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The surface tension of solid nickel 1957

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THE SURFACE TENSION OF SOLID NICKEL by EINO SAAREMAA A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE D E G K E E G E MASTER OF APPLIED SCIENCE i n the Department of Mining and Metallurgy at the University of B r i t i s h Columbia. 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' i n M e t a l l u r g i c a l Engineering Members of the Department of Mining and Metallurgy THE UNIVERSITY OF BRITISH COLUMBIA JANUARY, 1957. ABSTRACT The surface tension of s o l i d commercially pure nic k e l was determined by the force measurement technique using fine wires as proposed by Udin, Shaler, and Wulff. Grain boundary measurements were also made on the same metal. After finding experimentally that tests i n a vacuum -5 of approximately 5 x 10 y mm Hg were unsuccessful because of the high power vapour pressure of nickel at high temperatures, similar tests were made i n helium and argon atmospheres, the pressure being kept constant at 7&0 mm Hg during the experiments. The average surface tension of nick e l i n argon was found to be 2220 ± 300 dynes per centimeter f o r a temperature range from 1370°C to 1390°C. The r e l a t i v e grain boundary energy of s o l i d n i c k e l was determined by measuring the dihedral grain boundary groove angles of thermally etched n i c k e l . The interferometric method developed by H i l l i a r d and Harrold was used f o r t h i s purpose. An average value of l 6 l degrees was found f o r the dihedral angle. The grain boundary energy was calculated to be 7 4 0 ± 300 dynes per centimeter. Examination of thermally etched n i c k e l surfaces was inconclusive with respect to physical evidence f o r d i s l o c a t i o n . In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study. I further agree that permission f o r extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representative. I t i s under- stood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of M i n i n g and M e t a l l u r g y The University of B r i t i s h Columbia, Vancouver #, Canada. Date January 24th, 1957 ACKNOWLEDGMENT The author i s gra t e f u l to the National Research Council and Defence Research Board of Canada f o r f i n a n c i a l aid i n the form of a Research Assistantship granted during the past year. The work was carried out with the help of funds provided by the Defence Research Board of Canada. The experimental work was done i n the laboratories of the Department of Mining and Metallurgy, and the author wishes to thank Professor F.A. Forward, Head of the Depart- ment, fo r the f a c i l i t i e s and the assistance made available to him. Special thanks are extended to Dr. Vernon G r i f f i t h s , the d i r e c t o r of t h i s research, and to R.G. Butters f o r technical advice and encouragement. Thanks are extended to Professor A.M. Crooker, of the Physics Department, f o r providing the f a c i l i t i e s to prepare the p a r t i a l l y r e f l e c t i v e o p t i c a l f l a t s . TABLE OF CONTENTS Page I. INTRODUCTION . . . . . . . . . . . . . . . . . . -L I I . PREVIOUS WORK . . . . . . . . . . . . . . . . . . 3 Experimental Techniques . . . . » 3 Energy measurements . . . . . . . . . . . 3 Force measurements . . . . . . . . . . . 4 Grain boundary energy measurements . . . . 12 Theoretical Considerations. . . . . . . . . . 17 V i s c o s i t y IV Herring theory of d i f f u s i o n a l v i s c o s i t y . 19 Theories of surface tension of s o l i d s . . 22 I I I . EXPERIMENTAL . 23 Materials . . . . . . . • 23 Equipment 23 Procedures . . . . . . . . . . 23 D i f f i c u l t i e s 30 IV. RESULTS . . . . . . . . . . . . . . . . . . . . . 33 The Grain Boundary Energy of S o l i d Nickel . . 33 The Surface Tension of S o l i d Nickel 42 V. DISCUSSION . 48 Surface Energy Measurements . . . • 48 Thermal Etching . . . . . . . . . . 52 VI. CONCLUSIONS 58 VII. APPENDICES 60 A De f i n i t i o n s . . . . . dl B Nominal Composition of Commercially Pure Nickel of Type "A" Grade 63 TABLE OF CONTENTS-: (cont'd. ) Page 0 E l e c t r i c a l Power Supply, Thermocouple, and Vacuum Gauge C i r c u i t s . . . . . . 64 D Preparation of 50 Percent Transmitting L/Il 1 I* 01* k ^ Q o o w o O O O O O O O O O O O ^3 E Calculation of the Surface Tension of Sol i d N i c k e l . . . . . . . . . . . . 66 VIII. BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . 70 LIST OF ILLUSTRATIONS Fig. Page 1. The Stress Analysis of Loaded Wire . . . . . . . . 7 2. Surface Tension of Solid Copper as a Function of Temperature . . . , „ . . . . . . . . . . ./. 8 3. A Typical Experimental Stress-Strain Diagram . . . ? 4 . A Drawing showing the Marking of Reference Points on the Wire „ . . . , . . . . . . . . , 11 5 . Dihedral Groove Angle of a Thermally Etched Specimen vHcy-Mi,'. , . . . . . . . . . . . . . . 15 6. Vacuum furnace . . . . . . . . . . . . . . . . . . 24 7. Interior Arrangement of the Vacuum Furnace . . . . 26 8. Vacuum Furnace with Control Instruments . . . . . 33 9. Vacuum Furnace with Control Instruments . . . . . 33 10. Micrograph showing the Fizeau Fringes along the Grain Boundary of a Thermally Etched Nickel 11. Micrograph showing the Fizeau Fringes "'• . . 36 12. Micrograph showing the Fizeau Fringes. . . . . . . 37 13. Micrograph showing the Fizeau Fringes. . . . . . . 37 14. Elevation Profile No. 1 pf the Grain Boundary Groove Plotted from the Interferometric Data .,- 39 15. Micrograph showing the Mierosection of Grain Boundary Groove, of Thermally Etched Nickel 16. A Typical Experimental Stress-Strain Diagram. . > 45 17. Micrograph showing the 'Bamboo-like' Structure of Experimental Wires ... . . . , . 47 18. Thermally Etched Nickel Specimen showing a Sub-boundary. . . . . . . . . . . . . . . . . . . 54 19. Thermally Etched Nickel Specimen showing a Sub-boundary . . . . . ... '. . . .. , . , . . . 54- LIST OF ILLUSTRATIONS (cont'd.) F i g . 20. Thermally Etched Nickel 21. Thermally Etched Nickel 22. Thermally Etched Nickel 23. Thermally Etched Nickel 24. Thermally Etched Nickel 25. Thermally Etched Nickel 26. Thermally Etched Nickel 27. Thermally Etched Nickel Specimen . . . . . . . . . . 54 Specimen . . . . . . . . . . 54 Specimen . . . . . . . . . . 55 Specimen . . . . . . . . . . 55 Specimen 0 . . . . . . . . . . 55 Specimen . . . . . . . . . . . 55 Specimen ' . . . . . . . , . . „ 56 Specimen .• . . • . . . . , . . 56 LIST OF TABLES Table Page I. Horizontal Distances between the Fizeau Fringes Measured from the Experimental M i c r o g r a p h s . . 38 I I . Values of Dihedral Grain Boundary Groove Angle of Thermally Etched Commercially Pure Nickel. . • . . • 4-0 I I I . Experimental Stress and S t r a i n Measurements. . . . 43 IV. Experimental Values of the Surface Tension and Grain Boundary Energies of S o l i d Nickel. . 46 V. Calculated Values of the Surface Tension f o r S o l i d Nickel 50 VI. Calculated and Observed Values of Surface Tension f o r Some Metals 50 THE SURFACE TENSION QF SOLID NICKEL I. INTRODUCTION The present work on s o l i d n i c k e l was motivated by the growing need for numerical values of the surface tension of s o l i d s , as a re s u l t of increasing interest i n the role of interphase boundary phenomena i n m e t a l s . The existence of surface tension forces i n s o l i d s has been known f o r almost a century. Only very l i t t l e progress was made i n the study of t h i s phenomenon up to 1948 when C.S. Smith 1 extended to multiphase a l l o y s the idea of D. Harker and E.R. Parker that c r y s t a l growth i n heated metals i s due to grain boundary forces. Smith also showed the nature of the dependence of c r y s t a l l i t e shape on i n t e r - f a c i a l tensions between s o l i d i f i e d c r y s t a l s and s t i l l l i q u i d a l l o y . Since some of the mechanical properties of engineering al l o y s are structure se n s i t i v e , many metallurgists were alerted by t h i s paper. Hollomon and his co-workers r e a l i z e d that the phase 7 4 nucleation theory of Volmer, extended by Becker to s o l i d - state transformation, required some data on s o l i d surface tension to activate i t to p r a c t i c a l use. In the study of s i n t e r i n g of metal powders i t was found that a more basic understanding of the operating 5 mechanisms was needed fo r further development. Shaler showed that an exact value of surface tension of s o l i d metals i s necessary for solution of the k i n e t i c s of s i n t e r i n g , because t h i s force plays a p r i n c i p a l role i n d e n s i f i c a t i o n and strengthening of powder compacts« Welding metallurgists realized that surface forces determine the soundness of welded or brazed j o i n t s . G-.L.C. Bailey and H.C. Watkins were led to a study of the surface forces i n metals by t h e i r i n t e r e s t i n the joining of metals. Besides the metallurgists, chemical engineers interested i n c a t a l y s i s , mechanical engineers working on l u b r i c a t i o n problems had shown appreciation of the importance of the surface forces i n metals, but had been unable to determine experimentally the s o l i d : l i q u i d and solid:gas i n t e r - f a c i a l tensions. The surface forces ( d e f i n i t i o n s and terms used are given i n Appendix A) are a d i r e c t r e s u l t of chemical free energy. The i n t e r f a c i a l atoms are at energy states between those of i n t e r i o r atoms and vaporized atoms. The difference between the energy of surface atoms and i n t e r n a l atoms i s i n terms of the energy required to form a unit area of the surface, and i s c a l l e d surface energy. This excess energy i n the surface provides the d r i v i n g force f o r any process which may induce a decrease i n surface area. Such a process i s capable of doing work at the expense of the energy of decreasing surface. G-ibbs^ has proved that t h i s excess energy at the interface between two phases i s a p a r t i a l function of the - 3 „ i n t e r f a c i a l area, thus ^ Or = ^ OF ^ S " where - ^ j i s the rate of change of free energy of the system with changing surface area at constant temperature, pressure, and composition. V i s the i n t e r f a c i a l force per unit length or free energy per unit area, being expressed either i n dynes/cm or i n orgs/cm. Therefore, two approaches are possible to measure the surface energy of the s o l i d s . The surface tension may be deter- mined either by thermodynamic measurement of the surface energy or by a mechanical measurement of the surface force by the ap p l i c a t i o n of a balancing counterforce. At present the values of the surface forces of Cu, Au and Ag, a l l being metals of r e l a t i v e l y low melting points, nave been determined by the use of the second method. The present work to determine the surface tension of s o l i d n i c k e l i s an attempt to use the force measurement technique f i r s t developed by Udin and h i s collaborators, on a metal of higher melting point. II, PREVIOUS WORK Experimental Techniques Energy measurements. Thermodynamical measurements to determine surface energy have been carried out by the a p p l i c a t i o n of heat-of- solution method, Fricke and Meyer^ measured the heat of solution of massive gold i n iodine t r i c h l o r i d e , and found i t to be lower than that of f i n e gold p a r t i c l e s . Surface area was evaluated on the basis of cubic p a r t i c l e s , the diameter of the p a r t i c l e s being determined from the amount of broadening of the x-ray d i f f r a c t i o n l i n e s of the gold powder. This area and the excess heat of solution yielded a surface energy equal to 670 ergs per square centimeter. o H i i t t i g and his collaborators determined the i n t e r - f a c i a l tension copper:copper sulphate to be 40,000 ergs per square centimeter by measuring the electromotive force between electrodes of copper powder and massive copper respectively, i n a copper sulphate solution. Since copper tends to dissolve from the powder electrode and p r e c i p i t a t e on the massive metal, an equilibrium emf. due to the excess solu t i o n tension of p a r t i c l e s of radius r, and with an i n t e r f a c i a l tension y against the el e c t r o l y t e , can be measured. v frf whereto" i s the r a t i o of atomic weight to density of the p a r t i c l e F i s Faraday's constant. Certain errors are introduced i n estimating the surface area or p a r t i c l e size, and i n using the electrochemical method, due to the side reactions and non-equilibrium conditions, Force measurements The techniques f o r determining the force of surface tension have been more favored than those involving measurement of energy, because of fewer experimental d i f f i c u l t i e s and uncertainties. These methods involve the balancing of surface tension with a counterforce. Thin wires or f o i l s are loaded with weights of varying magnitude and heated to.a temperature below the melting point. The straixis (positive i f due to the counter- force, and negative i f due to the surface tension) are observed. The load which exactly balances the upward p u l l of the surface tension i s estimated from the plot of s t r a i n against stress. The interpolated load allows the determination of the value pf the surface tension i n a simple c a l c u l a t i o n . This technique has been known for many years, and surface tension values have been reported by Chapman and Porter,^ S c h o t t k y 1 0 and Berggren 1 1 i n the e a r l i e r part of the century. The filament technique was l a t e r used successfully to determine the 12 I J surface tension of viscous l i q u i d s . Tammann and co-workers, ' Sawai and c o l l a b o r a t o r s , 1 ^ ' 1 ^ and Mack^ reported good agreement between t h i s technique and more conventional methods of measuring the surface tension of glasses and t a r s . A l l these experimental results supported the b e l i e f i n r e l i a b i l i t y of the technique and soon i t was employed with success on s o l i d metals. Sawai and Nishida, ' and Tammann and 20 Boehme made quantitative measurements on gold f o i l s and attempted to calculate from t h e i r r e s u l t s the i n t e r f a c i a l tension of the non-equilibrium system gold:air. In experiments f o r determining the surface tension, loaded wires rather than loaded f o i l s may be used, although the s e n s i t i v t y of the t e s t i s somewhat., diminished due to the. unfavorable surface area to;, cross-sectional area r a t i o of t h i n wires. For example,. Sawai^s gold f o i l s being ' 7 . 7 x 10~^ centimeters thick.,,-.,,were subjected to a shrinkage stress of 19 kg. per square centimeter at a surface tension of 1500 dynes per centimeter, whereas i n 0,008-centimeter radius wire the stress i s only 0.57 kg. per square centimeter. Hence, the use of wires i n the experiments requires an operation very near the melting point f o r a reasonable duration. The i n s e n s i t i v i t y i n using wires i s counterbalanced by the following considerations. The c y l i n d r i c a l wire, i n equilibrium, i s i n a state of hydrostatic compression, the s t r a i n rate i n a l l directions being zero, which i s not true i n the case of f o i l s . Secondly, the grain boundary energy, undoubtedly one of the components of the force acting on the specimens, can be dealt with i n case of wires, i f the majority of. grain boundaries are perpendicular to the wire axis. F i n a l l y , s t r a i n anisotropy i s n e g l i g i b l e i n the case of wires as compared to f o i l s . Consider a wire of length 1 and radius r containing n + 1 grains with a cross-section equal to that of the wire with boundaries perpendicular to the axis of the wire (Fig, 1 ) . I f the shrinkage force of the surface tension i s just counterbalanced by a. suspended weight W , then the a x i a l stress (l) 'x - T i r 1 where if i s the surface t e n s i o n of the wire, the x-axis being along the wire with t e n s i o n p o s i t i v e . F i g . 1.. The S t r e s s Analysis of Loaded Wire The stresses perpendicular to the wire are due to the circumferential force of the surface tension and the l i n e forces due to each grain boundary. These forces are equal and can be expressed by where fc^is the grain boundary tension. - 8 - ' For the case of zero s t r a i n i n the x-direction, the s t r a i n w i l l also be zero i n the y- and z-directions, since the wire i s under hydrostatic stress. Hence ĜL =. (T̂  = and thus w = nc^ - 1 - n c V .... ^ \ From t h i s equation the surface tension can be. determined i n a single experiment i f the grain boundary tension i s known, or i f the r a t i o of grain boundary tension to surface tension i s a v a i l a b l e . 21 Udin, Shaler, and Wulff f i r s t applied t h i s technique to evaluate the surface tension of copper i n vacuum. Their value f o r the surface tension of s o l i d copper was 1670 dynes per 0 ' 22 centimeter at 1030 G, as corrected by Udin according to the Equation 3.. The res u l t s are summarized graphically i n F i g . 2, and a t y p i c a l s t r e s s - s t r a i n diagram i s shown i n F i g . 3. •i_ 2000 ! 1 . 1 'll 1 | — — _ _ _ _ _ A 1 O ____A 1 L egend A - 0.005" C io. Wire C 1 O - 0003" 0 io. Wire O 1 0_ Copper; I0/4E 3 . Cor reeled 10/50 — M el ri i 1200 I'600 T e m o e r o i u r e 0 K Fig. 2 Surface Tension of S o l i d Copper as a Function of Temperature (after H . UdinJ. - 9 - 2 3 Funk, Udin, and Wulff J experimented on s i l v e r i n a helium atmosphere ,• a f t er they found that the vapor pressure of s i l v e r was too hi'gh to a t t a i n the metal:vapour equilibrium i n th e i r experiments i n vacuum. The surface tension of s i l v e r i n helium was found to be 1 1 4 0 i 9 0 dynes per centimeter f o r the temperature range from 875°C to 952°C, which i s a true surface tension, because helium i s insoluble i n s i l v e r and w i l l not be phys i c a l l y adsorbed at high temperatures, and cannot chemisorb to a metal surface. 0.006 0.004 0.002 c S 0 fl -0.002 -0.004 -0.006 1 1 1 1 - X - X / - .—x - - 1 Test No. 13 i i 1 2 3 Siress, Dynes per Cm.1 5 « I 0 : F i g . 3. A Typical Experimental Stress-Strain Diagram (after H. Udin), 24 Alexander, Kuczynski and Dawson who worked with gold wires suspended i n tubular n i c k e l heaters i n vacuum, had sim i l a r d i f f i c u l t i e s . Since a complex gold:gpld vapor.nickel vapor system with no p o s s i b i l i t y of gold.gold vapor equilibrium was used, no reproducible values f o r the i n t e r f a c i a l tension of the system were obtained. - 10 - 25 Buttner, Udin and Wulff experimented on gold wires i n helium atmosphere to determine the surface tension of s o l i d gold. They determined a value of 1400-65 dynes per centimeter fo r the temperature range from-1017°C°to 1042°C. 26 Buttner, Funk ^nd Udin studied the effect of oxygen on surface tension. They found the i n t e r f a c i a l tension of the system gold:gold vapor:air at 1040°C to te 1 2 1 0 dynes per centimeter as compared to 1 3 7 0 dynes per centimeter f o r the system gold:gold vapor at the same temperature; and the i n t e r - f a c i a l tension of the system s i l v e r : s i l v e r vapor:air at 930°C only 450 dynes per centimeter as compared to 1140. dynes per centimeter for the surface tension of s i l v e r . Since the experimentally determined values are accurate only to about ±5%i no further experiments were carried out i n the case of gold. The e f f e c t of oxygen on surface tension of s i l v e r was studied more f u l l y . Tests were run i n various mixtures of p u r i f i e d helium plus dried e l e c t r o l y t i c oxygen. The r e s u l t s showed a l i n e a r relationship between surface tension and the logarithm of oxygen p a r t i a l pressure. I t was concluded that s i l v e r w i l l adsorb approximately 1 . 7 x l O 1 ^ atoms of oxygen per square centimeter of surface, the amount not being dependent on pressure i n the range investigated, and form a surface- s t a b i l i z e d f i l m of oxide. The experimental technique i s a d e l i c a t e one. The marking of reference points on the wire presents a d i f f i c u l t problem. Painting or p l a t i n g the wires outside a given gage length i s not satisfactory because anything adhering to the - 11 - surface lowers the surface energy. Knotting the wires at two points was used by Udin and co-workers i n the case of copper. The gage points are taken as i n t e r s e c t i o n of the inner loops of the knot with the straight portion of the wire (Fig.4). Later Udin and his group inscribed the marks by rotating the wire mounted i n a jeweler's lathe against a pair of razor blades spaced the required gage length apart. *1 F i g . 4 A Drawing Showing the Marking of Reference Points on the Wire. . The weights, ranging from zero to about two to three times the estimated weight required f o r balance, are fused to the ends of the gage-marked wires. Eight to sixteen wires used - 12 - i n each test are suspended from the l i d of the metal c e l l . The weights, l i d , and the box are made of the same metal used f o r the specimen. Thus at temperature the wires hang i n an equilibrium with metal vapor atmosphere. The wires are straightened and annealed at or above the experimental temperature f o r at le a s t one hour i n the vacuum or test atmosphere. A f t e r cooling, the specimens are removed to measure the gage length and count the grains. The assembly i s returned to the furnace and held at the test temperature f o r a time s u f f i c i e n t f o r measurable creep to take place, the time being dependent on the temperature and accuracy required. The furnace i s cooled to room temperature, the specimens are removed and re-measured. I t ha$ been found that no further grain growth takes place during the test because the thermally etched grooves produced during the anneal anchor the grain boundaries. The change i n length i s transformed to engineering s t r a i n and plotted against the stress due to the suspended weights, which^ are corrected by taking into account the weight of the wire below the point midway between the upper and lower gage marks. The intercept of t h i s plot with the .strain axis gives the value of the stress at zero s t r a i n or the value of w from Equation 3, Grain boundary energy measurements. In order to calculate the surface tension f o r the given metal, either the value of the grain boundary energy must be known, or the r a t i o between and assumed. Up to 194-8 no s a t i s f a c t o r y methods to determine the i n t e r f a c i a l energies, such as grain boundary energy, were ava i l a b l e . A new attack to the problem was proposed by G.S. Smith"1" who showed that the geometrical shapes of grains result from an approach to an equilibrium between phase and grain interfaces whose surface tensions geometrically balance each other at the points and along the l i n e s of contact when annealed at s u f f i c i e n t l y high temperatures fo r a s u f f i c i e n t length of time. This relationship between grain boundary tensions and grain 27 boundary geometry had been suggested e a r l i e r by Desch. Measure- ment of angles established between the interfaces when i n equilibrium, provides a method to determine the r e l a t i v e values of surface forces involved according to the r e l a t i o n s h i p JL = » J a (4) Sinvy, S i ^ a . 5 inv^ s Where E fs are the respective s p e c i f i c i n t e r f a c i a l energies, and y ' 3 a r e "the dihedral angles between the i n t e r - faces measured i n the plane perpendicular to the junction. However, t h i s relationship w i l l be complicated by the orienta- t i o n dependence of grain boundary energies. The evidence available concerning the orientation dependence of s o l i d i n t e r - face energies i s sparse but i t seems that both experimentally and t h e o r e t i c a l l y i t s e f f e c t may be expected to be small, except f o r a l i m i t e d number of cases when grain boundaries have cusp orientations. Physical measurement of microscopic dihedral angles - 14 - i s d i f f i c u l t because, by d e f i n i t i o n , they must be measured i n a plane perpendicular to the l i n e of junction of the interfaces, 2 which actual d i r e c t i o n i s unknown. Harker and Parker proposed a s t a t i s t i c a l method which, i f employed, would give the value of dihedral angle within the l i m i t s of ~5° when a large number of dihedral angles were measured at random. This method has been used i n some cases with c e r t a i n modifications by d i f f e r e n t researchers, A more subtle method i s the c a l i b r a t i o n of s o l i d : s o l i d i n t e r f a c i a l tension i n absolute units when the energies of such s o l i d : l i q u i d or solid:gas surfaces are known or can be determined by the methods described above. C a l i b r a t i o n with solid:gas tension i s preferred to the c a l i b r a t i o n with a s o l i d : l i q u i d tension because an i n e r t gas or vacuum can be used to avoid contamination of the interface under study. During the e q u i l i b r a t i n g heat treatment, shallow grooves on the free surface of the specimen at the grain boundaries are formed, the process being c a l l e d 'thermal etching'. I f the dihedral or groove angles ^ are measured and the surface tension of the s o l i d i n the gas i s known, the grain boundary energy can be calculated from the formula (Fig. J5) Ztfcos^ - (5) Or, subsequently, i f the surface tension and grain boundary tension are determined simultaneously, the two unknowns _? and tf* may be calculated from - 15 - . 2-*> cos | where a l l other values can be measured, (6) F i g . 5 Dihedral Groove Angle of a Thermally Etched Specimen. Because of the small size of the grooves (approx, 1 micron deep) and the obtuseness of the groove angle (usually 160 degrees or greater), a small experimental error i n measur- ing the angle produces a large r e l a t i v e error i n the calculated grain boundary energy, the experimental technique i s d i f f i c u l t . 28 Bailey and Watkins measured the groove angles of thermally etched copper i n microsections taken normal to the 29 surface,, Buttner, Udin and Wulff determined the absolute grain boundary energy of gold at 1300°K to be 365±j?0 dynes per centimeter by measuring dihedral angles, Greenough and King-^ used both microsections and taper-sections to measure dihedral groove angles i n s i l v e r , and checked t h e i r r e s u l t s successfully by a method of o p t i c a l gpniometry. The l a t t e r method, however, i s not s u f f i c i e n t l y accurate because of the very small size of the groove and continuous curvature of the groove surfaces and 31 of the magnitude of the wave length of l i g h t . Fullman determined the groove angle f o r copper coherent twin boundaries thermally etched i n lead vapor and demonstrated that there i s an optimum taper-sectioning angle for most accurate measurement of a p a r t i c u l a r groove. 32 33 Hess and others used the method of multiple-beam 34 interferometry developed by Tolansky to measure the groove angles of pure copper but found that large errors i n the calcu -r lated grain boundary energies r e s u l t , i f the reference plate and the surface deviate from p a r a l l e l i s m more than approximately 2 degrees. On the other hand, H i l l i a r d ^ has demonstrated that the interferometric method i s s u f f i c i e n t l y accurate, i f the reference plate and the experimental surface are p a r a l l e l , by measuring grain groove angles i n the Cu-Au system. This method consists i n matching an aluminized or si l v e r e d , o p t i c a l l y f l a t , glass surface of r e f l e c t i v i t y approximately 0,50 against the metal surface, and the use of monochromatic l i g h t i n the formation of Fizeau fringes which represent the contour l i n e s i n the surface topography. Knowing the necessary constants, the elevation p r o f i l e of the surface slope i n the grain boundary may be constructed and the dihedral angle measured. The equation governing fringes of minimuna r e f l e c t i o n from the i n t e r f erometric gap , i s e-n ^~ -*- £ = U M + I)TT (7) according to Harrold^ who used glass s l i p s coated with multiple d i e l e c t r i c f i l m s . When the objective i s focused i n the a i r gap between the r e f l e c t i v e surface of the o p t i c a l f l a t , the metal specimen reveals contour l i n e s at i n t e r v a l s of as S varies with i r r e g u l a r i t y of the metal surface and assumes successive i n t e g r a l values. £ , the sum of a l l relevant phase changes at r e f l e c t i o n , i s fixed i n th i s case. No c r i t i c a l evaluation f o r the interferometric method developed by H i l l i a r d and Harrold i s avail a b l e at the present. However, present work has demonstrated convincingly the a p p l i c a b i l i t y and the r e l a t i v e s i m p l i c i t y of the method with respect to other conventional techniques. This method makes i t possible to pick the grain boundary grooves suitable f o r the measurement of dihedral angles by observing the symmetry of the Fizeau fringes. No time-consuming work i s required as i n the case of microsections, or determining the dihedral angle by the use of s t a t i s t i c a l methods. Furthermore, the accuracy of the method i s believed to be higher than that of the other methods because of the fewer experimental d i f f i c u l t i e s and uncertainties involved. Theoretical Considerations. V i s c o s i t y Knowing the surface tension of s o l i d metals, i t i s possible to calculate t h e i r v i s c o s i t y , i f c e r t a i n assumptions are made. It i s assumed that the material contracts or extends uniformly along the length of the specimen, and also that i t flows i n a viscous fashion, i . e . that the s t r a i n rates are - 18 - proportional to the stress applied. I f viscous flow i s assumed, no change i n l a t t i c e energy should be present, and a l l the s t r a i n energy should appear as heat. The k i n e t i c energy of the moving weight being neglected, the time rate of heat generation equals the rate of change of pot e n t i a l energy by changing both the p o s i t i o n of the weight and the area of the metal surface. Therefore, under isothermal conditions, . , cLt v ds TJt - w d£ * "3* (8) where Q i s the heat of viscous flow. 37 According to Frenkel the.energy dissipated i n flow f o r a viscous rod under longitudinal s t r a i n i s dl ~~ jb* ^7jb> (?) where V) i s the c o e f f i c i e n t of v i s c o s i t y By making necessary substitutions, a s i m p l i f i e d relationship € - £ c c - f v (io, i s derived, i f the strains measured are small, <T* being stress. Udin has calculated the c o e f f i c i e n t of v i s c o s i t y from his experimental data on surface tension of copper using equation 10, and plotted the logarithm of the v i s c o s i t y against the r e c i p r o c a l of absolute temperature. The equation of the r e s u l t i n g l i n e i s OOQ 7 = Bo e, *r ( l l ) The a c t i v a t i o n energy of 59,000 c a l o r i e s per mole i s within the - 19 - range of values reported f o r s e l f - d i f f u s i o n of copper by Steigman,^ However, the constant i s 10^ times larger than 38 that predicted by Kauzmann according to the equation = V *GT ( } Udin concluded t h a t an atomic vacancy i s the unit of flow, but only a very small f r a c t i o n of vacancies can p a r t i c i p a t e i n the flow. Herring theory of d i f f u s i o n a l v i s c o s i t y . Since the correctness of the reported values for surface tension of various metals and the future experimental work i s based on the assumption that the specimens deform i n a viscous fashion, i t i s necessary that a mechanism be established that would explain both the viscous flow and the uniform deforma- 40 t i o n . Such a mechanism has been proposed by Herring who explains the deformation taking place under experimental conditions by means of a flow of vacancies between grain bound- aries and surfaces. This theory, which i s a d i r e c t but 41 independent extension of the theory put forward by Nabarro i n an attempt to explain the microcreep observed by Chalmers i n single t i n c r y s t a l s , suggests that any c r y s t a l can change i t s shape by s e l f - d i f f u s i o n i n such a way as to y i e l d to an applied shearing stress. This can cause the macroscopic behaviour of a p o l y c r y s t a l l i n e s o l i d to be l i k e that of a viscous l i q u i d . It is- assumed that t h i s phenomenon i s the predominant cause of creep at very high temperatures and very low stresses, though not under normal conditions. It i s pointed out that a poly- c r y s t a l l i n e s o l i d under a shearing stress, can, because of - 20 - s e l f r - d i f f u s i o n within the grains, y i e l d as a r e s u l t of d i f f u s i o n a l flow of matter within each c r y s t a l away from grain boundaries where there i s a normal pressure, and towards those where there i s a normal tension. The y i e l d i n g i s macroscopic- a l l y describable by an e f f e c t i v e v i s c o s i t y proportional to the square of the l i n e a r dimensions of the grains. This theory of so-called ' d i f f u s i o n a l viscosity'-provides a possible explana- t i o n f o r the behaviour which has been observed f o r f o i l s and wires being suspended with very small loads and held at an elevated temperature. The rate of y i e l d i n g of the specimen to the applied forces understandably depends on the detailed d i s t r i b u t i o n of the sizes and shapes of i t s c r y s t a l grains, and on whether or not the grain boundaries are able to withstand shearing stress f o r times as long as are necessary f o r measurable d i f f u s i o n a l flow to take place. I t has been found that shearing stresses across metallic grain boundaries are r a p i d l y relaxed at high temperatures, which seems to be a general property of grain boundaries. The rate of creep, u n t i l t h i s r e l a x a t i o n has become complete, w i l l usually be considerably f a s t e r than that due to d i f f u s i o n a l v i s c o s i t y . Compared to the stresses i n ordinary creep experiments, the low stresses used i n experiments to determine the surface tension of metals indicate that the mechanism proposed by Nabarro and Herring to explain the deformation of the specimen i s most probable, although i t i s , of course, quite conceivable that the mechanism of ordinary creep and microcreep are the - 21 - same, the threshold being lower and the rates f a s t e r at the higher temperatures. Herring's v i s c o s i t y equation f o r a wire with 'bamboo- l i k e ' structure i s 2 kTRL . . where T i s the absolute temperature R i s the radius of the grain L i s the length of the grain D i s the s e l f - d i f f u s i o n c o e f f i c i e n t B i s a function of grain shape L R SI i s atomic volume The theory was v e r i f i e d by the work of Udin and 42 others who determined that the v i s c o s i t y of 3 mil gold wires i s much higher than that of 1 mil wires. G-ood agreement with 43 theory was also found by Greenough who observed s t r a i n rates i n s i l v e r single c r y s t a l s . Opposed to these observations were 21 the r e s u l t s of Udin and his collaborators on s o l i d copper i n which case the v i s c o s i t y was found to decrease as grain size increased. A series of experiments were ca r r i e d out by Pranatis 44 and Pound with copper f o i l s of varying grain s i z e to confirm the Herring theory of d i f f u s i o n a l v i s c o s i t y . A good agreement between, calculated and observed values of the c o e f f i c i e n t of v i s c o s i t y , a c t i v a t i o n energy, and s e l f - d i f f u s i o n c o e f f i c i e n t was found. This additional evidence i n favor of Herring's theory of viscous flow indicates that the values of surface tension obtained by force measurement techniques may be regarded with confidence. The relationships determined between v i s c o s i t y and grain si z e , and v i s c o s i t y and temperature, as well as the values of observed v i s c o s i t i e s , strongly support the proposal that deformation under the experimental conditions develops almost e n t i r e l y by means of vacancy diffusion.. S l i p , kinking, o f f - s e t t i n g , and grain boundary s l i d i n g are assumed to make only l i m i t e d contributions. Theories of surface tension of solids Not only are the available data concerning the numerical values of the surface tension of so l i d s incomplete, but i n addition, no sa t i s f a c t o r y theory of the surface tension 45 of nonionic s o l i d s i s available at present. Stratton's electron theory of surface tension of s o l i d metals i s one of the best of the existing ones, but i t s predictions are consider- ably lower than the experimentally known surface tensions of 'l i q u i d metals, and hence has to be considered at least i n i t s quantitative predictions inadequate. Lately a new theory of surface tension of s o l i d s based on the elementary next-neighbor approach has been 4 6 presented by Skapski. This theory allows one to calcu l a t e the surface tension of non-ionic s o l i d s from the arrangement of next neighbors, from the heat of fusion, and from the surface tension of the l i q u i d substance at the melting-point. Comparison of the t h e o r e t i c a l values with experimental data obtained from f o i l - or wire-stretching experiments has given good agreement. I l l , EXPEEIMEOTAL Materials Commercially pure nickel of Type "A" grade was supplied by Johnson and Matthey Company Ltd,, London, England. This was i n the form of 1/2''* round bar for grain boundary energy measure- ments and 1/8'* wire f o r surface tension measurements. The analysis for t h i s material i s given i n Appendix B. In the l a t t e r stages of the experimental work, nickel wire of 0.010 inch size manufactured by Hoskins Alloys Canada Ltd., Toronto, became available. Equipment The vacuum furnace (Fig.6 ) used i n the experimental work was of mild s t e e l , with a diameter of 8 inches and a height of 20 inches, inside dimensions. It was evacuated by'a 275 l i t e r / s e c o i l d i f f u s i o n pump backed by a 140 l i t e r / m i n mechanical pump. Pressure was measured by a thermocouple vacuum gauge, and by an i o n i z a t i o n gauge. The vacuum measuring unit allowed _7 an e f f e c t i v e range from 1 mm to 10 ' mm Hg, and pressures of _5 2*10 - mm Hg were r e a d i l y obtained. One platinum thermocouple, used for c o n t r o l l i n g the temperature, and the power leads were inserted through the furnace arm which was s p e c i a l l y designed f o r that purpose. The other thermocouple was introduced into the furnace through the bottom plate of the furnace. The c o n t r o l l i n g thermocouple was connected to a Honeywell recorder-controller and the emf. of the - 24 - TAP 3 PEEP fe HOLES OKI TOP 12 HOLES IN BOTTOM J DRILL F i e 6 Vacuum Furnace other thermocouple was measured with a Tinsley potentiometer,, A heating c o i l of molybdenum 0 . 0 2 5 inch wire, wound on a grooved alundum tube 1 2 inches long and of 1 ^ inch diameter, supplied the furnace with power. Heat losses by ra d i a t i o n f rom the heated zone were decreased by the use of three concentric r a d i a t i o n shields made of molybdenum sheet. Figure 7 shows the i n t e r i o r arrangement of the furnace. A complete drawing of the e l e c t r i c a l power supply, thermocouple, and vacuum gauge c i r c u i t s i s given i n Appendix C. Procedures. Nickel wires of 0 , 0 1 0 inch diameter were prepared by drawing the 1 / 8 inch wire with jeweler's drawing plates. I t was found necessary to anneal the cold worked wires quite often. The best lubricant f o r the work was t h i n l u b r i c a t i n g o i l f o r s c i e n t i f i c instruments. Afte r every i n d i v i d u a l step of reduction, the wires were cleaned thoroughly with carbon t e t r a - chloride, and annealed i n hydrogen at about 1 2 0 0 ° G f o r approx- imately 3 0 minutes. Steel boats f i l l e d with f i n e alundum powder were used i n t h i s operation. The process was repeated u n t i l the minimum size of the wire was established. The Udin technique of force measurement was used. K Nickel wires of 0 . 0 1 0 inch diameter were cold worked about J>% by stretching and then annealed to grow s u f f i c i e n t l y large grains. The most e f f e c t i v e range of cold working was obtained - 26 - fONU/tt lON 4 A 0 E Fig. 7 Interior Arrangement of the Vacuum Furnace A - Radiation shells R - Refractory S - Specimen W - Winding tube T.C. - Thermocouple - 27 - experimentally by stretching wires at d i f f e r e n t rates of s t r a i n and determining microscopically the e f f e c t s . The wires, cut to suitable lengths, were knotted at two points leaving the approximately predetermined gage length between them. Great care was taken i n making the knots i n order to minimize cold working of the wire between the knots. A load of known weight was attached to each wire. The weight, ranging from zero tq about two to three times the estimated weight required f o r balance, had a hole d r i l l e d i n i t s center. The wire was pushed through the hole and then secured by making another knot below the load. The other end of the loaded wire was pushed through the hole i n the top of the ni c k e l box, and wedged there by a small n i c k e l plug. After seven or eight wires were mounted to the top of the box, the top was attached t i g h t l y to the box, a f t e r making sure that the wires and the weights touched neither each other nor the walls of the box. The box together with the top plus wires and the removable nic k e l bottom was introduced into the furnace using a small electromagnet. The wires were annealed and straightened at or above _5 the experimental temperature i n a vacuum of 5<LQ y mm Hg f o r approximately one hour. The furnace was cooled to room temperature, a process requiring about twelve hours. A f t e r the n i c k e l box was removed from the furnace with the electromagnet, the gage lengths of the wires, s t i l l attached to the top of the box, were measured with the t r a v e l l i n g horizontal microscope, and the grains were counted. The box was reassembled, returned - 2 8 - to the. furnace, and brought to the experimental temperature f o r a time s u f f i c i e n t l y long for a measurable creep to take place, t h i s time being longer the lower the temperature. The furnace was cooled again to room temperature, specimens were removed and re-measured. The change i n length was converted to engineering s t r a i n , and plotted against stress a r i s i n g from the suspended weights. To measure the g r a i n boundary energy of s o l i d n i c k e l , the following work had to be carr i e d out. Glass proof plates of 1 mm thickness were tested against a standard o p t i c a l f l a t , and the s a t i s f a c t o r y ones were picked out. These glasses were cleaned thoroughly i n n i t r i c acid and washed with d i s t i l l e d water and soap. They were dried c a r e f u l l y and mounted f i v e at a time on a s p e c i a l l y prepared support. Aluminum was evaporated i n a suitable apparatus. The determination of the thickness of the f i l m f o r f i f t y percent transmission and the weight of the coating material are shown i n Appendix D. A complete des c r i p t i o n of the apparatus and the 47 procedure used i n s i l v e r i n g the mirrors i s given by Strong. For t h i s work the apparatus from the Physics Department of the University of B r i t i s h Columbia was kindly made ava i l a b l e . Aluminum was used as a coating material since s i l v e r was considered to be unsatisfactory because of i t s tendency to form sulphides; the multiple d i e l e c t r i c films proposed by H a r r o l d ^ were not used because of the experimental d i f f i c u l t i e s i n preparing them. Optical f l a t s were produced with the following r e f l e c t i v i t i e s : 20%, 30%, 40%, 50%, and 70%, - 29 - Pieces approximately 1/2 inch long were cut from a commercially pure nickel bar of 1/2 inch diameter, cold worked three percent by compression and polished according t o the best conventional methods. These specimens were annealed i n vacuum for a few days i n the temperature range from 1200°G to 1300°C, The p r i o r cold work of the specimens allowed the g r a i n s t o grow to a s u f f i c i e n t l y large size, and to reach thermodynamical equilibrium at the experimental temperature dur i n g the time allowed. Afte r the furnace was cooled to room, temperature, the specimens were removed f o r microscopic observation of the thermally etched grain boundaries. The contour fringes of the grain boundary grooves were v i s i b l e i n monochromatic l i g h t , when the thermally etched metal surface was observed through the p a r t i a l l y r e f l e c t i n g o p t i c a l f l a t with i t s coated surface against the specimen, provided that the reference plate and the surface of the specimen were p a r a l l e l . The mercury green l i n e of wave length 5461 Angstroms used was obtained from an interference f i l t e r made by Barr and Stroud. The objective, focused i n the a i r gap, revealed the contour l i n e s at i n t e r v a l s of A/2 as the interferometric gap varied with the i r r e g u l a r i t y of the metal surface. The f i e l d of view was photographed to study the surface p r o f i l e s obtained from the fringes of equal chromatic order. By the a p p l i c a t i o n of experimental data i n the c a l c u l a t i o n s , the p r o f i l e of the gr a i n boundary groove was plotted, and the dihedral angle measured. A number of micrographs were taken to record some of the i n t e r e s t i n g phenomena i n the structure of the thermally r 30 - etched n i c k e l . D i f f i c u l t i e s . Temperature measurement and control. Temperature measurement inside the experimental zone, and temperature control were subject to considerable d i f f i c u l t i e s . In the f i r s t test when the platinum thermo-r couples were introduced into t h e n i c k e l box. It was observed that they became contaminated w i t h condensed n i c k e l vapor, making the readings completely unreliable. In the attempts which followed, the thermocouples had to be removed from the actual experimental zone and covered with suitable shielding material, thus introducing uncertainty i n the experimental temperature. The magnesium s i l i c a t e (AlSiMag2Z2Z) tube which served as the support f o r the nic k e l box containing the specimens, was found to be s a t i s f a c t o r y . Metal shields, such as titanium and molybdenum, were e n t i r e l y unsatisfactory because of t h e i r tendency to form al l o y s with condensing n i c k e l , and to melt around the thermocouple. The thermocouples were observed to pick up an induced A.C. voltage of up to 70 vol t s from the furnace wind- ing during the experiments at high temperatures. To correct the s ituation, thermocouples inserted i n the magnesium s i l i c a t e support were shielded with grounded molybdenum sheets. This method decreased the A.C. voltage i n the thermocouple to less than 1 v o l t , but the other thermocouple.used for control purposes and situated outside the furnace winding was never free from an A.C. voltage of at least about 15 v o l t s . - 31 - Temperature gradient. During the production of thermally etched nickel, metallographic specimens, as well as the thermal treatment of nickel wires to determine the surface tension of the metal, the presence of a considerable temperature gradient inside the furnace was observed, Although the effect of t h i s gradient would be somewhat diminished by the high thermal conductivity of n i c k e l , i t s undesirable effects on the experimental specimens were easily•detectable. Metallographic specimens prepared for interferometric technique to determine the grain boundary energy of s o l i d nickel were found to be unsatisfactory because of the sublimation of ni c k e l atoms on the polished surfaces. The o r i g i n a l wire size was increased up to 50 percent of i t s o r i g i n a l diameter because of the condensation of n i c k e l vapour i n the cooler regions of the hot zone, and p a r t i a l l y decreased at the hotter spots. Furthermore, the weights suspended on the wires were found to have increased up to 50 percent with respect to the o r i g i n a l l y determined loads for the same reason. To improve the thermal gradient i n the furnace the following corrective steps were taken: the o r i g i n a l gage length of the wires, which was approximately 8 centimeters, was decreased to about 2 centimeters; more rad i a t i o n shields were introduced, and the heating c o i l re-designed. Considerable improvement was observedo However, there was s t i l l some i n d i c a t i o n of a thermal gradient i n spite of t h i s , the n i c k e l box appeared to l i e i n a region of reasonably uniform temperature. - 32 - High vapour pressure of n i c k e l . To decrease the high rate of evaporation of nickel atoms at elevated temperatures, i t was decided to use inert gas atmospheres instead of vacuum. A helium atmosphere of 760 mm Hg was found to be unsatisfactory because of i t s very high thermal conductivity, the maximum temperature reached staying below 1200°C. An argon atmosphere of 760 mm Hg was used successfully, the maximum attainable temperature approaching 1400°G. The structure of experimental wires. The f i n e wires employed employed i n the force measure ment technique to determine the surface tension of s o l i d n i c k e l were two to three times greater i n diameter than those used by other workers on copper, gold, and s i l v e r . In the previous work the reported diameters of the wires used were 0 .003 and 0.005 inches compared to 0.010 inches i n the present case. The use of f i n e r wires i s considered to be a factor i n securing a sati s f a c t o r y bamboo-like structure of the wires during the experimental t e s t s . As a consequence, an i d e a l bamboo-like structure of the wires was seldom observed a f t e r experimental runs. Furthermore, much longer durations of the experimental runs had to be applied to achieve measurable s t r a i n s . Interferometrie work After many unsuccessful attempts to observe the Fizeau fringes according to the method described previously, o p t i c a l immersion o i l was employed between the specimen and F i g . 9 Vacuum Furnace with Control Instruments - 3 4 . - the coated s l i p , as proposed by H i l l i a r d . No fringes were detected i n the regions where o i l was squeezed between the specimen and the glass plate, but they were c l e a r l y v i s i b l e i n the regions where no o i l was present. I t i s assumed that t h i s t h i n layer of o i l carrying the weight of the specimen promotes an almost perfect p a r a l l e l i s m between the thermally etched surface and the reference plate, IV. RESULTS The Grain Boundary Energy of S o l i d Nickel. The r e l a t i v e grain boundary energy of s o l i d nickel was determined by measuring the dihedral g r a i n boundary groove angle of thermally etched commercially pure nic k e l specimens. The interferometric method as proposed by H i l l i a r d and Harrpld was used. Matching an aluminized o p t i c a l l y f l a t glass surface of r e f l e c t i v i t y approximately 0 . 3 0 , against the thermally etched specimen, and using monochromatic l i g h t (Hg green l i n e of 5461 Angstroms), Fizeau fringes representing the contour l i n e s i n the surface topography were observed and photographed at a magnification of 1170 times. Distances between the respective Fizeau fringes were measured to 0 . 0 0 5 cm by using a microscope. Knowing the magnification of the micrograph, the horizontal distances between the Fizeau fringes were calculated and plotted against the constant v e r t i c a l distances between the fringes, which were equal to or to 2.7305xlO~5 cms. From - 35 - the r e s u l t i n g elevation p r o f i l e of the surface slope i n the . grain boundary groove, the dihedral angle ^ was measured, A few of the micrographs (Figures 10, 11, 12, and 15) showing the Fizeau fringes along the grain boundaries of thermally etched nickel specimens, and the determination of dihedral angles from those are presented on pages 36 and 37. Table I gives the horizontal distances between the fringes, measured at the cross-sections as indicated on the micrographs, and calculated,taking into account the magnification. F i g . 14 shows one of the elevation p r o f i l e s of the gra,in boundary groove plotted from the attained i n t e r f erometric data. The tests to produce thermally etched specimens of -5 four hours duration were carr i e d out i n vacuum of 5 x 10 ^ mm Hg at 1375°C. Because of the experimental d i f f i c u l t i e s , only three d i f f e r e n t grain boundaries have been studied i n t e r f e r o - m e t r i c a l l y . - 36 - F i g . 10 HOOx Micrograph Showing the Fizeau Fringes along the Grain Boundary of a Thermally Etched Nickel Specimen. F i g . 11 HOOx Micrograph Showing the Fizeau Fringes. - 37 - F i g . 1 5 1 1 0 0 X Micrograph Showing the Fizeau Fringes. - 38 - TABLE I Horizontal Distances Between the Fizeau Fringes Measured from the Experimental Micrographs. P r o f i l e 1 2 3 4 5 a* bA a* b* a* b* ft a • b* • T b & 0.205 17.5 0.190 16.3 • 'i 0.195 16.7 0.160 13 = 7 0.190 16.3 0.155 13.2 0.250 21,4 0.275 23.5 0.190 16.3 0.170 14.6 0.145 12.4 0.220 18.8 0.240 20.5 o a 8 o 15.4 0 .. 0 0 0 0 0 0 0 0 0 0.175 15.0 0.160 15.7 0 ,230 19.7 0 .260 22.0 0.175 15.0 0.205 17.5 0.185 15.8 0 .260 22.2 0 .300 25.5 0.210 1 8 . 0 0.215 18.4 0.215 18.4 .0,265 22.7 0.245 21.0 A a gives the measured distance between fringes i n cm. _5 A b gives the calculated r e a l distances between fringes i n 10 cm. o gi g . 14 Elevation P r o f i l e No. 1 of the Grain Boundary ( Groove Plotted from the Interferometric Data. ^ - 40 - The values of dihedral angles determined from the above data are recorded i n Table I I . Table II i Values of Dihedral Grain Boundary Angles of Thermally Etched Commercially Pure Nickel. P r o f i l e Dihedral cos 4> Angle M> 2 2 1 159 79.5 0.182 0.364 2 156 78 0.208 0.41b 3 164 82 0,139 0.278 4 165 82.5 0.131 0.262 5 159 79.5 0.182 0.J64 The scattering of the valu es obtained f o r .the dihedral angle causes, as seen from the above table, a v a r i a - t i o n of the r a t i o up to ±30 percent with respect to the average value, 0.33. This large v a r i a t i o n cannot be explained by the orientation dependence of the grain boundary energies. It i s expected that the average grain boundary l i e s between grains so d i f f e r e n t i n orientation that the dihedral angle would be v i r t u a l l y constant. Hence, the reason f o r the scatter most probably l i e s i n the experimental technique or conditions. It i s not believed that H i l l i a r d ' s i n t e r f e r o - metric method has inherently low p r e c i s i o n , rather, i n the present case the scatter r e s u l t s from two contributing f a c t o r s . F i r s t , the very shallow nature of the grain boundaries revealed i n the specimens, i . e . , only two to four fringes i n depth did not permit the accurate p l o t t i n g of p r o f i l e . H i l l i a r d , however, - 41 - was able to get boundaries s i x to eight f r i n g e s i n depth and so was able to draw b e t t e r and more accurate p r o f i l e s . Secondly, only a very few boundaries were found where fringes could be observed, whereas,ideally,a f a i r l y large number would be necessary i n order t o get t r u l y representative values. An attempt was made to measure the dihedral grain boundary groove angles of thermally etched nickel i n micro- sections taken perpendicular to the surface. The thermally etched grain boundaries of the specimen were observed i n p r o f i l e and the suitable ones were photographed at a magnificat t i o n of 1100 times. The region containing the grain boundary groove was magnified (Fig. 1_?) from the p r i n t s and the dihedral angle measured. According to t h i s method the dihedral angle was found to be equal to 146 degrees. As stated on the previous pages (page 17)> the interferometric method i s considered to be more accurate than the microsectional one because of i t s r e l a t i v e s i m p l i c i t y and involvement of fewer experimental uncertainties. Hence, i n the following c a l c u l a t i o n s , despite i t s uncertainty because pf the small number of experimental determinations, an average value of l 6 l degrees i s used to determine the r e l a t i v e grain boundary tension of s o l i d n i c k e l according to the equation = 2 $ cos From t h i s equation the r a t i o of grain boundary energy to surface tension f o r s o l i d n i c k e l was found to be approximate- l y 0.33 for the average value of dihedral angle. This r a t i o has - 42 - been proposed and confirmed f o r copper, g o l d , and s i l v e r . F i g . X5 Micrograph Showing the M i c r o s e c t i o n of Grain Boundary of Thermally Etched N i c k e l at 1375°C f o r 4 Hours. O r i g i n a l M a g n i f i c a t i o n llOOx, enlarged 8 x . The Surface Tension of S o l i d N i c k e l S t r a i n and s t r e s s measurements w i t h i n the recorded temperature range f o r a determined l e n g t h of time are given i n Table I I I . The atmosphereic c o n d i t i o n s of the s u c c e s s f u l experiments are a l s o i n d i c a t e d . These r e s u l t s are p l o t t e d g r a p h i c a l l y i n a s t r e s s - s t r a i n diagram, to determine the load which e x a c t l y balances the upward p u l l of the surface t e n s i o n . F i g u r e 16 i s a t y p i c a l p l o t . A c t u a l l y the val,ue f o r s t r e s s at zero s t r a i n , as w e l l as the slope, was c a l c u l a t e d by using the method of l e a s t squares. Table IV gives the values of surface t e n s i o n and g r a i n boundary energy determined from the s t r e s s at zero - 43 - Table III Experimental Stress and Str a i n Measurements Test No. •1 2 4 Temp.°C. Time hours Atm. mm Hg Wire radius cm. 1300*5 20 5xl0~5 0.01425 1300-5 7 0 -5 5x10 ^ 0.0129 1391*5 5x10-5 0.0127 Spec. No. Stress dyne/cm^ x 10--5 S t r a i n cm/cm x 1 0 5 Stress dyne/cm.2 x IO-5 S t r a i n cm/cm x IO5 Stress _ S t r a i n dyne/cm cm/cm x IO--5 x IO5 1 2 3 4 5 6 7 8 1.75 3.23 4.07 -2.43 -1.61 0.328 0.568 1 .065 1.44 1 .91 4.26 5.41 -6.15 -3,18 -2.71 -1.01 -0.29 1.00 2.56 -5.77 3.75 - 6 . 4 5 4 .30 -4.05 5.77 -1.88 6.40 -0.405 Test No. 6' 7 10 Temp. °C Time hours Atm. mm Hg Wire radius cm. 1595*5 20 5x10"^ 0.0127 1350*5 5x10 ? 0.0127 1375±5 5x10 ^ 0.0127 Spec. No. Stress dyne/cm x 10"-5 S t r a i n cm/cm x 10 5 Stress dyne/cm, x IO"5 Strain cm/cm x 10 5 Stress S t r a i n dyne/cm cm/cm x IO--5 x 10? 1 2 3 4 5 6 7 8 2 .09 3.92 9.61 -6.02 -4.93 1.46 2.37 4 .36 6.02 7.97 10.82 12.27 13.67 -0.774 -0.543 -0.056 0.728 1 .32 1.94 1.79 0.92 -0.70 2.21 -0.115 4.04 0.97 5.42 1 .73 5.50 1 .98 7.46 3.41 - 44 - TABLE III (cont»d.) Test; No. 11 12 13 Temp.°C Time hours Atm, mm Hg Wire radius cm. 1327*10 73 760 (Argon) 0,0123 1300*10 96 760 (Argon) 0,0127 1380*10 6 760 (Argon) 0,0126 Spec. No, Stress S t r a i n dyne/cm2 cm/cm x 10 ̂  x 10 5 Stress dyne/cm x 10"^ St r a i n 2 cm/cm x 10 5 Stress dyne/cm x IO' 5 S t r a i n 2 cm/cm x 10* 1 2 3 4 5 6 7 0.749 -1,003 O.949 -0.843 1,58 0,076 2.63 0,327 4,06 1,006 1 .44 2 .92 3; 71 4.01 -1.38 -0.46 1.087 1.087 1.55 1.52; 2,52 3.87 4.18 0.115 0.972 1.752 1,977 Test No, 14 15 Temp„°C. Time hours Atm. mm Hg Wire radius cm. 1380±10 24 760 (Argon) 0,0126 1366±10 70 760 (Argon 0,0117 Spec. No. Stress S t r a i n dyne/cm^ cm/cm. x 10**-5 x 1 0 5 Stress dyne/cm x 10* 5 S t r a i n 2 cm/cm. x IO 5 1 2 3 4 5 6 t, 7 • 0.728 -1.32 0.930 -0 .19 1,525 0,382 2,52 1 .79 0. 836 1.05 / 1. ̂ 6 2,98 *™ *1» 0 2 2 - 0 .55 0,207 0.738 4.87 6,73 1.68 2.14 T  i - 46 - TABLE IV Experimental Values of the Surface Tension' and'Grain Boundary Energies of Solid Nickel, X Test No, .:"Atmos, Tempp°.0. Duration of test hours, " 1 11 " dyne/cm'r 10-5 Surface Tension dyne/cm Grain Bound, Energy , dyne/cm,- 1 Vaquum 130015 42 4V07 . 6760 2^53 2 Vacuum 1300+5 79 4,22 6370 2123 4 Vacuum 1391±5 19 6o97 10320 3440 6 Vacuum 1395+5 20 8,30 1230O 4100 3 Vacuum 139015 50 11130 mo 10 Vacuum 1375±5 50 - 2 .50 44160 13B7 Argon 1327iJ0 75 2„08 2990 997 12 Argon 1300+10 96 2,95 . 4410 1470 13 Argon 1380+10 6 ' 1.67 2470 825 14 . Argon ' 1380±10 24 1.33 I97O 657 i f - Argon 1366iio 70 2 .16 2920 973 - 47 • st r a i n , by use of the two simultaneous l i n e a r equations, w . i r r * - (^Trr 2 -*?* a/ = 2 if cos y where, as before w = suspended weight & = surface tension jf*= grain boundary energy r - radius of the wire = number of grains per unit length. The value f o r ft/1, the number of grains per unit length of the wire, was determined a f t e r every run by cutting pieces of the nick e l wire specimens, 1 cm. long, and mounting i n l u c i t e , polishing, etching, and observing these specimens under the microscope. The number of grains per cm of wire was determined to be 30110. A c h a r a c t e r i s t i c micrograph demonstra ti n g the structure of the nick e l wire i s given i n Figure 17 where the so-called 'bamboo-like 1 structure can be observed. 250x F i g . 17 Micrograph Showing the 'Bamboo-like* Structure of Experimental Wires. - 48 - V , DISCUSSION Surface Energy Measurements. The value of the dihedral grain boundary groove angle fo r thermally etched n i c k e l was found to be lf-1. degrees . according to the interferometric method* On the basis of the present work and as described e a r l i e r the accuracy of t h i s method i s believed to be. higher- than, that of. other .conventional techniques. Furthermore, the interferometric method i n comparison with the microsectional one was found to be simpler because of the r e l a t i v e ease i n picking v i s u a l l y the grain boundary grooves suitable f o r the measurement of dihedral angle by observing the symmetry of the Fizeau fringes'.. Hence, the above value f o r the dihedral grain boundary groove, angle of thermally etched nickel i s thought to'-be acceptable,although, of course, no attempt has been made to account f o r the o r i e n t a t i o n dependence of the grain boundary angle. Furthermore, t h i s value cannot be taken as f i n a l because of the small number of tests c a r r i e d out successfully. Substituting the determined value of the dihedral angle i n the equation X*= 2 Vcos ^ gives the r a t i o of grain boundary tension over surface tension f o r s o l i d n i c k e l . This r a t i o was determined to be approximately 0.33 which checks very well with the assumption f o r n i c k e l . Similar values had been postulated and confirmed experimentally fo r copper, gold, and s i l v e r . - 4? - The surface tension of s o l i d n i c k e l determined i n a vacuum of 5 x IO' 5 mm Hg i n the temperature range from 1300°C to 13950(3 w a s found to vary inconsistently from 4160 to 12,320 dynea per centimeter. The combination of the high vapour pressure of nic k e l under these experimental conditions, and the sharp thermal gradients within the hot zone, r e s u l t s i n excessive evaporation from some parts of the system and condensation onto other parts of the system. This produces changes i n the dimensions and weights of the test components. Therefore, l i t t l e r eliance may be placed i n the res u l t s obtained from tests i n vacuum. To overcome the d i f f i c u l t i e s due to the high vapour pressure of n i c k e l , an inert gas atmosphere was employed i n the experiments. The surface tension of s o l i d n i c k e l i n argon at 760 mm Hg i n the temperature range from 1300°C to 138o°C was determined to vary from 1970 to 4410 dynes per centimeter. The lower values, 1970 and 2470 dynes/cm are thought to be moat r e l i a b l e . It w i l l be seen that of the runs made with an argon atmosphere, those made with the longest times have the greatest surface tension value. The time f a c t o r may be coincid e n t a l , since the microstructures of the wires giving the larger values were not bamboo-like, and f o r t h i s reason alone these values can be rejected. However, since the vacuum runs are a l l high, i t may be that evaporation i s p a r t l y to blame. I t should be noted that the argon atmosphere does not prevent the evaporation of n i c k e l , i t merely reduces the p a r t i a l pressure of n i c k e l so that evaporation - 50 - i s diminished. The vapour pressure of nickel at 1357°C i s i n -3 the order 10 mm. 46 A recent paper by Skapski enables a c a l c u l a t i o n of surface tension to be made i f certain values are known. Appendix E shows the c a l c u l a t i o n for n i c k e l . It was necessary to extra- polate data by Norton^ i n order to get a value for the surface tension of l i q u i d nickel at the melting point since the c a p i l l a r y c o e f f i c i e n t of l i q u i d n i c k e l i s unknown. Calculated Values of"the Surface . a ^ Table V. ilues Tension f o r S o l i d Nickel (dynes/cm). 1450 1630 1671 1500 1680 1721' 1550 1730 1771 1600 1781 1822 1650 1831 1872 1700 1882 1923 The above table gives the calculated values and i t w i l l be seen that they agree i n order of magnitude with the observed surface tensions. It i s i n s t r u c t i v e to compare the r e s u l t s of Skapski as i n Table VI. Table VI Metal , (T t ? w Ag 1056 1130*60 Au 1267 1350±70 Cu 1417 1650-80 - 31 - A l l the calculated values are less than the observed ones, but the theory gives good general agreement and supports the chpice of the two lowest observed values, and t h e i r arithmetical average i s henceforth used 0 The p r e c i s i o n of the f i n a l value i s d i f f i c u l t to assess but i s probably better than 3 0 0 dynes/cm„ Hence, i t i s proposed on the basis of the present work that the surface tension of s o l i d n i c k e l i n argon within the temperature range from 1 3 7 0 ° C to 1 3 9 0 ° C i s 2220*300 dynes per centimeter. Thus the grain boundary energy of s o l i d n i c k e l within the previously stated experimental conditions would be 340*300 dynes per centimeter. However, these values cannot be considered as f i n a l and conclusive because of the small number of t e s t s c a r r i e d out successfully under s a t i s f a c t o r y experimental conditions. Nevertheless, i t i s believed that these values can be confirmed i n future t e s t s provided that the temperature control i n the experimental zone i s s u f f i c i e n t l y close, the temperature gradient i s n e g l i g i b l e , and the structure of the experimental wires approaches the so-called 'bamboo-like' structure, A f i n e r wire siz e , 0 , 0 0 3 to 0 , 0 0 3 inches diameter, would be desirable. It had been hoped that the value f o r the temperature- c o e f f i c i e n t of the surface tension of s o l i d n i c k e l , and the value fo r the v i s c o s i t y of s o l i d n i c k e l could be established experiment- a l l y , but the idea had to be abandoned because of the shortness of time and the i n s u f f i c i e n t l y close control of temperature during the t e s t s . Thermal Etching. During the experimental work to determine the grain boundary energies of s o l i d n i c k e l , the structures of the thermally etched metallographic specimens were examined f o r evidence in d i c a t i n g the presence of d i s l o c a t i o n s . The d i s l o c a t i o n concept has been found to be useful i n explaining c r y s t a l structure, d i f f u s i o n , s o l i d i f i c a t i o n , and deformation, but u n t i l l a t e l y d i r e c t experimental observations 48 have been scarce. According to Forty very l i t t l e work has been done on metals, most of the observations being ca r r i e d out on non-metals. From the calculations based on a d i s l o c a t i o n model of a small-angle grain boundary, i t i s expected that 'the spacing of i n d i v i d u a l dislocations along such a boundary i s within the resolving power of the.optical microscopet^ -^? Hence, i t i s conceivable that a microscope of s u f f i c i e n t l y high magnification power would reveal the d i s l o c a t i o n s . The di s l o c a t i o n s i n low-angle boundaries as well as i n the matrix can be revealed with methods based on the assumption that a d i s l o c a t i o n i s a s t r u c t u r a l discontinuity causing energy- difference i n the c r y s t a l l a t t i c e . The most useful method to reveal the dislocations i s the sensitive etching, which may be accomplished by chemical, electrochemical, ionic bombardment, or by thermal etching. A l l these techniques involve the removal of atoms from the high-energy d i s l o c a t i o n s i t e s , the r a t e of migration from these sites being higher than from the neighboring matrix. This method has been employed to show the presence of 54 55 dislocations xn s i l v e r and i n chromium. Commercially pure nickel thermally etched at 135°°C -5 and at a pressure of 5 x 10 mm Hg f o r four hours showed the following structure (Figs. 18-27). 55 The existence of dislocations i n chromium had been concluded from the following features revealed a f t e r a suitable thermal etch: 1. The existence of sub rboundaries developed as rows of p i t s . 2. The occurrence of generally spaced p i t s within the sub-grain. 3. The dihedral angles between sub-boundaries themselves and between the sub-boundaries and the grain bound- ar i e s . In the present work no sub-boundaries were detected as rows of etch p i t s , Tigs, 18 and 19 , however, show sub- boundaries but not as a row of p i t s . The continuous s o l i d boundary could be considered as a sub-boundary since the degree of mismatch across the.boundary is very small. The absence of rows of etch p i t s might be due to the decreased s e n s i t i v i t y of the thermal etch at the higher temperatures as i n the work on chromium. Si m i l a r l y , the occurrence of generally spaced p i t s - 54 - F i g . 18 1100 x Thermally Etched Nickel Specimen Showing a Sue-boundary. Fi g . 20 2J00 x F i g . 19 2300 x Thermally Etched Nickel Specimen Showing a Sub-boundary. F i g . 21 2300 x Thermally Etched Nickel Specimen Showing the Configuration around an Inclusion. Thermally Etched Nickel Specimen Showing the Configuration around an Inclusion. 2J00 x gift. 23 550 X Thermally Etched Nickel Specimen Showing the Configuration around Two Inclusions. (0) F i g . 24 2300 x Thermally Etched Nickel Specimen Showing the Symmetrical Deep P i t from the F i g . 23. Thermally Etched Nickel Specimen Showing a Symmetrical Deep P i t . 2300 x Thermally Etched Nickel Specimen Showing an Octahedral Deep P i t . F i g . 26 1100 x US* 27 1100 x Thermally Etched Nickel Specimen Showing some Symmetrical Features. Thermally Etched Nickel, Chemically Etched before Thermal Treatment. within the grains was dubious,, However, i n t e r e s t i n g configura- tions around the inclusions were rioted i n the metallographic study of thermally etched n i c k e l . Such patterns around an i n c l u s i o n are demonstrated i n Figures 2 0 , 21 and 2 2 . Figures - 2 3 to 25 show the structures s i m i l a r to those noted i n the chromium etched at higher temperatures only ( 1500°G). The r e l a t i v e l y deep p i t , shown i n Figure 2 5 , exhibits octagonal symmetry at i t s base. The i n c l u s i o n of Figure 23 magnified i n Figure 24 shows hexagonal symmetry at i t s base and dodecahedral i n the second and t h i r d l e v e l s from the base. It has been postulated that t h i s symbolizes the reverse process of s p i r a l 56 growth as proposed by Frank and d i r e c t l y observed i n cadmium iodide c r y s t a l s by Newkirk, Thus, screw dislocations may be operative i n permitting sublimation to take place at temperatures at which the vapour pressure corresponds to low supersaturations. 57 Danko and Griest observed s i m i l a r sublimation figures on the surface of pure n i c k e l , copper, and zinc specimens. They found that the geometry of the sublimation figure of n i c k e l and zinc corresponded to t h e i r respective c r y s t a l structures, the figures on the copper specimens being of c i r c u l a r nature. In the course of the present work no cubic sublimation figures were observed on the thermally etched n i c k e l specimens, as reported by Danko and Griest. However, triangular, hexagonal, and c i r c u l a r configurations were exhibited i n some cases (Fig, 2 6 ) . The specimen seen i n the Figure 27 was chemically etched prior to the thermal treatment. - j>8 - VT. CONCLUSIONS 1. The surface tension of s o l i d n i c k e l i n argon at 760 mm Hg and within the temperature range from 1370°C to 1390°C was determined to he 2220*300 dynes per centimeter, 2, The grain boundary energy of s o l i d J i l c k e l i n argon within the previously stated temperature range was found to be 740*300 dynes per centimeter, assuming the r a t i o of g r a i n boundary energy to surface tension f o r ni c k e l to be 1/3. 3„ The dihedral grain boundary groove angle of thermally etched s o l i d n i c k e l was determined i n t e r f e r o m e t r i c a l l y and measured to be l 6 l degrees, 4. The r a t i o of grain boundary energy to surface tension f o r s o l i d n i c k e l , using the measured dihedral, angle, was calculated to be approximately 0,33 which checks well with the assumption of 1/3, 5. The physical evidence f o r disl o c a t i o n s i n ni c k e l was inconclusive on the basis of observations on the structures of the thermally etched n i c k e l specimens i n vacuum of 3 x 10"^ mm Hg and at 1350°C. Nevertheless, a more sen s i t i v e thermal etch at lpwer temperatures i s expected to produce more cpnvincing physical evidence of dislocations i n n i c k e l . t>„ Udin's technique to determine the surface tension of metals by force measurement on stretched wires wag found to be s a t i s f a c t o r y on a metal of higher melting point, provided - 59 - precautions were taken to reduce the high vapour pressure of the metal. 7. H i l l i a r d ' s interferometric method to measure the dihedral grain boundary groove angles was found to be applicable and simple. The accuracy of the method was not very high i n the present work, but t h i s could be blamed on the small number of tests carried out. It i s believed that the p r e c i s i o n of t h i s method i s not inherently low. - 60 - VII. APPENDICES - 61 ^ APPENDIX A DEFINITIONS Interface • . ...... i . M Aneihterface i s def ined; to . be: a .bounding,;surface across which a discontinuity can be observed, - v . . Grain boundary In the s o l i d state the observed interfaces due to a discontinuity may be caused by d i f f e r e n t changes. I f the interface i s due to a sudden change i n l a t t i c e o r i e n t a t i o n within a single phase, i t i s c a l l e d a grain boundary, I n t e r f a c i a l energy Atoms i n the discontinuous regions or interfaces do not have t h e i r normal number of neighbors at normal distances, they are i n higher energy states compared to the atoms i n neighboring homogeneous regions. The t o t a l excess energy of these i n t e r - f a c i a l atoms due to the abnormal s t r u c t u r a l arrangement, i s c a l l e d i n t e r f a c i a l energy. Grain boundary energy Grain boundary energy i s the excess.energy of the i n t e r - f a c i a l atoms due to the abnormal crystallographic arrangement. Sp e c i f i c i n t e r f a c i a l energy The s p e c i f i c i n t e r f a c i a l free energy may be defined as the increase of free energy of a system per unit increase of i n t e r f a c i a l area under conditions so that the new inte r f a c e has i t s minimum energy configuration. - 62 - I n t e r f a c i a l force The f i c t i t i o u s force per unit length which i s assumed to replace the free energy per unit area of interface i n calculations has the same physical dimensions and i s considered to be a tension as the i n t e r f a c i a l energies are always p o s i t i v e . Surface tension. The surface tension i s defined to be the i n t e r f a c i a l force, i f a s o l i d or l i q u i d i s i n equilibrium with the vapour i n a vacuum or i n a noble gas atmosphere. - 63 - APPENDIX B Nominal Composition of Commercially Pure Nickel of Type "A" Grade Ni (Co) Ee 99.4% 0.15% Mn Cu 0.20% 0.10% c 0.10% S i S 0.05% 0.005% £1 3> F R . C . UOv. v . a E D HO V . ^0 -Q- JMUULgJJUU H CO o c+ H- O P ' I—1 P *d o o d S: 6 Q CO P d d *c( 01} w CD H *< O x . K3 iy CD e g O O O d •d i—1 CD P d hj H o no v. A - Ammeter L - Lamp R - Relay P - Potentiometer S - Specimen T - Time elapse meter V .G. - Vacuum gauge control IsG. - Ionization gauge T.C.G. -• Thermocouple gauge T.C. - Thermocouple C.J. - - Cold junction O.P. - Diffusion pump M.P. - Mechanical pump R.C. - Recorder-controller - 65 - APPENDIX D Preparation of 50 Percent Transmitter^ Mirrors. Determination of the thickness of the f i l m f o r 50 percent transmission. Formulas: 0 . 6 9 5 4-n \ = y /*» \ Data: Absorption Refractive nk index k index n f o r green Hg spectral l i n e of 5461 A S i l v e r 1 7 . 9 0 .175 3 . 3 0 Aluminum 3 . 4 8 1 .16 4.04 Calculation: 4(1.14)(4.04) = Q o 9 2 8 x 1 Q 6 c m - l ' 5461 x 10-° x = v = 0.747 x 10" 6 cm x 0 . 9 2 8 x l O b Determination of the weight of metal needed f o r 50 percent transmission to coat the glass,' Formula x. - m t 4-npcx where i s the density of the metal. r i s the distance between source of coating metal and glass . m i s the weight of metal needed Data: r i s 15 cm. Density Thickness of Film S i l v e r 1 0 . 5 0 . 9 H x 1 0 " § cm Aluminum 2.7 0.747 x 1 0 - b cm Calculation: m A 1 = 4(3.14) ( 2 ,7) ( 1 5 2 ) ( 0 .747 x 1 0 " 6 ) = 27 mg. - 66 - APPENDIX E Calculation of the Surface Tension of S o l i d Nickel 53 According to Skapski the surface tension of metals can be calculated from the arrangement of t h e i r next-neighbors, from the heat of fusion, and from the surface tension of the l i q u i d at the melting point. The following equation: holds for the c a l c u l a t i o n of the surface tension of a s o l i d i n i t s most densely populated planes to reveal the minimum surface tension, where (1) Zo,= number of next-neighbors on the surface ~ 2j= number of next-neighbors i n the l a t t i c e . Qp= heat of fusion P£ = density of s o l i d at melting point, fl = density of l i q u i d at melting point Ct = s p e c i f i c surface tension of l i q u i d TM= melting point i n °K A£ = configurational entropy f o r the l i q u i d L u " i " surface. = configurational entropy f o r the s o l i d *u>*^ surface. f\ = molar area. where | = density f a c t o r N = Avogadro's number - 67 - M = molecular weight where Oj gravity acceleration X = height of the lower surface of the plate above the undisturbed l e v e l of the l i q u i d . Following Skapski, f o r n i c k e l , which i s a face-centred cubic metal, the minimum surface tension can be calculated i n the most densely populated planes, I l l - p l a n e s . For n i c k e l : 12 V 9 ~o Is Z„= 1 2 - 9 5 = 3 / 1 2 = 1/4 \ = 1 . 0 9 M = 5 8 . 6 9 T = I 4 5 5 ° c . rn = 4 , 4 8 x 1 0 ^ ergs/degree U«Kt 7 AS = 3 . 9 0 x 1 0 ' ergs/degree 6^ = I 8 . l 8 x l 0 1 0 ergs/gram-atom (Metals Handbook) j>6 i s calculated from the l a t t i c e parameter of n i c k e l . el = 3.517x10"^ cm, and from the cubical thermal expansion c o e f f i c i e n t ^ - 6 59 jb - 3 8 . 1 x 1 0 cm/m, according to Mott and Johnson a ?1455°C = ( 3 . 5 1 7 x 1 0 " 8 ) 3 ( l + 3 . 8 l x i o " 6 ( 1 4 5 5 - 2 0 ) ) c u 3 l 4 5 5 ° G ~ 4 d X 1 0 ~ 2 4 <?m^ p . - ^(58.69) . 8 o 4 8 g / c m 3 J $ °c 6 . 0 2 3 5 x l 0 2 3 ( 4 6 x i o j - 68 - Since no d a t a i s a v a i l a b l e c o n c e r n i n g t h e density of l i q u i d n i c k e l at t h e m e l t i n g p o i n t , t h e d e c r e a s e i n density of s o l i d nickel at the same temperature i s assumed to be s i m i l a r to that of copper. Hence, the density of l i q u i d n i c k e l at the melting point i s taken t o be equal to 8.26 g/cm.3 fa = 8 . 2 6 g/cm? From equation (2) ? , 1 . 2 n o A g = 1 . 0 9 ( 6.0235xlO ° J 3 ( 5 8 . 6 9 ) 3 = 3 3.4x10 'cur Because the c a p i l l a r y constant for l i q u i d n i c k e l i s unknown, the values of G[ has been extrapolated from the data 58 presented by Norton et a l . Owing to the uncertainty of t h i s extrapolation, calculations were made for six d i f f e r e n t values of and results are presented below: From equation (1) £ . l ( % l 8 x l ^ ) + ( S ^ S j ^ x ^ o ) + 1455±273 .(4.48^ 3 . 9 0 ) x l o 7 V p 4 33.4x107 8.26 y 2(33.4x10/ = 1 3 6 + 1 . 0 1 8 ( 1 5 5 0 ) + 2 . 5 9 ( 0 . 5 8 ) = 1 3 6 + 1 5 7 9 + 15 q! = 1730 dynes/cm at 1455°C. Assuming the thermal c o e f f i c i e n t for surface tension of s o l i d nickel be equal to that of s o l i d copper, 0 . 5 5 dynes/cm/°C ^ v w c = 1 7 3 0 + ° » 3 5 (1455 - 1380) = 1730 + 41 „ = 1771 dynes/cm at 1 3 8 0°C. TABLE V. Calculated Values of the Surface Tension for Soli d Nickel. 1450 1500 155Q 1600 1650 1700 1630 1680 1730 1781 1831 1882 1 6 7 1 1 7 2 1 1 7 7 1 1822 1 8 7 2 1 9 2 3 - 7P - VIII. BIBLIOGRAPHY 1... Smith, C.S., Trans. A.I.M.E. , v o l . 175, 194-8, p. 15. 2. Harker, D. and Parker, E,R., Trans. A.S.M., v o l . 54, 1945, p. 156. 5. Volmer, M. and Weber, A., Zei t s . fur phys. Chemie, v o l . 119, 1926, p. 277. 4. Becker, R., Ann. der Physik, v o l . 32, 1958, p. 128. 5. Shaler, A.J. , Seminar on Kin e t i c s of Sintering, A.I.M.E., Met. Tech., Dec. 1948. 6. Gibbs, J.W., Collected Works I, 1951, N.Y., Longmans, Green and Co. 7. Fricke, R. and Meyer, F.R., Zeits. fur phys. Chemie, vol.A l 8 l , 1958, p. 409. 8.. Hut t i g , G.F., Zei t s . f u r anorg. und allgem. Chemie, v o l . 247, 1941, p. 221. 9. Chapman, 'J.C. and Porter, H.L. , Proc. of Royal S o c , , v o l . A 185, 1909, p. 65. 10. Schottky, H., Gessel. Wiss. Goettingen, Nachr. Math. Phys. 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