i t h e x p e r i m e n t a l d a t a f o r v a l i d i t y . a The purpose o f t h i s i n v e s t i g a t i o n i s : TABLE I 13 •ELECTRICAL CONDUCTIVITY EQUATIONS FOR TWO-PHASE COMPOSITES Equation for Two-phase Equation for Porous Comments System Body Reference A. m 2 a + 2a 0 m 2 V g l - p 2 } °1 + 2°2 a = m 2 a 2 ( l - V L) 2 + V, Spherical Maxwell Inclusions (16) ° m - a 2 V a l " a 2 } a + a„ m 2 a l + °2 a = a 2 ( l - V L) m 2 + V, C y l i n d r i c a l Juretschke Inclusions (17) a - a, m 1 V 2 ( a 2 - cjj) a + Ka„ a„(l + K) m I I a = m K a 2 ( l - V x) K + V, K i s a Function of a2/al Doebke (18) cr - a_ m 2 a + K ( a 2 + a 2 ) 1 ^ m 1 "2 ; V 2 ( a 2 - a 2) ? ? 1 / a L + K(OJ + a 2 ) X / 2 a m Ko 2(l - V p K + V, Inter-Penetrating Phases Torkar (19) E l . a 2 ( A V 2 - B) % _ (C- V„) Same,as two-phase Tetrakaideca-system hedron Model for V2>0.25 E2. a = a.-m 1 (i - v 2) Same as T w o - D h a s e Tetrakaideca- G r e k i l a system hedron Model and Tien (20) for V 2<0„25 A, B and C = constants. a = e l e c t r i c a l conductivity of the compact, m ' and a 2 = e l e c t r i c a l conductivity of the pore and matrix phases, and V„ = Volume f r a c t i o n of the pore and matrix phases, 14 CHAPTER 2 I I . THEORETICAL DEVELOPMENTS. II.1. GEOMETRIC RELATION AND CURRENT NETWORK In the development of the theory, i t i s assumed that the p a r t i c l e s i n a compact are monosized spheres (because a sphere i s the simplest and most symmetric shape) and that they are arranged i n a regular three-dimensional array. During hot-pressing,they deform p l a s t i c a l l y at the points of contact and form f l a t faces. The compact density change as a r e s u l t of this deformation, with respect to contact radius, has been derived by Kakar and i s given by D = 2 1 2 3/2 " ( 7 ) g:(RZ - a V ' where :D i s the bulk-density of the compact at contact radius 'a'. R i s the instantaneous radius of the p a r t i c l e at neck radius 'a' and 5 i s a geometric constant which depends on the packing configuration. F i r s t , ' c o n s i d e r the geometry of deformation of two spheres i n contact (Figure 11). Let the two spheres constitute an e l e c t r i c a l path. The equivalent resistances are shown. In the i n i t i a l period of deformation, the neck region w i l l have a much higher resistance, r 2 , (neglecting contact resistance) as compared to the resistance r^ of the spheres. Hence,the conductivity of the c i r c u i t w i l l depend upon the neck area ira and the thickness of the neck G (boundary width). The t o t a l current path i s (G + 4y'). The boundary width G can be assumed to remain constant during neck growth and as G << 4y', the t o t a l path length 4y ' - 4y. 15 DEFORMATION OF TWO SPHERES EQUIVALENT ELECTRICAL NETWORK Figure 11. Geometry of deformation of two spheres i n contact and equivalent e l e c t r i c a l network. TWO DIMENSIONAL (CUBIC) ARRAY OF SPHERES Figure 12. . Spheres i n two-dimensi e l e c t r i c a l network. j 1 I • i i F ^ 1 I J j : r i i • * 1 1 ] I EQUIVALENT ELECTRICAL NETWORK 1 cubic array and equivalent 16 The spheres can be arranged i n a two dimensional cubic network, as shown i n Figure 12 . The corresponding resistance c i r c u i t i s also shown. If r i s the resistance of each neck region, a v e r t i c a l column of spheres i n Figure 12 w i l l have a resistance of 2r (neglecting resistance of the sphere). If there are n spheres i n a h o r i z o n t a l row, they constitute n p a r a l l e l paths. As there i s no flow of current along a h o r i z o n t a l d i r e c t i o n , the resistance of the network becomes 2r. If there are N stacking of spheres, each column w i l l have a resistance n of (N - l ) r - Nr (since N >>1, N - 1 = N). The c i r c u i t resistance becomes N_ r. The p r i n c i p l e can be extended to spheres arranged i n a three- n dimensional array. Each sphere can now be contained i n a unit c e l l . The s p e c i f i c resistance of this unit c e l l i s the same as for the spheres packed i n three dimensions. II.2. THEORETICAL MODELS The basic systems of packing which give r i s e to s p a c e - f i l l i n g (29) unit c e l l s can be summarised as follows: (1) simple cubic (Z = 6), (2) orthorhombic (Z = 8), (3) body-centered cubic (Z = 8), and (4) rhombohedral (Z = 12), where Z i s the coordination number. Of these the b.c.c. packing i s an unstable arrangement i n a u n i d i r e c t i o n a l f i e l d of force ( i . e . g r a v i t a t i o n a l f o r c e ) . However, the b.c.c. packing gives r i s e to a tetrakaidecahedron unit c e l l which has been extensively used i n the t h e o r e t i c a l models for s i n t e r i n g ^ \ grain growth etc.. For t h e o r e t i c a l purposes, i t i s assumed that each type of packing i s stable and maintains i t s symmetry on a p p l i c a t i o n of pressure, and that the material at the points of contact spreads symmetrically during deformation to maintain the s p h e r i c i t y of the p a r t i c l e . The d i f f e r e n t modes of packing are shown i n the Appendix. 17 II.3. GEOMETRIC RELATIONSHIPS a) Simple Cubic Packing Consider a cubic array of spheres, deformed under uniform hydrostatic pressure along the three mutually perpendicular d i r e c t i o n s . Each sphere w i l l have s i x f l a t faces formed as shown i n Figure 13. 2 2 1/2 The unit c e l l i n this case i s a cube of side 2y,where y = (R - a ) The number of current-paths through the unit c e l l = 1. 2 The area of current-flow = Tra and The path-length = 2y. = 2(R 2 - a 2 ) 1 / / 2 . The conductivity of the unit c e l l , l / (where R = r e s i s t i v i t y ) i s given Rm m by J L = L_ + i _ ( 8 ) m c i where 1_ _ c o n c j u c t i v i t y Qf s o l i d , v i z . sphere, C a n <^ -jL = conductivity of the i n s u l a t i n g phase f i l l i n g the 1 rest of the unit c e l l , which i s a i r i n the case . of porous compacts. Now, , A A. f = ̂  a c + ^ - a. (9) m c 1 where A and A. are the areas of current-flow, L and L. are the path-lengths, and o c and are the s p e c i f i c c o n d u c t i v i t i e s . Subscripts c and i stand for conducting and i n s u l a t i n g phases, r e s p e c t i v e l y . As o, + o (for a i r ) , equation (9) reduces to 1 i - = ̂  0 q (10) m c 18 1 9 For the cubic c e l l , , 2 1 _ ira R ~ 2y CTc • The s p e c i f i c conductivity of the unit c e l l a i s L a = ^ x i - (11) m A R s m where L g i s the length of the unit c e l l , and A g i s the area of the unit c e l l normal to current-flow. Substituting for _1__ i n equation (11) from equation (10), we get R L A s c a = — • -— • cr m A L c s c a L A m s c -, i . e . — = — • — _ _ _ _ _ _ _ _ _ _ _ _ ,12) a L A c c s 2 For the cubic c e l l , L g = 2y and A g = 4y . Hence the r e l a t i v e conductivity i s a 2 m _ _2y Tra "a 2y 4y 2 c J a 2 m TT a a 4 R 2 - a 2 c (13) b) Orthorhombic Packing Each sphere a f t e r deformation w i l l have eight faces. The deformed sphere for this model and i t s unit c e l l are shown i n Figure 14. F i g u r e 14. Geometric r e l a t i o n s h i p of orthorhombic model and current-path (x - x). 21 In this case, the number of current-paths = 1, 2 the area of current-flow = ira = A , and path-length = 2y = L . For the unit c e l l , L = 2y s A„ = 2/3y 2. s Hence, a L A m _ s , c o ~ TT ~A~ (equation 12) c c s 9 2 2y . Tra 2y 2/3y 2 a 2 m ti a a c 2/3 (R - a 2) (H) c) Rhombohedral Packing* (F.C.C. and H.C.P Packing) Each sphere has twelve points of contact and forms twelve f l a t faces. The resultant unit c e l l i s a rhombic dodecahedron shown i n Figure 15. The number of current-paths through the unit c e l l = 3 - 2 Therefore, the area of current-flow = 37ra x cos 0, where 6 i s the angle between the centre to centre l i n e of spheres i n two d i f f e r e n t planes and the d i r e c t i o n of current-flow. The distance between * Alternative annroach in r h p A n n p n r l i Y Figure 15. Geometric r e l a t i o n s h i p of rhombohedral model and current-paths 23 centres of any two spheres i n contact during deformation = 2y ; and cos 0 = /2 //I , from geometry. 2 r - i— H e n c e , = 3ira x . /2 //3 . The length of current-flow through the sphere = AB = y = L /3 C For the unit c e l l , length of current-flow = y = L /3 Area projected normal to current-flow = 2/Jy = A g . a L A m s c / . 1 1 \ — = -— . -— (equation 12) a L A c c s — y /3 /6~ T r a 2 x /8 2/3 y 2 ~ y /3 2 JL_ _ ( 1 5 ) /— • 2 2 v /2 (R Z - a Z) d) ( i ) B.C.C. Packing^ The shape of the deformed sphere i s schematically represented i n Figure 16. There are eight points of contact during the i n i t i a l stages of deformation. The number of current-paths = 4. 2 Area of current-flow = 4Tra cos 0 , where 0 i s the angle between centre to centre l i n e of two spheres i n contact and the d i r e c t i o n of current-flow. a i . e . m a c * A l t e r n a t i v e approach i n the Appendix. 24 F i g u r e 16. G e o m e t r i c r e l a t i o n s h i p o f b . c . c . model and c u r r e n t - p a t h s . 25 cos 0 = , from geometry. /3 The l e n g t h o f c u r r e n t - f l o w A = 4ira x _1_ . /3 = AB = -2y- y = L /3 F o r the u n i t c e l l , L = /3 A r e a n o r m a l t o c u r r e n t - f l o w 16 2 A = — y s 3 J m L A s _ _ c L ' A c s ( e q u a t i o n 12) 2 1 3 4ira x — x — /3 1 6 v 2 m /3 — ix 2 R - a (16) ( i i ) T e t r a k a i d e c a h e d r o n P a c k i n g T h i s i s the same as b . c . c . p a c k i n g , w i t h s i x a d d i t i o n a l p o i n t s o f c o n t a c t g i v i n g r i s e t o f o u r t e e n f l a t f a c e s . The u n i t c e l l i s shown i n f i g u r e 17. There i s an a d d i t i o n a l c u r r e n t - p a t h t h r o u g h f a c e r a d i u s a^. Number of c u r r e n t - p a t h s = 4 + 1 2 L e n g t h s o f c u r r e n t - f l o w a r e L 1 /3 y 1 and L = 2y, where y = ( R 2 - a ^ ) 1 7 2 and y 2 = ( R 2 - a 2 V / 2 L = AB c l 2 73 y l L = CD = 2y„ c 2 y2 Figure 17. Geometric r e l a t i o n s h i p of tetrakaidecahedron model and the a d d i t i o n a l current-path. 27 Areas of current-flow are A = 4Tra. x cos 9 and A = T r a 0 , c, 1 cn 2 where cos 0 = — , from geometry. S3 For the unit c e l l , = •— y^ ; = 2y '1 S3 s 2 ' 2 16 2 Area of unit c e l l normal to current-flow i s A = --— y, . s 3 1 a L S l C l ̂  S2 C l S C2 47ra T r a , 1 * S3 ""2 16 2 16 2 3 y l 3 y l Now, y x = ^ Y 2 Substituting for y^ and s i m p l i f y i n g , m _ IT a 4 c /3 l l + i , 2 2 2 2 R - a^ R - a 2 I I . 4. DEDUCED RELATIONSHIPS a) When R = R n In a l l four cases, the f i n a l form of the equation can be represented by a 2 m a a B 2 2 a R - a c (17) 2 8 where a i s a constant,dependent upon deformation geometry. The compact density change with respect to contact radius, as given i n equation ( 7 ) ^can be modified by using a = o when D = D Q , i . e . the i n i t i a l packing density before deformation. This gives D R R 2 - a 2 3/ , ( 1 8 ) o L Where R i s the i n i t i a l p a r t i c l e radius before deformation. As the o ( 2 6 ) deformation proceeds, the value of R increases, as shown by Kakar (Figure 1 8 ) . I t can be seen from the figure that for a / R < 0 . 2 5 , i . e . for small deformations, R remains approximately constant (R - R ). Then, D o R TJ 2 2 R - a o 72 ( 1 9 ) i . e . (r) 7 3 - 1 = R 2 - a 2 o ( 2 0 ) Substituting equation ( 2 0 ) i n equation (L7)we get ( 2 1 ) b) When R^ R p A more rigorous d e r i v a t i o n , taking into account the v a r i a t i o n of R, can be obtained as follows:  30 KR 2 - a 2 ) 3 / 2 (equation 7) . 3 2 / 3 D 2/ 3 = L _ R 2 - a 2 M u l t i p l y i n g both sides by . R , we get D2/3 ,2/3 R 2 = R - a D 2 / 3 g 2 / 3 R 2 - 1 = 2 2 R - a (22) Substituting equation (22)in equation Q.7)we obtain a< L 2/3 6 2 / 3 R 2 _ ! (23) The above geometry of deformation i s v a l i d only t i l l a c r i t i c a l stage i s reached when the f l a t faces formed on the spheres begin to touch each other. Table II shows the values of a, 8 , Do> and (a/R) c r i t i c a l for the d i f f e r e n t packing geometries.The r e l a t i v e conductivity values for d i f f e r e n t r e l a t i v e densities have been calculated using the re l a t i o n s h i p (23) for d i f f e r e n t packing geometries. The r e s u l t s are plotted i n Figure 19. Equations (21) and (23) are the same equation at the i n i t i a l stages of deformation. .8 R E L A T I V E DENSITY Figure 19. The o r e t i c a l r e l a t i o n s h i p of r e l a t i v e conductivity vs. Relative density for the proposed models. 32 TABLE II Type of a g D , . . Packing . ° ( a / R > • . , i n percent C r i t i c a l Simple cubic TT/4 8 52.36 1/V2 Orthorhombic •n/2/3 4/3 60.46 1/2 Rhombohedral TT//2 4/2 ' 74.05 1/2 b.c.c. 32/3 9 68.02 X /2 II.5. PRESENT VS. PREVIOUS CONDUCTIVITY EQUATIONS , (16 ) Figure 20 shows the t h e o r e t i c a l curves of Maxwell and (19) Torkar calculated for s p h e r i c a l pores i n a continuous matrix. A (19) value of 0.72, as suggested by Torkar i n h i s paper i s used f o r the s t r u c t u r a l constant K (equation D Table I ) . The t h e o r e t i c a l curves of the present i n v e s t i g a t i o n are superimposed f o r comparison. The present theory predicts much lower conductivity values for the compacts. This i s due to the fac t that equation (21)predicts zero conductivity when D = Dq, i . e . f o r the 'green compact'. This need not be so, as discussed i n a l a t e r chapter. The conductivity a' of. the green compact causes the zero point of the curves to be s h i f t e d upwards. Consequently, the th e o r e t i c a l curves l i e closer to the Torkar equation. The z e r o - s h i f t has to be experimentally determined by measuring the conductivity of the green compact, as the r e l a t i v e conductivity of the green compact may vary from almost zero for non-metallic materials to a value between 0.2 to 0.3 for m e t a l l i c systems.  34 CHAPTER 3 I I I . EXPERIMENTAL VERIFICATION OF THEORY To test the t h e o r e t i c a l equation (21), i t i s necessary to determine the e l e c t r i c a l r e s i s t i v i t y (or conductivity, a ) of a compact as a function of r e l a t i v e density. I n i t i a l attempts to test the models with porous oxide compacts were not successful, as the r e s i s t i v i t y of porous A^O^ or MgO compacts was found to be higher (>10^ ohm-cm below 1000°C) than any die materials that can be used for hot-pressing. For this reason, the models are tested with spheres of s o d a - l i m e - s i l i c a 4 2 glass, having a r e s i s t i v i t y between 10 to 10 ohm-cm i n the temperature range 550 to 650°C. The glass spheres used are of two sizes - 0.70 mm and 0.42 mm average diameter (Figure 21, a & b), the nominal composition of which i s S i 0 2 70%, A l ^ 2%, CaO 12%, MgO 2%, and Na 20 10%. The glass spheres were supplied by the 3 M company, St. Paul, Minnesota. The model i s also tested on n i c k e l spheres of 0.65 mm average diameter (Figure 21c), supplied by the S h e r r i t t Gordon and Company, Fort Saskatchewan, Alberta. I I I . l . EQUIPMENT It was necessary to measure the conductivity of glass at a s u f f i c i e n t l y elevated temperature(550 - 650°C),as the room temperature conductivity was greater than 1 0 ^ ohm ^ - c m 1 > The equipment was b u i l t to measure the conductivity and the bulk density of the glass compacts simultaneously during hot-pressing. A schematic diagram of the equipment i s shown i n Figure 22. Stainless s t e e l plungers were used as electrodes. Small sections of transparent s i l i c a glass tubing of 17 mm diameter were used as the die material, as s i l i c a glass has a very high 35 (c) Figure 21. Photographs of a) glass spheres, 0.42 mm average d i a . , b) glass spheres, 0.70 mm average d i a . , c) n i c k e l spheres, 0.65 mm average dia. Magnification x 10. ure 22. Schematic diagram of the equipment used for r e s i s t i v i t y measurements of glass spheres. 37 e l e c t r i c a l r e s i s t i v i t y ' b e l o w 700°C (>IO 1 6 ohm-cm). The die and the electrodes were inserted into a r e c r y s t a l l i s e d alumina tube and held by a s t a i n l e s s s t e e l p in, which also served as an e l e c t r i c a l lead. The alumina tube was lowered into the furnace. Shrinkage was measured with a d i a l gauge having a s e n s i t i v i t y of 0.0002" per d i v i s i o n . This gauge was mounted on the alumina tube with i t s p i n r e s t i n g on a p o r c e l a i n rod f i t t e d into the upper electrode. E l e c t r i c a l resistance was measured with an Impedance bridge -3 7 (range 10 to 10 ohms), operated at IK c/s and 6v. A simple loading device, made up of a lever arm and pans, was .used, as indicated i n the f i g u r e . A maximum of 25 lbs, could be loaded on the pans, which was s u f f i c i e n t to hot-press the glass compacts to a pressure of 50 p . s . i . , i n the temperature, range 550 - 650°C. III.2. PROCEDURE The glass spheres were cleaned with d i l u t e h y d r o f l u o r i c a c i d , washed and dried with isopropyl alcohol to eliminate adsorbed water. A weighed amount of the glass spheres was loaded into the die, tapped and well-shaken i n order to obtain a uniform packing. The as-compacted density was calculated from the i n i t i a l volume of the compact and true density of the glass which was determined by the pycnometric method. I t took 20 to 25 minutes for the specimen to reach the furnace temperature, a f t e r the assembly was introduced into the furnace. During this h e a t i n g - up period no appreciable shrinkage was recorded on the d i a l gauge. The experiments were c a r r i e d out i n dry a i r , as presence of water vapour and low p a r t i a l pressures of oxygen have been known to a f f e c t the (28) e l e c t r i c a l conductivity of glass. A.C. r e s i s t i v i t y was measured i n 38 preference to D.C. to avoid electrode p o l a r i z a t i o n , as glass i s usually an i o n i c conductor. A c a l i b r a t i o n experiment was i n i t i a l l y performed (without the glass spheres) to standardize the shrinkage curves. 111.3. MEASUREMENT ON NON-POROUS GLASS In order to test equation (21), i t was necessary to know the conductivity of the non-porous glass. For t h i s , the following procedure was adopted: a batch of glass spheres was melted i n a platinum c r u c i b l e at 1500°C and held at this temperature f o r 24 hours to eliminate pores. A f t e r 24 hours, the c r u c i b l e was r a p i d l y withdrawn from the furnace. The glass was cast into pre-heated s t a i n l e s s s t e e l moulds. I t was annealed i n a i r at 600°C for eight hours and furnace cooled. Thin specimens of 17 mm diameter and 5 mm thick were cut with an u l t r a s o n i c v i b r a t o r and s i l i c o n carbide suspension i n petroleum l i q u i d . Both faces of the specimens were ground f l a t and as p a r a l l e l as p o s s i b l e to each other on the diamond-polisher. To ensure good e l e c t r i c a l contact with the electrodes, gold was vapour deposited on the faces of the glass with a guard-ring on one side. The purpose of t h i s guard-ring i s to prevent surface leakage. The glass specimen was e q u i l i b r a t e d at d i f f e r e n t temperatures for d i f f e r e n t periods, for the conductivity determination. 111.4. PROCEDURE FOR NICKEL SPHERES The n i c k e l spheres were i n i t i a l l y cleaned and then held i n a stream of cracked ammonia at 650°C for 4 hours to reduce surface oxide. The hot-pressing was c a r r i e d out i n a P h i l i p s 12 KW induction unit. An Inconel die with a graphite sleeve and Inconel plungers were used 3 9 for hot-pressing (Figure 23). A weighed amount of spheres was tapped into the die,with graphite spacers between the spheres and the rams to prevent adherence between n i c k e l and the Inconel. The die and plunger assembly was mounted on a hydraulic press and enclosed i n a quartz tube. An i n e r t atmosphere was maintained around the die during hot-pressing which was car r i e d out under 7200 p . s . i . and at 800°C. The shrinkage was measured with a d i a l gauge. The sample was cooled to room temperature i n the die and them removed. The apparent density was determined from the weight to volume r a t i o . True density was determined by the pycnometric method. Specimens of d i f f e r e n t r e l a t i v e densities were produced following the above procedure. D.C.- conductivity f o r the hot-pressed specimens was measured at room temperature using a Ke l v i n bridge (Pye Cat.no. 7415). As the change i n e l e c t r i c a l r e s i s t i v i t y of n i c k e l compacts i s i n the order of several micro-ohms, the following procedure was adopted. The compact was mounted ins i d e a v i s e made of l u c i t e . A steady current of 7 amps, was passed through the compact. To obtain uniform current d i s t r i b u t i o n across the cros s - s e c t i o n a l area, thin lead discs (reagent grade) were pressed against the sides of the compact; the current terminals were held against the lead discs by screws provided on eit h e r side of the v i s e . Two p o t e n t i a l probes were introduced as shown i n Figure 24 at a fixed distance apart. The probes were made of n i c k e l (to eliminate contact p o t e n t i a l s with the specimen). Their ends were narrowed to small hemispherical t i p s to ensure good e l e c t r i c a l contact. The n i c k e l probes were set i n two reamed holes to prevent any 40 Inconel plungers ^Graphite susceptor « lncon-1 die — - G r a p h i t e sleeve Figure 23. Schematic diagram of the die used for hot-pressing n i c k e l spheres. 4 1 Figure 24. Photograph of the v i s e used to measure the e l e c t r i c a l conductivity of n i c k e l compacts. 42 sidex^ays movement. A number of measurements were made using the Kelvin bridge and a d e f l e c t i o n galvanometer (Tinsley Type S.R. 4/45 ). I t was observed that any v a r i a t i o n i n the pressure of the probes against the specimen produced i n s i g n i f i c a n t changes i n the readings. The e l e c t r i c a l conductivity was calculated f or each specimen from the measured resistance values. CHAPTER 4 IV. RESULTS AND DISCUSSION IV.1 . CONDUCTIVITY VS. DENSITY FOR GLASS The e l e c t r i c a l r e s i s t i v i t y of the compacts of glass spheres was measured continuously as a f u n c t i o n of d e n s i t y during h o t - p r e s s i n g under isothermal c o n d i t i o n s . The volume of the compact was c a l c u l a t e d using the height of the compact at any stage of d e n s i f i c a t i o n ( i n d i c a t e d by the d i a l gauge) and the diameter of the d i e , which remained constant. The e l e c t r i c a l c o n d u c t i v i t y a , was c a l c u l a t e d from the r e s i s t a n c e J m measurements. The cm values are p l o t t e d as a f u n c t i o n of r e l a t i v e density f o r three d i f f e r e n t temperatures as shown i n Figure 25a and b. The c o n d u c t i v i t y of the compact increased w i t h increase i n r e l a t i v e d e n s i t y . IV.2. CONDUCTIVITY VS. RELATIVE DENSITY FOR NICKEL Compacts of n i c k e l spheres were hot-pressed at 800°C under a pressure of 7200 p . s . i . f o r d i f f e r i n g periods to o b t a i n d i f f e r e n t r e l a t i v e d e n s i t i e s . These compacts were subsequently used f o r D.C. c o n d u c t i v i t y measurements at room temperature. The r e s u l t s are p l o t t e d as a f u n c t i o n of r e l a t i v e d e n s i t y , as shown i n Figure 26. IV.3. TEST OF THEORETICAL MODELS The t h e o r e t i c a l models were tested by comparing the r e s u l t s w i t h the general .equation 44 15 6 6 5 -7 7 5 - 8 R E L A T I V E DENSITY Figure 25.a. Conductivity vs. Relative density f o r glass compacts at d i f f e r e n t temperatures. 45 Figure 25.b. Conductivity vs. Relative density for glass compacts at d i f f e r e n t temperatures. 46 R E L A T I V E DENSITY Figure 26. Conductivity vs. Relative density for n i c k e l compacts at room temperature. 47 Where and D are experimentally measured v a r i a b l e s ; a = a material constant and a = a geometric constant, c ° i s the i n i t i a l bulk density, which i s constant for a given experiment. Hence, log i s p l o t t e d vs. l o g ^ ^ — ^ ^ - l ^ j i n Figure 27, which shows that at every temperature the p l o t i s l i n e a r with a slope close to unity f o r most of the data. This agreement between the t h e o r e t i c a l p r e d i c t i o n and the experimental data confirms the v a l i d i t y of the above equation. The deviation from the predicted curve,in the i n i t i a l stages of d e n s i f i c a t i o n , i s due to p a r t i c l e rearrangement i n the density range 60 to 66%. The density increase i n this range i s caused by p a r t i c l e s l i d i n g as w e l l as by p l a s t i c deformation, rather than by p l a s t i c deformation alone. This p a r t i c l e s l i d i n g changes the coordination number and establishes a greater number of current paths. Beyond 66%, the density increases mostly by neck growth at the points of contact and the t h e o r e t i c a l equation i s obeyed. IV.4. ELECTRICAL CONDUCTIVITY OF GLASS As the temparature i s r a i s e d , the conductivity of s o l i d glass r a p i d l y increases and over a considerable temperature range, the conductivity can be represented by an Arrhenius type of equation, -Q/RT a = a e ^ o where = a temperature independent constant Q = experimental a c t i v a t i o n energy for conduction. R and T have t h e i r usual meanings. (29) Terai has pointed out that i n sodium al u m i n o s i l i c a t e glasses of composition s i m i l a r to that used i n t h i s study, the e l e c t r i c a l 10 3 2 . 3 48 9 8 7 6 5 4 10 . 4 9 8 7 6 5 4 3 10 10 . 2 550°C • - 3 5 + 5 0 mesh o - 2 0 + 3 0 mesh J I i I I I I _ L J L 4 5 6 7 89 10' 1^3 I 3 4 5 6X10 ( D Do Figure 27. Log vs. log compacts. - I ) (If3 - 1 for glass 49 conductivity i s e l e c t r o l y t i c . The current i s mainly c a r r i e d by the sodium-ions moving through the 'holes' i n the structure, and the transport number of sodium ions i s close to unity for sodium alum i n o s i l i c a t e glasses. To study the temperature dependence of e l e c t r i c a l conductivity, the logarithm of the conductivity of the non-porous glass i s plotted against the r e c i p r o c a l of absolute temperature i n Figure 28. The conductivity of a glass reaches an equilibrium value at a given 30 temperature only a f t e r a long i n t e r v a l of time (Kaneko and Isard ). However, the conductivity values measured i n this work a f t e r a fixed time i n t e r v a l at d i f f e r e n t temperatures, l i e on a s t r a i g h t l i n e . Figure 28 shows that the log conductivity vs. 1/T r e l a t i o n s h i p s a f t e r constant i n t e r v a l s have nearly the same slope. In order to measure the equilibrium conductivity at 600°C, the specimen was held for 72 hours at this temperature t i l l i t attained a steady value, which was -3 -1 -1 3.5 x 10 ohm- cm . This value i s now substituted for o"c i n equation (21^. Thus the only unknown parameter a i n equation (21} can be calculated. The value of a was found to be 0.9. Equation (21)can be rewritten as, (24) log ac = log am - log a - log Jf|~) 3 - 1 Using a = 0.9 and equation (24),log a c values at d i f f e r e n t temperatures were calculated and plotted i n Figure 29. This figure also includes the logarithmic conductivity values for a porous glass 50 4xl0 3 [- o 6 Hours • I 2 Hours 10 .-3 9 8 7 6 2 h I 0 10 5 O E q u i l i b r i u m value ( at 600 °C ) I 2 -4 l /TX I 0"^(° K"1) 12*5 Figure 28. Log a vs. — for the non-porous glass at d i f f e r e n t times. 51 Figure 29. Log (calculated) vs. — for non-porous glass and log a vs. 7 f f ° r porous glass. 52 ( r e l a t i v e density = 0.7). For the porous glass, equation(21)can be -Q/RT. rewritten, replacing o £ by O q e a = a o m o Ik) - Q/RT e At a constant r e l a t i v e density, log o"m vs. 1/^ gives the experimental a c t i v a t i o n energy for porous glass, which has been found to be the same as for the non-porous glass. The l i n e s i n Figures 28 and 29 have a slope of (0.67 ± .05)x 10 degrees Kelvin and the corresponding a c t i v a t i o n energy for e l e c t r i c a l conductivity i s 30 ± 3 kcal/mole. This value i s s i m i l a r to the values reported i n l i t e r a t u r e (25 to 30 kcal/mole) for sodium-aluminosilicate glasses. IV.5. RELATIVE CONDUCTIVITY VS. RELATIVE DENSITY Figure 30 shows the p l o t of r e l a t i v e conductivity vs. r e l a t i v e density for glass spheres of two d i f f e r e n t sizes at 550°C. The calculated a values from Figure 29 were used for c a l c u l a t i n g the r e l a t i v e conductivity (0 m/cj ) at d i f f e r e n t temperatures. The r e l a t i v e conductivity values follow the general trend of the t h e o r e t i c a l curves, which are superimposed i n the figure for comparison. Figures 31 and 32 show s i m i l a r plots for the r e s u l t s obtained at 600°C and 650°C. The deviation i n the range .60 to .66 r e l a t i v e density i s again i n d i c a t i v e of the contribution from p a r t i c l e rearrangement. IV.6. ELECTRICAL CONDUCTIVITY OF THE 'GREEN COMPACT' -4 The t h e o r e t i c a l l y derived equation(21)has the boundary condition that when D = D„, a = o, i . e . the e l e c t r i c a l conductivity of o» m ' J 0.8 RELATIVE DENSITY igure 30. Relative conductivity vs. r e l a t i v e density for glass at 550°O 0.8 RELATIVE DENSITY Figure 31. Relative conductivity vs. Relative density for glass at 600°C. RELATIVE CONDUCTIVITY 56 the compact before deformation i s zero. However, the p a r t i c l e s i n contact with each other,before hot-pressing*do have a c e r t a i n conductivity. As a r e s u l t equation(2l)should be modified as a = a + ao m c [ i y - •] (25) where a' i s the e l e c t r i c a l conductivity of the compact before deformation. Equation (25) can be rewritten as, a • m a' = a / a a 1 c c - - 1 (26) Thus, the experimental r e s u l t s have to be corrected for a zero-error along the ordinate. For glass t h i s error i s n e g l i g i b l e , as the r e l a t i v e o' conductivity o f the green compact, — i s < .005, below .63 r e l a t i v e °c density. The r e s u l t s for compacts of n i c k e l spheres demonstrate the e f f e c t , when o'/oQ i s rather large (Figure 33). The r e l a t i v e conductivity of the green compact ( .60 r e l a t i v e density) was found to be 0.18. The r e s u l t s were corrected f o r this error and re p l o t t e d i n Figure 34. This corrected data f i t the t h e o r e t i c a l models more c l o s e l y . IV.7. VERIFICATION OF THEORETICAL MODELS WITH PREVIOUS RESISTIVITY DATA As discussed previously, a large amount of data are a v a i l a b l e on the r e s i s t i v i t y of metal powder compacts. Although most of these compacts were made-up of random-shaped p a r t i c l e s , i t would be i n t e r e s t i n g to see i f any of these data f i t the equation derived i n t h i s work. In  RELATIVE DENSITY Figure 34. Corrected values of r e l a t i v e conductivity vs. r e l a t i v e density for n i c k e l compacts. 59 order to do t h i s , the work of Grootenhuis appears to be the l o g i c a l choice, as both density and r e s i s t i v i t y values were reported i n his work. Other r e s i s t i v i t y data were mostly reported as a function of temperature and pressure and thus could not be tested with the th e o r e t i c a l equation. Some of the re s u l t s of Grootenhuis shown e a r l i e r i n Figures 4 to 6 are replotted i n Figure 35 and compared with the curves drawn from the present theory. The curve for orthorhombic packing i s corrected for the zero-error and s h i f t e d upwards. A reasonable agreement with the experimental data can be seen, which again confirms the v a l i d i t y of the derived equation. IV.8. PACKING GEOMETRY INSIDE THE DIE It i s evident from the Figures 30 to 34, that the experimental data follow closely the t h e o r e t i c a l l y derived equation for the hexagonal prism model i n the i n i t i a l stages of d e n s i f i c a t i o n . This agreement indicates that the o v e r a l l packing geometry of spheres i n s i d e the die may be s i m i l a r to the orthorhombic packing. When a die i s randomly f i l l e d with a number of monosized spheres, with intermittent shaking and tapping i n order to achieve a uniform packing, the spheres tend to spread l a t e r a l l y to achieve the most stable configuration. However, the die-wall o f f e r s resistance to l a t e r a l spreading. As a r e s u l t , a c e r t a i n degree of s t a b i l i t y of (31) packing i s maintained i n s p i t e of an unstable configuration. McGeary studied the various modes of f i l l i n g the die and the e f f e c t of container s i z e on the packing density. His re s u l t s are shown i n Figure 36. At I) values greater than 10, the packing density of the compact reaches d  6 1 7 0 Figure 36. E f f e c t of container si z e on the e f f i c i e n c y of packing one-size spheres (After McG e a r y 3 ^ ) . 62 a maximum of 62.5% of the t h e o r e t i c a l density. This value i s close to the as-compacted density f or orthorhombic packing. (32) Smith, Foote and Busang studied the coordination number of spheres i n a die r e s u l t i n g a f t e r shaking and tapping. Their r e s u l t s showed a Gaussian d i s t r i b u t i o n of the number of spheres with a given coordination number. The average coordination number of the spheres was close to 8. Thus these r e s u l t s confirm that the i n i t i a l packing of the spheres inside a die i s close to orthorhombic, as was observed i n this study. IV.9. DEFORMATION GEOMETRY INSIDE THE DIE The width of the die does not permit an i n t e g r a l number of spheres across the diameter; hence the four i d e a l modes of packing discussed i n section II do not e x i s t across the diameter of the die, since a c e r t a i n number of spheres are l i g h t l y held against the die-wall (due to the die-wall e f f e c t ) . On i n i t i a l a p p l i c a t i o n of the load, the loosely held spheres rearrange, giving r i s e to a higher r e l a t i v e density, (33) ( 34) D u f f i e l d and Grootenhuis , and Kakar and Chaklader reported a volume change of 1 to 4% for glass spheres of 0.5 mm average diameter, xrtien a load of 1000 p . s . i . was applied at room temperature. As no p a r t i c l e (sphere) fragmentation was observed, t h i s volume change was att r i b u t e d to the p a r t i c l e rearrangement discussed above. On further loading, the p a r t i c l e s begin to deform. Kakar (34) and Chaklader studied the deformation geometry of spheres i n randomly f i l l e d dies. Although i n d i v i d u a l colonies of rhombohedral or tetragonal deformation were observed, the majority of the spheres showed 63 a hexagonal prism mode of deformation. The mean coordination number of spheres during deformation was between 8 and 9. This indicates that the orthorhombic packing configuration i s maintained during the i n i t i a l stages of deformation, as was observed i n the present study. IV.10. EFFECTS OF OTHER PARAMETERS ON CONDUCTIVITY MEASUREMENTS a) Surface E f f e c t Surface conduction has been known to a f f e c t the e l e c t r i c a l conductivity measurements of d i e l e c t r i c materials. At temperatures below 300°C, the e l c t r i c a l conductivity of glass i s greatly influenced by i t s surface condition. Traces of water on the surface d r a s t i c a l l y increases the conductivity of glass , Above 300°C, the contribution of the surface conductivity to the t o t a l conductivity becomes less s i g n i f i c a n t , as the i o n i c conductivity of glass increases markedly. The surface to volume r a t i o also a f f e c t s the surface conductivity, but for spheres t h i s r a t i o has the minimum value and hence, the surface conductivity can be expected to be minimum for the compacts of s p h e r i c a l p a r t i c l e s . The 2 s p e c i f i c surface area ( i . e . Cm /gm) of a p a r t i c u l a t e compact has the most s i g n i f i c a n t e f f e c t on the surface conductivity., as compacts of very fin e p a r t i c l e s w i l l have a large surface area a v a i l a b l e f o r conduction. In order to minimize the contribution from surface conduction on the o v e r a l l conductivity, large spheres (0.4 to 0.6 mm diameter) of glass were used for conductivity measurements. b) Orientation E f f e c t As previously discussed i n the t h e o r e t i c a l models, 64 .0 A . L f - t - f <"> C S C This equation reduces to a A m _ _ c , a " A c s since i n a l l four models, L = L . The d i r e c t i o n of current-flow i n c s the models discussed has been considered to be perpendicular to the face of the unit c e l l s . In p r a c t i c e , however, one has to consider cases when the current-flow i s p a r a l l e l to the face-diagonal or the cube-diagonal of the unit c e l l s . When the orientations of the unit c e l l s are varied with respect to the d i r e c t i o n of current-flow, the number of paths through the unit c e l l changes, but so do the e f f e c t i v e area of current- flow and the area of the unit c e l l normal to current-flow. The t o t a l e f f e c t of the change of o r i e n t a t i o n with respect to the d i r e c t i o n of current-flow i s such,that the conductivity of the unit c e l l i s unaltered regardless of i t s o r i e n t a t i o n , as shown i n the appendices. Hence, the r e s i s t i v i t y of systematically packed spheres does not change with the d i r e c t i o n of current-flow. However, the packing of spheres in s i d e a die i s not systematic, and t h i s could cause large v a r i a t i o n s i n conductivity depending on the d i r e c t i o n of current-flow. Any change i n the p o s i t i o n of neighbours of a given sphere from the i d e a l configuration could increase or decrease the number of current-paths and the e f f e c t i v e area of current-flow. The conductivity i s a very s e n s i t i v e function of packing geometry. The r e s u l t s obtained i n the present-work on glass and n i c k e l only i n d i c a t e that any such deviations due to random f i l l i n g 65 apparently n u l l i f y one another. Thus, the o v e r a l l conductivity-density r e l a t i o n s h i p follows the orthorhombic model reasonably w e l l . 66 V. SUMMARY AND CONCLUSIONS The e l e c t r i c a l r e s i s t i v i t y of a powder-compact during hot-pressing i s calculated using the geometry of deformation of p a r t i c l e s under load. The p a r t i c l e s are assumed to be monosized spheres. An equation r e l a t i n g the e l e c t r i c a l conductivity to the r e l a t i v e density for d i f f e r e n t i d e a l packing arrangements has been derived, which i s 2/o 2/T 2 a /°" = ct (D 3 J R - 1 ) . The t h e o r e t i c a l equation was compared with hot-pressing data on glass and n i c k e l spheres. The following conclusions can be made. 1) The general t h e o r e t i c a l equation proposed i s found to be obeyed by the p a r t i c l e s during the i n i t i a l stages of hot- pressing (in the range 0.65 to 0.75 r e l a t i v e density). 2) A deviation i n the range 0.6 to 0.65 r e l a t i v e density was encountered, which could be due to p a r t i c l e rearrangement at the beginning of hot-pressing. 3) The r e l a t i v e conductivity vs. r e l a t i v e density p l o t s showed that the o v e r a l l packing of spheres inside the die i s close to orthorhombic, i n agreement with the observations of previous studies. 4) The t h e o r e t i c a l equation has been modified i n order to take into account the r e l a t i v e conductivity 'of the green compact. The modified equation f i t s the data on metal-compacts obtained i n the present study and i n previous i n v e s t i g a t i o n s . 67 VI. SUGGESTIONS FOR FUTURE WORK 1) I t would be i n t e r e s t i n g to derive the resistance of a sphere i n terms of the contact areas from p o t e n t i a l theory and La Place's equation i n three-dimensions. This w i l l give an equation which would predict the resistance of the green-compact as w e l l as the resistance during deformation. 2) The coordination change i n the i n i t i a l stages of hot-pressing could be c l o s e l y followed by evaluating any change i n the parameter 'a' i n this range. 3) The t h e o r e t i c a l equation could be tested with more r e s i s t i v i t y data of random-shaped p a r t i c l e s . S i m i l a r l y , the p a r t i c l e s i z e e f f e c t on the r e l a t i v e conductivity should be experimentally determined and i t s e f f e c t on the t h e o r e t i c a l predictions should be evaluated. b « APPENDICES APPENDIX I 69 There a r e f i v e s i m p l e and s y s t e m a t i c modes o f p a c k i n g o f (35) u n i f o r m s p h e r e s . These a r e shown i n F i g u r e 37 (from Morgan ) and can be d e s c r i b e d as f o l l o w s . 1 . C u b i c P a c k i n g T h i s p a c k i n g i s c o n s t r u c t e d by p l a c i n g s p h e r e s i n s q u a r e f o r m a t i o n ( f i g u r e 37a). Spheres i n second and subsequent l a y e r s a r e p l a c e d v e r t i c a l l y o v e r t h o s e i n p r e c e d i n g l a y e r s . Such an arrangement makes t h i s p a c k i n g most open and l e a s t s t a b l e . 2. O r t h o r h o m b i c P a c k i n g F i g u r e 37 c and d shows t h a t t h i s t y p e o f p a c k i n g can be o b t a i n e d e i t h e r be s t a c k i n g s p h e r e s i n second l a y e r h o r i z o n t a l l y , o f f s e t w i t h r e s p e c t t o t h o s e i n the f i r s t l a y e r by a d i s t a n c e R ( s p h e r e r a d i u s ) a l o n g t h e d i r e c t i o n of one s e t o f rows, o r by s t a c k i n g v e r t i c a l l y o v e r t h o s e i n f i r s t s i m p l e rhombic l a y e r . I t t u r n s out t h a t t h e s e two ways o f p a c k i n g a r e i d e n t i c a l i n n a t u r e , though o f d i f f e r e n t o r i e n t a t i o n i n s p a c e . 3. B o d y - C e n t r e d - C u b i c P a c k i n g F i g u r e 37 b shows t h i s t y p e o f p a c k i n g . I t can be seen t h a t i f the s p h e r e s i n the t h i r d l a y e r have t o l i e v e r t i c a l l y o v e r t h o s e i n the f i r s t , i t forms a v e r y u n s t a b l e arrangement i n a u n i d i r e c t i o n a l f o r c e f i e l d . 4 . T e t r a g o n a l P a c k i n g T h i s i s c o n s t r u c t e d by p l a c i n g s p h e r e s i n second l a y e r h o r i z o n t a l l y , o f f s e t w i t h r e s p e c t t o t h o s e o f the f i r s t l a y e r by a d i s t a n c e R, a l o n g the 70 • • * d i r e c t i o n of one of the sets of rows. A l l the layers are simple rhombic i n this case. 5. Rhombohedral Packing. As i n the case of orthorhombic packing, this can be constructed either from square-layer type base, or from simple rhombic- layer type base. But i n this case, the spheres i n second layer are h o r i z o n t a l l y o f f s e t with respect to those i n the f i r s t l ayer. This o f f s e t i s i n a d i r e c t i o n b i s e c t i n g the angle between two sets of rows by a distance of R/2~ i n the case of square layer formation, and of 2R//S i n the case of simple rhombic layer formation. These two ways of packing are i d e n t i c a l i n nature, though of d i f f e r e n t o r i e n t a t i o n i n space. Table III summarises the basic methods of simple and systematic packing of uniform spheres. TABLE III Basic Methods of Packing and th e i r Construction Method of Coordination Density Packing Number % Simple Cubic 6 52.36 Orthorhombic 8 60.46 Body-Centred-Cubic 8 68.02 Tetragonal 10 69.81 Rhombohedral 12 74.05 7 1 iff T 'Ĵ ^^^B ^ 4 (a) Cubic (c) Orthorhombic (from cubic base). (e) Rhombohedral (from cubic base; face-centred cubic). (g) Rhombohedral (from rhombic base; face-centred cubic). (b) Body-centered cubic (d) Orthorhombic (from rhombic base). (f) Tetragonal. (h) Rhombohedral (from rhombic base; close-packed hexagonal) Figure 3 7 . Basic systems of s p h e r i c a l packings (After Morgan ) 72 APPENDIX II A. Theoretical Calacuations The t h e o r e t i c a l values are calculated using equation(23)for d i f f e r e n t packing arrangements. m ( D 2 /3 B 2/3 R 2 _ i) (23) Equation(21) i s the same equation as above f or < 0.25. K m = a (21) Values of R and D for d i f f e r e n t — r a t i o s are read from Figure 18 R (which gives R values) and Figure 38 (which gives D values), respectively. (Ref. 26). R has a r b i t r a r y u nits. Values of a , 3, and D for d i f f e r e n t packings are given i n Table II (Chapter II)  1. Simple Cubic Packing 74 it R 0 0.05 0.10 0.15 0.20 0.25 0 .30 0 .35 0 .40 0 .45 0 .50 0.55 0 .60 0 .65 0 .70 R 0.620 0.620 0.620 0.620 0.620 0.621 0.622 0.624 0.627 0.631 0.637 0.646 0.660 0.679 0.710 /a 52.36 52.56 53.15 54.15 55.57 57.42 59.75 62.58 65.94 69.87 74.39 79.59 85.09 90.84 95.94 a m a c 0 0.002 0.008 0.018 0.033 0.052 0.078 0.110 0.150 0.199 0.262 0.341 0.442 0.575 0.755 75 2. Orthorhombic Packing R / O 0 0.620 60.46 0 0.15 0.620 62.51 0.021 0.20 0.620 64.12 0.038 0.25 0.622 66.21 0.061 0.30 0.623 68.77 0.090 0.35 0.625 71.82 0.127 0.40 0.629 75.34 0.173 0.45 0.635 79.27 0.230 0.5 0.643 83.51 0.302 a m a c 3. Rhombohedral Packing R D % _m a c 0 0.620 74.05 0 0.15 0.620 76.53 0.051 0.2 0.621 78.44 0.093 0.25 0.622 80.84 0.148 0.30 0.624 83.70 0.220 0.35 0.628 86.91 0.310 0.40 0.633 90.32 0.432 0.45 0.642 93.65 0.564 0.50 0.656 96.41 0.741 •a R 77 4. B.C.C. Packing •a R m 0 0.620 68.02 0 0.15 0.620 70.33 0.031 0.2 0.620 72.14 0.057 0.25 0.622 74.48 0.091 0.30 0.623 77.37 0.135 0.35 0.625 80.80 0.190 0.40 0.629 84.76 0.259 0.45 0.635 89.18 0.345 0.50 0.643 93.95 0.454 78 B. EXPERIMENTAL RESULTS 1. Glass at 550 C a = 0.70 mm average diameter; b = = 0.42 mm average diameter D = o 0.632 ; a = 1.27 X 10  2 ohm - 1 - Cm"1 CT X m io" 5 1 -1 cm D 1 n r-- 1 x 10 -2 ! m ohm- . ° - CT c a b a b a b a b 1.54 1.60 0.658 0.645 2.94 1. 58 0.012 0.013 2.76 3.57 0.661 0.650 3.25 2. 11 0.022 0.028 3.71 7.34 0.664 0.662 3.57 3. 36 0.029 0.058 5.80 9.31 0.670 0.671 4.19 4. 29 0.046 0.073 7.33 10.70 0.673 0.677 4.50 4. 91 0.058 0.084 8.67 11.88 0.679 0.687 5.12 5. 94 0.068 0.094 13.41 14.80 0.701 0.703 7.38 7. 58 0.106 0.117 14.09 15.89 0.712 0.707 8.50 7. 99 0.111 0.125 16.76 17.34 0.725 0.721 9.82 9. 41 0.132 0.137 18.04 21.47 0.733 0.75 10.06 12. 33 0.142 0.169 18.53 24.13 0.74 0.765 11.33 13.52 0.146 0.189 20.23 27.09 0.751 0.78 12.43 14. 31 0.16 0,213 I "1 2. Glass at 600 C D = o 0.64 a = 3. c 5 x 10" ohm ^-cm a x m ohm i o " 4 1 -1 cm D l j x 10~ 2 a m c c a b a b a b a b 0.40 0.19 0.657 0.644 1.71 0.46 0.011 0.005 1.13 0.81 0.660 0.661 2.07 2.21 0.031 0.023 2.56 1.76 0.678 0.67 3.92 3.1 0.071 0.049 3.03 2.20 0.687 0.675 4.84 3.56 0.084 0.062 3.81 3.09 0.692 0.690 5.35 5.16 0.106 0.087 4.79 4.81 0.724 0.728 8.57 8.97 0.133 0.136 5.44 5.28 0.737 0.738 9.87 9.91 0.151 0.15 80 3. Glass at 650°C D Q = 0 . 6 3 a c = 6 . 4 5 x 1 0 3 o h m " 1 - ™ " 1 a x 1 0 4 m ohm-1cm 1 2 / 3 - 1 x 1 0 - 2 m 2 . 4 3 3 . 6 5 4 . 8 5 6 . 4 2 7 . 7 1 8 . 6 5 9 . 2 6 1 0 . 7 5 b 1 . 6 2 3 . 4 7 5 . 9 5 6 . 7 2 7 . 7 4 8 . 3 4 1 2 . 3 5 1 2 . 6 1 0 . 6 6 7 0 . 6 7 6 0 . 6 8 4 0 . 6 9 5 0 . 7 1 4 0 . 7 2 4 0 . 7 3 1 0 . 7 5 2 b 0 . 6 6 5 0 . 6 7 8 0 . 6 9 1 0 . 7 0 0 0 . 7 1 2 0 . 7 2 2 0 . 7 7 8 0 . 7 9 3 . 8 8 4 . 8 1 5 . 6 4 6 . 7 7 8 . 7 0 9 . 7 2 1 0 . 4 2 1 2 . 5 3 b 3 . 6 7 5 . 0 2 6 . 3 6 7 . 2 8 8 . 5 0 9 . 5 1 1 5 . 1 1 1 6 . 2 9 0 . 0 3 7 0 . 0 5 6 0 . 0 7 4 0 . 0 9 8 0 . 1 1 8 0 . 1 3 2 0 . 1 4 1 0 . 1 6 4 b 0 . 0 2 6 0 . 0 5 5 0 . 0 9 5 0 . 1 0 7 0 . 1 2 3 0 . 1 3 3 0 . 1 9 6 0 . 2 0 0 81 4. Nickel at Room Temperature ac = 14.62 x 10 4 ohm-1 cm 1 a m -1 -1 ohm _cm ° 4 D x 10 cr 2.63 0.602 3.79 0.646 4.87 0.688 4.95 0.692 0.180 0.259 0.333 0.339 5.34 0.715 0.365 5 - 9 5 0.720 0.407 9 - 8 0.877 0.67 82 APPENDIX I I I THEORETICAL MODELS FOR DIFFERENT ORIENTATIONS OF THE UNIT CELL 1) Simple Cubic Packing Consider the current-flow along a face-diagonal (Figure 39a), Number of current-paths = 2 . 2 1 Area of current-flow = 2iTa x — = A„ ST 2 Area of the unit c e l l normal to current flow = 4 / 2 v a A m _ c V - A~ c s / 2 i r a 2 4/2 2 2 a 2 2 4 R - a 2) Orthorhombic Packing Consider the current-flow normal to a prism-face (Figure 39b) Number of current-paths through the unit c e l l = 3; one path l i e s completely in s i d e the unit c e l l and the other two are shared by three unit c e l l s a-piece. 83 84 Hence e f f e c t i v e area of current-flow through the unit c e l l 2 2 2 = Tra + — -na cos 0 , where cos 6; - ig,from geometry. 2 1 2 Hence A = ua + — ira c 3 4 2 - 3 Tra . A = ^ x 2y s /3 a A m _ c a ~ A c s 2 2 4 Tra TT a 3 W 2/3 R 2 - a 2 /3 3) Rhombohedral Packing Consider an F.C.C. unit c e l l . Let the current-flow be p a r a l l e l to the edge (Figure 39c). E f f e c t i v e number of current-paths = 8 . Area of current-flow i s 2 = 8Tra cos 6 , where cos Q = — }from geometry. Si A = 8y 2 s ' . 39 (c) 39 (d) Figure 39. (c & d).Geometric relationships for the u n i t - c e l l s i n d i f f e r e n t orientations with respect to current path Hence a A m c a - A c s a m Q 2 1 1 OTra x — x x.e. ~ ,- 2 a Si 8y c 2 a /2 R 2 - a 2 4) A l t e r n a t i v e Approach for H.C.P. Packing From Figure 39d, Number of current-paths = 9.. 2 A = 9ua cos G , where _ /2 ^ cos 6 = — , from geometry ; /3 2 and A = 6/3 y . s J • a A m c Q 2 /2 1 9Tra — a A /3 6/3 y 2 c s * a 2 /2 R 2 - a 2 87 A l t e r n a t i v e Approach for B.C.C. Packing Consider the current-flow along the edge (Figure 39e) Number of current-raaths = 4 . 2 A = 4ira x cos 0 , where c 1 _ cos 0 = — from geometry; /3 and A = -~y-s 3 a A 2 , m _ _ c _ 4-rra 3 a A /3 16y' c s ^ /3 a 2 = — Tf 2 2 R - a Z Consider current-flow along the face diagonal (Figure 39f), Number of paths through the unit c e l l = 2 2 /2 A^ = 2ira cos 0 , and cos 0 = — , from geometry. /3 _ 4y_ /2 2y • 8/2 y 2 S /3 /3 3 88 39 (e) 39 ( f ) F i g u r e 39. (e & f ) . G e o m e t r i c r e l a t i o n s h i p s f o r the u n i t - c e l l s i n d i f f e r e n t o r i e n t a t i o n s w i t h r e s p e c t t o c u r r e n t p a t h . A 9 2 /2 Z T r a x — /3 8/2 /3 a* — TT — 90 BIBLIOGRAPHY 1. " S i n t e r i n g and Related Phenomena". Eds. G. C. Kuczynski, N. A. Hooton,and C. F. Gibbon, (1965). 2. P. Murray, D. T. L i v e y , and J . W i l l i a m s , "Ceramic F a b r i c a t i o n Processes", Ed. W. D. Kingery, 147-170 (1963). 3. W. Trz e b i a t o w s k i , Z. Physik. 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Foote, and P. F. Busang, Phys. Rev., 3^, 1272 (1929). 33. A. D u f f i e l d and P. Grootenhuis, Special Report no. 58, pp.96, Symposium on Powder Metallurgy (Iron and Steel I n s t i t u t e ) , (1954). 34. A. K. Kakar and A. C. D. Chaklader, J . App. S e i . , 38(8), 3223 (1967). 35. V. T. Morgan, Special Report no. 58, pp. 86, Symposium on Powder Metallurgy,(Iron and Steel Institute),(1954).

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