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Size effects in copper whiskers Moore, Zelma Esther 1961

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SIZE EFFECTS IN COPPER WHISKERS by .ZELMA ESTHER MOORE A THESIS SUBMITTED IN ;PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE IN THE DEPARTMENT OF MINING AND METALLURGY We accept t h i s t h e s i s as conforming to the standard r e q u i r e d from candidates f o r the degree of MASTER OF APPLIED SCIENCE Members of the.Department of Mining and M e t a l l u r g y THE UNIVERSITY OF BRITISH COLUMBIA December 1961 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia, Vancouver 3, Canada. Date ~~"\fV^- ABSTRACT Whiskers were grown by the hydrogen reduction of cupric chloride. Tensile tests were performed on the whiskers, some of which were long enough to divide into two or three parts. After the whisker yielded ("primary tests") they were retested ("secondary tests") after removal of the deformed region. In order to see i f length had any. effect on y i e l d stress, a norm-a l i z i n g proceedure was established to convert the y i e l d stress measured at any diameter d to an•equivalent whisker with d = 1 0 j a . No observable length dependence of y i e l d stress was found for either primary tests or secondary t e s t s . The diameter dependence of y i e l d stress was founk to depend on the type of t e s t s . For primary t e s t s , the y i e l d stress was inversely proportional to d"*~'^ , while for secondary tests, 1 inversely proportional t o d . A disloc a t i o n mechanism to explain t h i s was proposed i n terms of only a part of the cross-sectional area (ai small annular r i n g at the periphery of the whisker) taking part i n the deformation. This mechanism was suitable only i f the whiskers were assumed to be i n i t i a l l y free of disl o c a t i o n s . -A decrease i n the value of Young's Modulus of about 30$> from the normal values was observed for whiskers subjected to a secondary t e s t . ACKNOWLEDGEMENT The author wishes to thank her research director,. Dr. • E. Teghtsoonian, for his advice and encouragement given during this investigation. Thanks are extended to'fellow graduate students, especially Dr.- K. G. Davis, for many-helpful discussions. Indebtedness is also acknowledged to Mr. W. • J . - Allday for his technical assistance. This project was financed by Defence:Research<Board Grant Number 7510-33-i i . TABLE OF CONTENTS page INTRODUCTION 1 1. General 1 2 . Previous Work . . . . 2 a. Strength, of Whiskers • . . 2 b. -Elastic Behaviour of'Whiskers 3 c. Plastic Behaviour of Whiskers . . .'. 6 •a. Size-Effects-in-.Whiskers 10 i . Crystal Perfection 13 i i . Effect of Surface Defects l 6 i i i . Grip Effects 17 e. Effect of Surf ace Films on Whiskers . 17 f. Effect of Impurities in Whiskers 18 3- Purpose of Present Investigation 19 EXPERIMENTAL PROCEDURE 20 1. Growth . . 20 2. Selection . 22 3- Specimen Mounting 2h k. Measurement of Specimen Length 25 5- Determination of Cross:Sectional Area . . . . . . . . . 25 6. Tensile Testing.Apparatus 26 a. Construction . 26 b. Calibration . 29 7- Annealing Furnace . 29 8. Experimental Procedure 29 a. Tensile Tests 29 i i i . TABLE OF CONTENTS CONTINUED Page b. Annealing Tests 30 EXPERIMENTAL OBSERVATIONS AND RESULTS . . . 31 1. Growth Observations 31 2. Cross Sectional Area 33 3. Tensile Tests 33 a. Reliability of the Quantitative: Measurements . . . 33 b. • Statistical Treatment of the Data 3^ c. Primary Tests 3^ d. •• Annealing Tests h3 e. Secondary Tests hh f. Variation of Young' s Modulus h$) g. Summary 56 DISCUSSION 57 1. Growth 57 2. Comparison of Results With Previous Work 58 3- Diameter Dependence.of.Stress 69 h. Variations of Young's Modulus . . 82 SUMMARY AND CONCLUSIONS 8U RECOMMENDATIONS FOR FURTHER WORK . . . 85 BIBLIOGRAPHY 86 APPENDIX I 88 APPENDIX. II 97 APPENDIX. T i l 108 APPENDIX IV . . . . I l l APPENDIX V . . . 115 APPENDIX VI 123 iv. FIGURES Page 1. •Stress-Strain.Curves of Iron and Copper Whiskers Pulled in Tension 5 2. Effect of Creep :on the Force-Elongation Curve (before yield) 7 3- Stress-Strain Curves of Copper Whiskers 8 k. Propagation of a Luders Band in a Copper Whisker 9 5- •Strength Versus Diameter for (a), Iron and (b) Copper Whiskers 11 6. Dependence of Crit ical Shear Stress ~ t j . 0 on .Whisker Diameter- 12 7« Recovery of Yield Point of Copper Whisker After.Removal of a Slipped Portion 13 8. Dependence of the Strength of Silicon Whiskers on Temperature 15 9. Tensile Strength of.Silicon Rods and Whiskers 15 10. Whisker Growing Furnace. 20 11. Schematic Diagram of Whisker Growing Furnace 21 12. Whisker Handling-Apparatus . . . 23 13- Schematic Diagram,of the Whisker Tensometer 28 1^. Whisker Tensometer 27 15. An Odd ..Shaped.Whisker , 32 16. Primary Yield Stress as a Function of Diameter 35 17. Log Plot of Primary Yield Stress Against Diameter . . . . . 36 18. The Average Strength of Copper Whiskers as a Function of the Reciprocal of the Diameter 37 19. Primary Yield Stress Against Diameter 39 20. Primary Yield Stress as a Function of Length kO 21. Diagram Illustrating Derivation of the Normalizing Procedure ^1 22. Normalized Primary Yield Stress as a Function of Length . . h2 23. Secondary Yield Stress as a Function of Diameter i+5 2k. Log Plot of Secondary Yield Stress Against Diameter . . . . k6 V. FIGURES COWTIMJED Page -2-5 25. Secondary -Yield-Stress- Against. Diameter d . . . . . . . . . . . 47 26. Secondary-Yield Stress as -a Function.of Length . U8 27. Normalized Secondary Yield-Stress as a Function of Length . . 50 28. Plot of the Ratios E/Ej_ of Young' s Moduli Against. E i . . . . . 51 29. Stress-Strain Curve of Whisker'BB 52 30. Stress-Strain Curve.of.Whisker.GG . . 53 31. - Ratio. . .E/E^ of the Young's Moduli Against.Initial $ Elongation 5U 32. Ratio E/E^_ of the Young's Moduli Against Primary, Normalized Yield.Stress 55 33- Critical-Resolved .Shear. Stress-Against Diameter 59 3U. Crit ical Resolved .Shear Stress for £l00] Whiskers.Against Diameter . . . 6 l 35- Cri t ical Resolved..Shear-Stress for-."[|.llo3 Whiskers Against Diameter . . . . . . . . 62 36. Cri t ical Resolved Shear Stress for C 111] Whiskers-Against Diameter 63 37- Log Plot of the Crit ical Resolved.Shear. Stress-Against Diameter 6U 38. Dependence of Flow.."Stress on -..Diameter . . . . . . . . . . . . 66 39- Comparison of Crit ical Resolved.Shear Stress.With Crit ical Resolved Flow Stress 67 kO. Stress-Elongation Curve of A Whisker Tested on the Instron '. 68 Ul. Model for Estimating the Maximum Number of Loop Sites . . . . 71 U2. Diagram of a Whisker Containing Dislocation. Loops 72 U3- Diameter dQ of - Annular-Ring ..Against k f o r d = 2 p..-A = kd 1 '^ . 75 UU. Diameter dQ of.. Annular-Ring Against k f o r d =10^u. A = kd 1 '^ 76 1 6 U5. Diameter dQ of Annular Ring.Against k for d = 20 ji. A = kd 77 ' ' 2.5 Ub. Diameter d,0 of Annular Ring.Against k for d = 2 p.. A = kd 79 2 5 7. CLQ g . 10 p 80v i . FIGURES CONTINUED Page 2.5 k8. Diameter dQ of-Annular Ring Against .k for d=20 p.. A=kd . 8 l U9. Variation of Young's Modulus With .Film Thickness 83 50. Calibration of Large Helical Spring 92 51. Calibration of.Small Helical Spring '• • • 93 52. Calibration of Restoring Force of Suspended Rod 9^ 53- Current-Force Relationship at Low Current- 95 5^. Current-Force Relationship. . . . . . . 96 55- Ratio of Primary Yield Stresses ^ ^ ^ O " ^ Against Diameter for L2=1000 ^ and L^OOO p.. . 121 56. Ratio of Secondary Yield Stresses V J ° T <^>T Against L2j L 3 Diameter for 1^=500 p. and L3=3500 p. . 122 v i i . TABLES Page I. Tensile : Strength of Whiskers . . . . . . . 2 II. • Shear Strength of Whiskers 3 III. Maximum.Elastic Strains of Whiskers h IV. Comparisons Between Measured and Calculated Values of Area. 9 9 V. Primary Tests . . 1 0 1 VI. Secondary Tests 10k VII. Saimoto's Results ±2k VIII. Brenner's Results 1 2 5 IX. Calibration of Impedance Transducer 9 0 INTRODUCTION 1 . General One of the scientific problems that has puzzled investigators for almost half a century is the discrepancy between the actual and the theoret-ical strengths of solids. As yet, because of the nature of the binding forces in metals, the theoretical strength of metal crystals has not been accurately calculated. However, various estimates of the theoretical shear stress ^ t h ' required to nucleate slip in the absence of dislocations have been made^ . Mackenzie^ arrived at a value of as G/30 where G is the elastic shear modulus in the shear direction. Also, by means of bubble raft studies, Bragg and Lomer^  tended to confirm that the theoretical shear strength of perfect metal crystals is about G/30. In practice, bulk single crystals exhibit only a fraction of their ideal strengths deforming at stresses of less than 10~^G. This large gap between the real and the ideal seemed to suggest that the theoretical strengths were too high. It was not until 1952 that Herring and Galt^ finally resolved this problem. -They found that tiny filamentary growths or "whiskers" of tin had strengths of that predicted by theory. These whiskers were about 2 microns (u) in diameter and a few millimeters (mm) in : length. From simple bend tests they found that these whiskers could withstand an elastic strain as high as 2$, while in bulk t in, flow began at strains lower than 0.01$. Since then, many people have entered the field of whisker research. Many important contributions have been made in the field of crystal growth and in the study of the solid state as certain measurements can be made that are otherwise impossible with bulk crystals. - 2 -2. Previous Work a. Strength of Whiskers I n determining the st r e n g t h o f whiskers, t e n s i l e t e s t s are more s u i t a b l e than bending t e s t s . The disadvantage of a bending t e s t i s t h a t the s t r e s s i s nonuniform both•across and along the whisker. S p e c i a l techniques f o r the t e n s i l e t e s t i n g of whiskers have been devised by Gyu la i5 , and E i s n e r ^ , and Brenner?. Gyulai. t e s t e d sodium c h l o r i d e whiskers and he rep o r t e d a maximum 2 2 s t r e n g t h of llOKg/mm . E i s n e r found a maximum st r e n g t h of 39° Kg/mm f o r s i l i c o n whiskers. This s t r e n g t h i s about 2$ of the Young's Modulus f o r the <^111^ d i r e c t i o n . T e n s i l e t e s t s were performed by Brenner on whiskers o£ i r o n , copper and s i l v e r . The maximum.strengths of the whiskers and t h e i r s i z e s are given i n Table I where f V equals the maximum s t r e s s the whisker ^ max sust a i n e d before f r a c t u r e or y i e l d occured a n d \ ^ m a x equals the c a l c u l a t e d r e s o l v e d shear s t r e s s . TABLE I Te n s i l e Strength of Whiskers Bulk crystals 1 Material 2 d (») 3 C m ax (kg/mm2) 4 Tntax (kg/mm s) 5 T o r l t i o a l (kg/mm') 6 Ultimate tensile strength (kg/mm*) Fe 1.60 1340 364 4.5" 16-23" Cu 1.25 300 82 0.10b 12.9-35.0" Ag 3.80 176 72 0.06b Reproduced from Reference 7. - 3 -Compared with the reported strengths of bulk crystals ^'9 (columns 5 and 6), the yield point shear strengths of the whiskers are 80 to 1200 times greater than those of the large crystals. The ratios between the tensile strengths (columns 3 and 6) were lower. Brenner also found, as Table II shows, that the highest shear strengths of the whiskers were either close to or above TABLE II Shear Strength of Whiskers 1 Whiskers 2 Whisker axes 3 Slip system 4 G(kg/mm») 5 T m i i G 6 Tth — (estimated) G Fe Cu Ag [HI] '111' ;ioo; (110) (111) (Ul) [111] •101" ;ioi; 6100 3700 2300 0.060 0.022 0.031 0.033-0.19 0.033-0.13 0.033-0.13 Reproduced from Reference 7-'. the estimate of the theoretical strength of perfect crystals. To account for this high strength of whiskers, Gait and Herring^ have formulated two theories. One is that whiskers are free of dislocations. The other theory postulates that whiskers contain only a few dislocations and these are insufficient to cause multiplication. b. Elastic Behaviour of Whiskers The most outstanding characteristic of whiskers is their elastic be-haviour. The tensile stress-strain curves of a variety of whiskers have been determined. Brenner? investigated the stress-strain behaviour of copper, iron and silver whiskers while Coleman, Price and Cabrera 1 0 performed stress-strain tests on cadmium and zinc whiskers. Evans, Marsh and Gordon11 found stress-- k -strain curves for sodium and potassium chloride whiskers. The stress-strain curves for silicon whiskers were obtained from bending tests conducted by-Pearson, Read and Feldman1^. The whiskers listed in Table III, except for some of the whiskers grown by precipitation, exhibit a maximum elastic strain TABLE III Maximum Elastic Strains of Whiskers Max. M.-ili:ri;il clastic Method of Method of Mrain testing growth IV •1,9 Tension Halide reduction Cu 2.8 Tension Halide reduction Aj; •1.0 Tension Halide reduction Ni l.H Tension Halide reduction Si 2.0 Tension Halide reduction Zn 2.0 Tension Vapor condensation NaCl 2.<i Tension Precipitation SiO- 5.2 Tension Vapor condensation Al.Cn :<.o Tension Vapor condensation MoO, 1.0 Tension Vapor condensation <:: 2.0 Tension Vapor condensation Sri 2 to 'A Bending Growth from solid (ir. l.H Bending Halide reduction ZnO 1.5 Bending ZnS 1.5 Bending Vapor condensation l.iF 3 Bending - Cleavage MgSO, • 71I=0, Precipitation hyciro(|uinonc, <:tc. > 2 . Reproduced from Reference 29;. of at least 0.01, which is 100 to 1000 times greater than that of annealed bulk crystals. The elastic part of the stress-strain curves are reversible for fast strain rates, but for slow strain rates -Gabrera"'-0'1-^ reported that for some zinc whiskers the strain was not reversible. Brenner? observed large deviations from Hooke's Law for iron whiskers but not for copper whiskers. The true stress-strain curves of two iron whiskers are shown in Fig. 1. Hooke's Law was obeyed up to about 2$ but beyond that, Youngs Modulus, E, was no longer a constant. E was calculted from the in i t ia l slopes of the stress-strain curves. For one whisker this was close to - 5 -1200 / / 1 100 _ / y / / 1000 - / / F , [ , H J / / 9 0 0 - / A / / eoo / E E // 700 t/> tr 600 </» 50O - /° ^' Ft [loo] / / 400 _ ^ O INCREASING STRAIN / A DECREASING STRAIN 300 200 o/ y& C u [lOO] 100 0 -°1 l l l l l l l l .005 .010 .015 .020 .025 .030 .035 .010 .045 STRAIN Figure 1• Stress-strain Curves of Iron and Copper Whiskers Pulled in Tension. Reproduced from Reference l U . the value for a <C.100^ > direction. E was close to the value for a <slll^> direction for the other whisker. The orientation of the whisker, appears to have l i t t l e effect on the elastic behaviour. Coleman et al"^ tested zinc and cadmium whiskers in the diameter range of l-10u. In a l l cases the in i t ia l elastic strain was always linear and reached values of 1-2$. The cri t ical shear stress was several hundred times that observed in macroscopic crystals. They also found, that the calculated moduli were consistently lower than the accepted values by a factor of about 0.7• In a later paper, Cabrera and Price 13 observed that the elastic curve exhibited a deviation from linearity which was detectable at about O.k'fo ' strain. The amount of ..this deviation at 1$ strain varied between 0.02$ and 0.12$ strain. This implied the presence and. motion of dislocations. - 6.-These whiskers also exhibited creep when subjected to a high stress for several hours. They found that after creep at a constant stress, the deviation from.linearity was considerably reduced and that there was no change in the in i t ia l slope (Fig. 2). To explain these results, they postulated, a dislocation network containing Frank-Read sources. Under a suitable stress, dislocation loops wil l be generated. This wil l result in large elastic strains provided the surface is a strong enough obstacle to hold the dislocations inside. Upon removal of this applied stress, most of the loops wi l l collapse into the source. In the case of a large crystal, the Frank-Read source could be far from the surface. Then the surface could not prevent slip at low stresses since a large enough pile-up of dislocations could be formed to multiply the applied stress at the head of the pile-up to ah';'• ImSuntoheces'sa^yT i^dt"break through the barrier. However, in order to produce slip in a small, crystal, the applied stress would have to be increased considerably. If i t is assumed that during creep the high stress destroys the sources leaving only a certain number of loops already created, then the elastic stress-strain curve should "show a smaller deviation from linearity, but the same slope. This is what was observed by Cabrera and Price. c. Plastic Behaviour of Whiskers When the elastic limit of a whisker is exceeded, either fracture or plastic deformation occurs. Brenner1^" has found that in the case of thin copper and iron whiskers, fracture wil l occur with very l i t t l e plastic de-formation i f their elastic limit is very high. One reason is that, following a large elastic strain, the plastic strain, rate is extremely high. In the case of ductile whiskers, Brenner15 observed that.their stress-strain curves, - 7 -CLOMSATIOtt IN MICRONS Figure 2. Effect of Creep on the Force-Elongation Curve (before yield). Reproduced from Reference 13.' (a) Elastic behaviour of a typical whisker up to 0.8l$ strain before creep. (b) Creep strain as a function of time at a constant stress of about U.5 x 10 dyne/cm^. (c) Elastic behaviour after creep. The deviation from linearity begins at strains much higher than i t did before creep. The yield point is increased at least 50$' ElTEIOOft (tl Figure 3»~' Stress-Strain Curves of Copper Whiskers Reproduced from Reference 2,9. as shown in Fig. 3> are characterized by an extremely sharp yield point and an extensive "easy glide" region (a - b). The "easy glide" region was followed by a work-hardening region (b - c). Deformation in the "easy glide" region occurred by the propagation of Luders bands shown in Fig. k. After yielding, one or more.small deform-ation zones were observed. Upon reloading, the slipped region travelled along the whisker until the ends of the whisker were reached. The small, constant stress necessary to propagate this slipped region is called the flow stress Ratios between the yield stress cr'y and the flow stress GTfi as high as 80 to 1 were measured by Brenner. Upon further reloading, work hardening occurred. In explaining these results, Brenner showed that the sharp: yield point cannot be due to dislocation pinning by impurities as postulated by Cottrell"'". He concluded that slip in whiskers was initiated either by the activation of very small dislocation sources already present in the system or by means of some other unknown mechanism. In a later revaluation, Brenner-^ explains that - 9 -Figure k. Propagation of a Luders Band in a Copper Whisker, Mag. 80 X. Reproduced from Reference lk. the flat part of the stress-strain curve is not primarily to "easy glide" but rather is due to the fact that a certain stress is necessary to propagate the front of the Luders band. Brenner compared the flow stress to the shear stresses that produce equivalent amounts of deformation, as in the Luders bands, rather than to the cr i t ical shear stress of bulk crystals. Coleman et al 1^ 1 found that for zinc and cadmium whiskers, yielding occurred at a particular region of the whisker. The propagation of Luders bands at a flow stress 30 times smaller than the yield stress was observed. They felt that this formation of slip bands eliminated any possibility of the whisker being a nearly perfect crystal. Price^" deformed zinc whiskers in tension inside an electron micro-scope and studied the motion of individual dislocations. The whiskers were found to be in i t ia l ly free of dislocations and possessed sharp yield points whose values were determined by stress concentrations at large surface steps - 10 -or at the grips. After a very small amount of plastic strain, Price found dislocations in a narrow zone a few microns wide. The zone extended a l l the way across the crystal. These dislocation's were essentially of two types: (i) long dislocations which were easily immobilized by obstacles, and ( i i ) short screw dislocations some of which broke up into long narrow loops which then split up into circular loops. With further strain the density of dis-locations in this narrow zone increased and a large number of loops was produced. These loops blocked the motion of the long dislocations which in turn acted as obstacles to the glide of the screw dislocations. When the density of the loops was very high, a l l the dislocations became entangled. This resulted in further glide occurring at the edges of this deformed region where only a few loops were present. Thus the width of the deformed region was increased. The propagation of a deformation front which was optically observed as a Luders band was the result of this process. d. Size Effects in Whiskers In 192k, Taylor1? reported that- the tensile strengths of very fine wires of antimony were between 18.0 to 22.0 Kg/mm2. This compared to a tensile strength of 1.10.Kg/mm2 for bulk antimony1^. This was the f irst report of variations in the properties of crystals with diameter. These variations have been termed "size effects". The strengths of whiskers as a function of size have been deter-mined for sodium chloride whiskers by Gyulai^ and by Brenner^ for copper and iron whiskers (Fig. 5)- Brenner found, despite a high scatter, that the strength of a copper or iron whisker was inversely proportional to the diameter. - 11 -\ 4°° o \ \ \ o o 1 J -I 1 1 I I 1 L_ -1 1 I 1_J e to i2 4 « (a) V \ \ e to 12 F i g u r e ^• Strength Versus Diameter f o r (a) I r o n and (b) Cppper Whiskers 1 Dashed Curves are f o r O 3 p r o p o r t i o n a l t o d Reproduced from Reference 7 • In c o n t r a d i c t i o n to Brenner, Eder and M e y e r 1 9 found a r e l a t i o n s h i p between y i e l d s t r e s s ~ r J 0 and diameter as shown i n F i g . 6 . These measured values of T J l a y w i t h i n a s c a t t e r e d r e g i o n whose boundaries showed a l / d ^ r e l a t i o n . I t i s i n t e r e s t i n g t o note t h a t a l l Brenners measurements l i e completely outside the s c a t t e r e d r e g i o n of t h e i r measurements. Eder and Meyer o f f e r e d no e x p l a n a t i o n f o r these d i s c r e p a n c i e s . No dependence of str e n g t h on diameter was observed by Coleman • et al"*" 0 i n cadmium and z i n c or by Pearson et a l 1 2 i n s i l i c o n whiskers. How-ever, i n a l a t e r paper, Evans and M a r s h 2 0 r e p o r t a s i z e - s t r e n g t h r e l a t i o n -s h i p f o r s i l i c o n whiskers of diameters smaller than those of Pearson. They suggested t h a t the reason f o r t h i s was t h a t the s t r e n g t h depends more on surface c o n d i t i o n s r a t h e r than dimensions since f o r various reasons a rough surface i s more probable i n l a r g e whiskers than i n s m a l l ones. - 12 o Eder & Meyer • Brenner Figure 6. Dependence of Crit ical Shear Stress on Whisker Diameter. Reproduced from Reference 19-- 13 -Brenner? also found that i f , after the whisker.yielded, the un-yielded portion was remounted, the strength of the whisker increased as the length decreased (Fig. 7) Eisner^ reported that for silicon whiskers which were remounted after fracture, their fracture stress was somewhat larger than before. This might be the result of nonumiformity of cross-section along the length of the whisker, the thinnest section fracturing f irs t . ///////// - 40 a 30 20 10 WHISKER - MOUNT AT POSITION I X I WHISKER REMOUNTED 1^  AT POSITION 2 0 —1 1—1 lJflL.lilT—L_T_ 1 f I I ( 0 1 2 0 J Z 3 4 5 6 7 6 9 O H I 2 Figure 1. Recovery of Yield Point of a Copper Whisker After Removal of a Slipped Portion. Reproduced from Reference 15. To account for these size effects, various explanations have been proposed. i . Crystal .Perfection Although whiskers have exhibited the potential strength of perfect crystals,it has not been established whether they are structurally perfect. Structural perfection implies here only the absence of extended defects, in particular, dislocations. This does not include point defects such as vacancies,.impurities etc. Brenner'? concluded from his results on strength versus diameter that the whiskers contained a small number of defects which were distributed statistically in a rather complex manner both on the surface - Ik -and in the interior of the whiskers. He thought that the internal defects were probably dislocation sources of the type postulated by Frank and Read. The resolved- shear stress necessary to operature this type of source is given by ~C=£b where G is the shear modulus, b the Burger's vector of the dislocation segment and L the length of the pinned dislocation segment. In the copper and iron whiskers tested by Brenner, the length of the dislocation sources-must be of the order of 0 . 1 microns. However, in a later paper, Brenner 15 stated that i t was uncertain i f a dislocation source of such short length could operate. Also, i t was not clear why the dislocation sources were only a small fraction of the whisker diameter. Tests on silicon whiskers and silicon rods cut from bulk silicon were performed over a range of temperatures by Pearson, Read and Feldmann-'-2. According to Cottrel l l , the yield stress of a perfect crystal would vary in-significantly over the temperature range 600°C to 800°C. Pearson et al found that the yield stress of silicon whiskers at 800°C was less than half the yield stress at 650°C (Fig. 8 ) . Thus, Pearson et al concluded, these high-temperature tests on the whiskers showed that room-temperature fracture strength was not an adequate criterion of crystal perfection. It was also found (Fig. 9 ) that for small enough diameters of silicon, their room-temperature fracture stress was the same as for the whiskers, but the yield stress was the same as bulk silicon at 800°C. This was further evidence that the room-temperature fracture strength was not due to a low dislocation density. - 15 -0 ,0.002 0.004 0.006 0.008 0.01 002 0-04 0.06 Figure 8. Dependence of the Strength of Silicon Whiskers on Temperature. Reproduced from Reference 12. 2 O 4 a — o 0 RODS • WHISKERS • ->.. . \ t c V C R O S S SECTION IN S Q U A R E C E N T I M E T E R S Figure 9. Tensile Strength of Silicon Rods and Whiskers Reproduced from Reference 12. - 16 -PP On the other hand, Gorsuch , who measured the density and distribu-tion of dislocations in iron whiskers by means of X-ray rocking curvesj found that the more perfect whiskers have dislocation densities below 10°" dislocations per cm2. Therefore whiskers of less than 10^ in diameter would contain, at most, only a small number, of dislocations and should behave as perfect crystals. Various attempts have been made to resolve this question of whether whiskers are strong simply because they are small or whether high strength is peculiar to whiskers. Costanzo2^ attempted to determine i f there was a size effect with fine polycrystalline copper wires. Wires down to 50p. in diameter were tested at room temperature and at -195°C• At room temperature, Costanzo found no definite size effect, "bjftat -195°C he found that the yield,, stress decreased with diameter. Shlichta 2^ also tried to compare metal whiskers with other types of filaments. He used Taylor-process wires1? and electropolished drawn wires. These exhibited an increase in strength with decreasing diameter comparable to that observed for whiskers. However, in a later letter to Costanzo, he stated that these results were fortuitous. The tentative conclusion made by Shlichta was that the relation between size, strength, growth mechanism and crystalline perfection was more complex than originally thought. i i . Effect of Surface Defects It could be assumed that the yield stress is determined by some other type of imperfection rather than free dislocations or dislocation sources. Brenner1^ thought i t more likely that dislocations are nucleated at submicroscopic imperfections on or near the surface. Experimentally, Brenner15 found that yielding could not be induced in an elastically strained iron whisker by rubbing another iron whisker over i t . This would indicate that i f there are any defects, they must be of a specific nature. Pearson - 1 7 -1 2 et al found that the fracture stress in bulk silicon could be raised by • (/. etching which suggested that surface irregularities were the important imperfections. i i i . Grip Effects Fleischer and Chalmers25 calculated the average shear stresses caused by the bending moment applied by the grips during a tensile test of a single crystal. It was shown that^ in general, the resolved shear stress on the various crystal planes was the sum of the contributions from the applied stress, and the grip s t r e s s ^ : where m^ = cos 9 * cos X. . , the Schmid factor for the j th slip system, J J j = the angle between the specimerit axis and the j** 1 slip direction, 0^= the angle between the specimen^ axis and the j*'*1 slip plane normal, 'X^= the grip stress resulting from slip on the i ^ 1 system, n .^= the-fraction of the grip stresses arising from slip on ^ ' system i that is resolved on system j . In case of slip on the primary system, the resultant stress '"^p i - s : ^ ~m ( o * - £ E (a/L) 2 t a n 2 A ) 'P a where a = crystal diameter L •» crystal length It can be- seen that the effect on yield stress, since i t is developed at small strains, is virtualy negligible. e. Effect of Surface Films on Whiskers It was thought that the "presence of a thin oxide film on metal whiskers might contribute to their strengths. It was found by Roscoe , and Cottrell and Gibbons2?, that an oxide film increased the strength of a crystal and also that the thicker the oxide, the greater the strengthening - 1 8 -effect. This increased strength was explained as resulting from the hin-derance of dislocations moving out of the crystal by the film. 13 Cabrera and Price J :found that the yield points of zinc and cadmium whiskers were increased by a factor of about 2 by the presence of an oxide layer. They found an optimum- thickness of the oxide the order of tens of angstroms. Beyond this optimum value, i t is possible that nonuniform oxidation wil l weaken the whisker at several points. Brenner"? formed continuous oxide films on copper whiskers by heating in air at 1 0 0 ° C to 1 5 0 ° C , but found that their strengths were not 2 8 significantly changed. Saimoto , tested copper whiskers in dilute sulphuric acid and also found that the oxide coating did not contribute appreciably to the strength of the whiskers. f. Effect of Impurities in Whiskers Most of the whiskers which have been tested are not exceptionally pure. Brenner^ found that copper whiskers grown from Cul contained about 3 0 ppm of silver while iron whiskers grown from FeBrg. contained about 1 0 0 ppm. Impurities could strengthen the whiskers by pinning the few dislocation sources that may be present in the whiskers. Brenner'? stated that in the case of copper, dislocation pinning does not'contribute to the strength of the whisker. If the reverse was true, a strong temperature and time dependence on strength-would be expected. This was not found. By using purified Cul, Brenner"^ grew copper whiskers containing less than 1 ppm of silver, but nq; signif-icant change in strength was observed. However, i f a few percent of silver halide was added to the Cul from which the whiskers were grown, Brenner-^ reported that their strengths were about l/3 that of pure whiskers. Hence, during whisker growth, impurities may weaken the whisker by forming dislocation sources. 3. Purpose of Present Investigation The main purpose of this investigation was to extend the study of size effects in copper whiskers both for length and diameter. - 20 -EXPERIMENTAL PROCEDURE 1. Growth The copper whiskers used in this investigation were grown "by the method of halide reduction hy hydrogen ^9. Standard reagent grade anhydrous cupric chloride was used. The maximum limits of impurities as stated by the Allied Chemical and Dye Corporation are: Insoluble Nitrate (HOg) Sulphate (SO )^ Iron (Fe) Substances not precipitated by H^ S (as sulphates) 0.01$ 0.005$ 0.005$ 0.01$ 0.20$ The whiskers were grown in a tube furnace as illustrated in Fig. 10 and 11. The procedure in making a growth run was as follows. The Figure 10. Whisker Growing Furnace Figure 11. Schematic Diagram of Whisker Growing Furnace - 22 -Vycor tube was cleaned with nitric acid, then rinsed and dried with acetone. The hydrogen was f irst passed through a catalytic purifier, and then through a drier of molecular sieves which removed any water that might be present. The helium was passed through a liquid nitrogen cold trap and then, 'for convenience, also through the molecular sieves. The flow of gas was opposite to the. direction in which the boat was inserted in order to prevent air from entering and contaminating the hot furnace. The helium was f irst turned on and then a fireclay boat, which had previously been dried to remove water, was f i l l ed with cupric chloride and pushed into- the cooling chamber. The furnace was then flushed with helium for about 20 - 30 minutes after which the flow of helium was replaced by hydrogen. The boat was then pushed into the furnace... After the reduction, which lasted anywhere from 5 minutes to one hour depending on the temperature, the boat was drawn from the furnace to the cooling chamber. After the boat had cooled down,- the flow of. hydrogen was replaced by heliium. The cool boat was then removed and placed in a dessicator. 2. Selection When a suitable boat of whiskers was obtained, i t was examined with a Reichert stereoscopic microscope at 12 and 36 magnification (Fig. 12). Upon location of a good whisker i t was removed at the substrate by a pair of very fine straight tweezers. The insides of the claws of the tweezer were met-allographically polished and degreased in order to prevent the whisker from sticking to the claws. The whisker was then; placed on a white card and examined under a Reichert metallograph!c microscope at 110 and 390 magnif-ication. The criteria for whisker selection were these: • (l) straight, untapered and at least 3 long; (2) no surface defects such as pits or short branch growth; F i g u r e 12. Whisker Handling Apparatus - 2k. (3) the surface facets must not change along the length of the whisker. If the whisker met the above requirements, i t was stored in a de'ssicator for future use. 3» Specimen Mounting A pyrex' probe with a U-shaped tungsten filament, the current through which was controlled by a foot switch, was used to transfer the whisker from the card to the tensile machine. On the tip of the probe a blob of glue was melted. This mounting compound was diphenyl carbazide which melts at 173°C• The whisker was picked up with the tip of the probe by melting and then freezing the glue. . The current to the two grips of the tensile machine was adjusted until the temperature was well above the melting point of the glue. The glue was prevented from boiling off by subjecting i t to a stream of cold dry, helium. This produced a thin, tacky skin of glue on each filament. The whisker was mounted by touching the free end to the movable filament, A. The probe was then lowered until the whisker came into contact with the glue on the fixed filament, B. At this point the probe was removed by heating the tip and carefully withdrawing i t from the surrounding glue. The whisker was then ready for its in i t i a l tensile test. These in i t ia l tests wi l l be referred to as "primary tests". With this method i t was possible to mount successfully about two out of every five whiskers without contaminating the whisker with glue, yielding the whisker or forming a thick oxide coat of the whisker. For a test, in which the unyielded portion of a whisker was re-mounted, the following procedure was used. After the whisker yielded, i t was "examined under the Reichert stereoscopic microscope to see where the formation of Luders bands had occurred. If yielding had started at one point near one - 25 -of the grips, then i t was possible- to,remount the whisker. The filament nearest the Luders bands was heated until the glue began to sag away from the whisker The whisker was then removed by quickly screwing away filament B. This left the whisker attached to one filament. The whisker was then remounted by either slightly raising or lowering filament B and then screwing i t back until the yielded portion of the whisker rested on the glue. A blob of glue was then carefully dropped over the yielded portion. Tensile tests now performed were called "secondary tests". This remounting method combined with the in i t ia l mounting technique and also taking into account where the whisker yielded, gave about one suc-cessful test out of. every ten attempts. k. Measurement of Specimen Length The length of the whisker was measured by a graticule in one of the 6X eyepieces of the Reichert stereoscopic microscope. The hairline in the graticule was divided into ten main divisions each one of which was itself divided into ten small divisions. The length of a small division was 80u with the 2 X objective, 26.7u with the 6X objective and l6p. with the 1 0 X objective. In a given measurement, the error in length was + 1 . 0 division for each end of the whisker. Hence, for the lengths tested the maximum error in measuring the length was 6$. 5- Determination of Cross Sectional'Area An in i t i a l cross sectional area of a whisker was determined by measuring the diameter of the whisker with a microscope equipped with a travelling eyepiece. A circular cross section was assumed since for the cross sections observed this approximation is satisfactory. This in i t i a l value was then used in computing the stress on the whisker, and a stress-- 26 -strain curve for the whisker was plotted. From this curve, a correction to the in i t ia l value of c^oss sectional area was made in the following manner. The slope from the stress-strain curve gave E , Young's Modulus, and this value was compared to the values of Young's Modulus E c for the three main directions of growth, the [ i l l ] , [lio] and [lOcQ directions?. These values of E c were computed from the formula | = S l l - 2 ( S l l - S 1 2 - S ^ / 2 ) ( V 1 + t | * \ + where Sj_j are the elastic compliance constants which were computed from Overton and Gaffney's 3 0 values of C , , , C n „, and C, , . ^ - , 0 0 are the cosines of the '11' -12' kk 1>2>3 angles formed hy the axis of the specimen with the three edges of the unit cube. In almost a l l cases F^ was close to one value of E c . Using this value of E c , a new stress-strain curve was plotted. From any point on this new plot, a new value of stress for that point could be calculated since stress = E c (strain). In turn, from this new stress, a more correct value of cross sectional area was calculated.. 6. Tensile Testing Apparatus a. Construction The whisker tensometer used in this investigation was the same one used.by Saimoto, and hence only a brief description of the tensometer wi l l be given here. This tensometer was constructed following a design similar to that of Brenner-^ and is illustrated schematically in Fig. 13 and is shown in Fig. lk. Basically i t consists of a fixed mount to which one end of the whisker is attached. The other end of the whisker is attached to a suspended rod in which an Alnico permanent magent is imbedded at a suitable position. When the solenoid surrounding the magnet is activated, the magnet and .rod are pulled towards the solenoid centre thus importing a force on the whisker. The - 2 7 -extension of the whisker i s measured by means of an impedance transducer. An important f e a t u r e of t h i s apparatus i s a micrometer-stop which i s used as a brake to i n t e r r u p t p l a s t i c f l o w a f t e r y i e l d i n g . I f the l o a d i n g f o r c e i s not reduced immediately a f t e r y i e l d i n g , f l o w occurs so r a p i d l y t h a t the deformation cannot be f o l l o w e d . I t i s a l s o used t o c a l i b r a t e the. impedance transducer. Figure lk. Whisker Tensometer Impedance Transducer Micrometer Stop EX Fixed Pyrex Arm, B Mobile _> Pyrex Arm, A -Damping Device T.C._ Bridge Ammeter Figure 13. Schematic Diagram of the Whisker Tensometer ro CO - 29 -b. C a l i b r a t i o n The tensometer was. r e c a l i b r a t e d us ing the same method as desc r ibed by Saimoto. The r e s u l t s of t h i s r e c a l i b r a t i o n agreed very w e l l w i th the r e s u l t s o f Saimoto, the d i f f e r e n c e be ing 2$. A l l the c a l i b r a t i o n curves are found i n Appendix I. . . 7- Anneal ing Furnace An anneal ing furnace was const ructed out of a g lass T - j u n c t i o n . The top of the T was wound with Chromel A, l/8" r i bbon . The range of temp-eratures i n which t h i s furnace could be used was.'. l imited by the low mel t ing po int of the g lue , i e . 173°C. The temperature was c o n t r o l l e d by a thermo-couple which was embedded i n the glue, on the f i x e d g r i p . The furnace r e s t e d on a s t r i p of i n s u l a t i n g m a t e r i a l on an aluminium b l o c k and was p o s i t i o n e d over the f i x e d g r i p c l e a r i n g the end of the g r i p by about 2 i n c h e s . I t was then p o s i t i o n e d over the whisker which was annealed i n an atmosphere' of hel ium that .entered the furnace through the bottom of the T. 8. Exper imental Procedure a . T e n s i l e Tests A f t e r a s u i t a b l e whisker had been obtained and p laced on a white ca rd , a smal l drop of glue was p laced on one end of the whisker which secured i t to the c a r d . Th is was found to be necessary as otherwise the whisker could e a s i l y - b y blown away by a smal l gust of a i r . The diameter was then measured. A p o r t i o n of the whisker was then removed by c u t t i n g i t w i th a razor b lade which had been degreased with acetone. I t 'was then mounted on the tensometer and p u l l e d . Most whiskers were cut i n t o two p i e c e s , but a few were long enough to d i v i d e in to three or fou r p i e c e s . - 30 -"b. Annealing Tests Annealing tests were performed in two ways: (i) A whisker was mounted and pulled until i t yielded and flowed. The whisker was then detached from grip A in the manner described in the section on specimen mounting, and the annealing furnace was slipped over the whisker. It was annealed in a helium atmosphere for 12 - lk- hours at 100°C. After annealing the whisker was remounted and pulled. ( i i ) After the whisker was detached from grip A as above, i t was removed from grip B with a pair of tweezers. It was then placed in a porcelain boat and the glue was dissolved from the end of the whisker with acetone. The boat was then wrapped in aluminium f o i l which was perforated with a needle. The boat was then placed in the whisker growing furnace and annealed under an atmosphere of hydrogen for 1 - 2 hours at'400°C. Afterwards the whisker was remounted and pulled. - 3 1 -EXPERIMENTAL OBSERVATIONS AND. RESULTS 1. Growth Observations Cupric chloride reacts with water in the following manner: CuCl2•+ H20 - CU(0H)2 + 2HC1 The presence of this Cu(0H)2 in the Cucl 2 has a very disastrous effect on the growth of whiskers. If as received anhydrous cupric chloride was used, in almost a l l instances, no whiskers were produced. It was thus found necessary to purify the CuCl 2 by passing dry HCI gas through i t . The quality and quantity of the whiskers grown from the purified CuCl 2 decreased rapidly with increasing exposure to the air as CuCl 2 is very hygroscopic. Whiskers were grown in the temperature range of 550°C to 800°C. The reproducibility of whisker growth for apparently the same conditions is not very good and hence only geaeral tendencies are listed below. (i) At temperatures between 550°C and 700°C the whisker growth was heavy with most of the whiskers being less than 15 p in diameter. Some boats of whiskers contained very long ( l - U cm) whiskers, but these usually were tapered or contained surface irregularities such;as-very short branch growths. ( i i ) At temperatures above 700°C the whiskers were usually coarse (up to 100 p. in diameter) and the growth was not as profuse as in ( i ) . ( i i i ) The hydrogen flow rate did not seem to effect the growth results provided i t was above a minimum rate of about lOOcc per minute. The flow rate was usually between 200-300cc/min. - 3 2 (iv) Aside from the presence of any Cu(0H)2 in the CuCl 2 , a very important factor determining the quality of the whiskers was the cleanliness of the reducing atmosphere and surround-ings. If the Vycor tube was cleaned before each growth run, much better whiskers were obtained than i f several growth runs were done in the tube before cleaning i t . Most of the whiskers were straight, usually with either short branches or other types of surface defects. However, as also reported by 29 2 8 Brenner ' and Saimoto , a large variety of other shapes such as polygonal spirals, circular and polygonal helices, twists, kinks and many others such as the one shown in Fig. 15 were observed. Figure 15. An Odd Shaped Whisker, Mag. UOOX. 2. Cross Sectional Area Table IV, Appendix II, compares the results, in measuring the area optically and in calculating the area, by the method previously described, of the whiskers. This calculated value of area was used in a l l the stress calculations. In most cases, the agreement between the' two values of area was good. . In the optical measurements, of diameters a minimum error of 2 divisions or 0.7 P- was possible. 3 . Tensile Tests a. Reliability of the Quantitative Measurements In measuring the yield stress of the whiskers, the two major sources of error are the force calibration of the apparatus and the cross sectional area determination of the whiskers. The maximum error•introduced due to the compensation for restoring force is about lOmg. This would give an error in the yield stress of, at the very most, 2$, since the smallest load measured at yield was about ^OOmg. Hence the probable error for the . load measurements, which includes calibrating (5$) and.compensating errors, is about 7$-The error in measuring the cross sectional area was discussed in 28 some detail by Saimoto . He showed that the,assumption of Young's Modulus being the same for both whiskers and bulk crystals is reasonable. The error is due to the assumption that the whiskers have orientations of t i l l ] [lOO] , or [ l l 0 3 . Saimoto found in his•investigations that the whiskers which possess axes off the low index ones do so by about 15°. This would introduce a maximum error in the Young's Modulus of 10$ (Appendix III). Therefore the values of stress may have an error of as much as 16$. However, i t should be noted that most whiskers do have low index axes and hence this error in stress is an extreme. - 3h -b. Statistical Treatment of the Data •Since a comparatively.large number of results was obtained,-a .statistical .treatment .of these results was'made'. The methods used, are dis-cussed in Appendix IV. Also, the probable error of the calibrations in Appendix: I was computed.statistically and .is shown on.the graphs. c. Primary Tests The results of the primary tests are listed in Appendix II, Table V'. From these results,, the yield, stress, O"3 , was plotted.against the diameter, d (Fig. l6) . In order to determine the best line to draw through these points, log 0 ° m w a s plotted against log d (Fig. 17) since CP was n as.sumed to be a function ; of d . The slope, n,. of the band containing the points was found to be -i.6. The method used in calculating this slope - •> 1.6 is found in-Appendix XV.. Thus C T m was inversely proportional to d In comparison, as was previously mentioned in the introduction, Brenner found the yield stress of his whiskers proportional to l / d . Brenner determined,his relationship by plotting the average value of yield stress of whiskers of approximately the same diameter against l / d a v e (Fig. 18). This method was tried for the results obtained in this investigation, but i t proved very unsatisfactory since approximate straight lines could be drawn both for CT > m versus l / d and l / d 2 . l 6 A plot ofC5* against l /d " (Fig. 19) gave that C T L = lj?71 •+ 2.8 Kg/mm2 (d in microns) (1) 7T76 In this equation, was replaced by <yT>c for convenience in future cal-culations. -The line drawn through.the points on Fig. 16 was calculated using the above equations. - 37 -Figure.18. The Average Strength-of Copper Whiskers as a Function of the Reciprocal of the Diameter. Reproduced from Reference 7. - 38 -The graph of against length, • L, is shown in Fig. 20. The m various symbols denoting primary tests, secondary tests and annealing tests were used to distinguish between whiskers which were.later retested and those that were not. -As can been seen.for Fig. 20, no particular dependence of CT1 on L is apparent. However, since this plot was for whiskers of various diameters as well as lengths, any trend could well be. hidden by the dependence . of 0 ° m on d. Therefore it was decided to use a normalizing procedure to convert the strengths of whiskers of various diameters.to comparable strengths for whiskers of one diameter. The normalizing procedure was as follows. Consider a plot of yield stress against diameter d as shown in Fig. 21. Let a whisker of diameter d^  and of strength be chosen as a standard S a-whisker. If a whisker of diameter dQ with strength was how considered to be a whisker of diameter d s , its strength would then be In the case of a whisker with the same diameter d„, but now of strength c ' . m (position A), a comparable strength of a whisker of diameter d s (position is given by = ° ° m . ( 2 ) n s A standard whisker of 10 p. in diameter was chosen since many of the whiskers had diameters around this value. Using equation (1) to calculate the value of O^g for d = 10 p., equation (2) becomes = i+2.3 ^ m (3) The values of ^5"^. for a l l the whiskers was calculated using equation (1) and then O"^ was calculated from equation (5). These results are.listed in Appendix II, Table V. Fig. 22 shows the plot of against L and again these is no apparent trend. Since the scatter was quite high i t was possible that any trend could be masked by this scatter and so 320 o 280 ^ £ = 1 5 7 1 + 2 . 8 (Kg/mm2) ,1 .6 2U0 200 o 160 o o 120 80 ho o o . 0 o o o o o 0 0 8 •10 i / d 1 - 6 ( l o - V 1 - 6 ) 12 11+ 16 F igure 19- Primary Y i e l d S t ress Against Diameter. 320 o 280 240 Q Primary Tests Q Whiskers Used in Secondary Tests k?200 — Q Annealing Tests o o »160 !H •P CO £1201 80 o 0 0 o 0 • °Q> ° o • i *o|_ o Q • O o • Q O O o o D o o o o • 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.it-Length L (mm) Figure 20. Primary Yield Stress as a Function of Length. - 1+1 -•Diameter d Figure 21. Diagram Illustrating Derivation of the Normalizing Procedure. O Primary Test Q Whiskers Used in Secondary Tests O Annealing Tests Theoretical Volume Dependence Curve ^5o| \ Theoretical Surface \ - Dependence Curve 20d_ 1 2 0 l _ \ \ 8 d V \ 4ol • • O ^ ^ ^ V v • 0 .4 0 . 8 1 .2 1 .6 2 . 0 2 . 4 2 . 8 3 . 2 3 . 6 4 . 0 4 . 4 4 . 8 5 . 2 Length L (mm) Figure 2 2 . Normalized Primary Yield Stress as a Function of Length-- U3 theoretical curves> assuming either a volume dependence or a surface de-pendence for yield stress, were calculated and.plotted on Fig. 22. The method of calculating these curves is given in Appendix V, Part A. In the case of a volume dependence, for whiskers of diameter 10 u and varying in length from 1000 p. to 5000 u, a variation in of approximately 3 [Appendix V, Part B], would be expected between the two limiting values of L. However, since the scatter is of this order, • any volume dependence would not be observed. Similarly, in.the case of a surface dependence, a variation of about 10 would be expected. This was not observed. It would thus seem that ttf there is a length dependence for "yield stress, i t is so small as to be unobservable. d. Annealing Tests The annealing of whiskers mounted on the tensometer was very unsatisfactory. Only one test out of about twenty attempts was successful (whisker GG). The major problems were the low melting point of the mountin compound and the tendency of this compound to contaminate the whisker by forming a coat on its surface. Three other successful annealing tests were obtained by method (ii) as described on page 30. Since the results of the tests on these•annealed whiskers did not vary significantly from the results of secondary tests on ordinary whiskers they wi l l be included in the next section. - kk -e. - Secondary Tests Appendix II, Table VI l ists the results of the secondary tests. The primes refer to the number of times that each whisker was retested. As can be seen in Figs. 2 0 and 2 2 , the in i t ia l values of yield stress for secondary tested whiskers show no particular patterns of their own. These results were treated in a similar manner as the results in the previous section. However, the area used in calculating the yield stress was the same as the area "calculated from the primary test and not from the secondary tests on the same whisker since the diameter of the whisker was assumed not to change. - As before, yield stress } w a s plotted against diameter, d (Fig. 2 3 ) and log 0^m w a s plotted against log d (Fig. 2k). The slope of. the band containing the points was found to be - 2 . 5 , so that 0 ° m was now inversely proportional to d 2 ' - \ The plot of 0 ° m against l / d (Fig. 2 5 ) gave O 3 = MJ-25 + 3 9 . 0 Kg/mm2 (d in u) (k) d 2'5 with O"3 again replacing C 7 ° m for convenience. .The line drawn through the points on Fig. 2 3 was calculated by this equation. The graph of 0 ° against length L is shown in Fig. 2 6 . In this m case there seemed to be an increase of strength with a decrease in length. However, as this plot did not take into account the fact that the whiskers were of various diameters, these Whiskers were normalized in the same manner, as in the previous section, to whiskers of 1 0 u in diameter. The normalizing procedure gave that - ^ 5 -2 3 ^ 6 8 i o 20 30 Diameter d ( u) Figure 2k. Log Plot of Secondary Yield-Stress Against Diameter. l U O ^ — 120 bO to <u u -P cn rH 100 80 L _ 6 o L _ ko 20 C^r» = ^ 2 5 +39.O (Kg/nun2 d "2 Figure 23, 12 lh l / d 2 " 5 ( ^ ) 2 - 5 10  k 16 Secondary Yield Stress Against Diameter d . 18 o o o o o o o o o o o o (P o o o o o o 0.0 3.6 - h9 ~ and the.resuits both for values of and are given in Appendix II, Table VI. The plot of C T > n against L is given in Fig. 21. The values of , 2 for most of the points l ie between about 3 0 - 7 0 Kg/mm . This compares reasonably well with the results obtained for the primary tests (Fig. 2 2 ) where the points have values of between 2 0 - 8 0 Kg/mm .^ -Again theo-n retical curves for the volume and surface dependence of yield stress were calculated (Appendix V, Part A). .For the case of volume dependence for whiskers of 1 0 y. in diameter with the limiting values of L being 5 0 0 p. and 3 5 0 0 p., a variation of around 2 (Appendix V, PartB) would be expected. From Fig. 2 7 i t can be seen that any volume dependence on stress would be masked by the scatter. In the case of a surface dependence, the variation expected is about 6 . This is not observed. Thus the results for secondary tests are about the same as those of primary tests, ie. any length dependence of yield stress was so small as to be unobserved. f. Variation of Young's Modulus A rather surprising result obtained from the secondary tests is that there is an apparent change in the Young's Moduli of these whiskers. Appendix II, Table VI, l ists the values of E/E^ where•E. is the value of the Young's Modulus obtained from the primary test on the whisker and E is the value-obtained from the secondary test. Fig. 2 8 is a diagrammatical re-presentation of the variation of E/E^ with E^. Figs. 2 9 and 3 0 show stress-strain curves for two whiskers, both of which were retested. The stress-strain curves of both the primary and the secondary tests were reversible. Initial ly, values of E and the ratio: . . of E/E^ were plotted against such parameters as the in i t ia l elongation of the whisker (Fig. 3 1 ) , diameter,, in i t ia l length, etc, but no correlation was found. -However, when the ratio E/Ej_ was plotted against the normalized value of primary yield, stress, Theoretical Volume O Dependence Curve o o o CO Q ? ° Q O o Theoretical Surface Dependence Curve - o 0 . 8 1.2'- 1.6 2 . 0 2 . 4 2 . 8 3 . 2 3 - 6 Length L (mm) Figure 27. -Normalized Secondary Yield Stress as a Function of Length. 1.1-i.o. 0.9. 0.8. 0.7 0.6 0.5 o.k 0.3 0.2 o o o o o • o o E = Young's Modulus Obtained from Secondary Test. E^ = Young's Modulus Obtained from Primary Te st. E± = 0.68(10^) Kg/mm2 for [100] Whisker E ± = 1.34(10^) Kg/mm2 for [110] Whisker Average Value 0.1. 0.0 0.68 1.3U E ± (lO^g/mm2) Figure 28. Plot of the Ratios E/Ej_ of Young's Moduli Against E ± . 20.0 o.o 0.2 o.h o.6 0.8 l.o $ Elongation G Figure 30. Stress-Strain Curve of Whisker GG. o 1.0 o 0 . 8 0.6 o.U 9 o o o o o o 8 o o o o o o o o o o o o 0.2 o o . d 0 . 0 0 . 2 0.1+ 0.6 0 .8 . 1.0 1.2 1.1+ I n i t i a l $ Elongation €EL m . F i g u r e 31. R a t i o n E/E^ of the Young's Moduli Against I n i t i a l E l o n g a t i o n . - 5 6 -(Fig- 3 2 ) , i t was found that E/E^ increased with an increase of 0 ° n up to a value of O - 3 equal to about 8 0 Kg/mm2 where the curve seemed to level off. The points in the squares were determined by averaging values of E/E± over intervals of 0 ° n from 1 0 - 2 0 , 3 0 - kO, 5 0 - 6 0 , 6 0 - 7 0 and 9 5 - 1 1 5 Kg/mm2. g. Summary The results of the primary test showed the following: (i) the yield strengths of the whiskers were proportional to d"^ "' ( i i ) no observable dependence of length of yield strength. The results of the secondary tests showed the following: -2 (i) the yield strengths of the whiskers were proportional to d" ( i i ) no observable dependence on length of yield strength, ( i i i ) an apparent decrease in the Young's Modulus of about 3 0 $ . - 57 -DISCUSSION 1. Growth While the study "of whisker growth was beyond '.the scope of t h i s i n v e s -t i g a t i o n , a b r i e f d i s c u s s i o n of the e f f e c t of the presence of water on whisker growth w i l l be given f o r the sake of completeness. The - c l a s s i c a l theory of c r y s t a l growth .considered a c r y s t a l t o be s t r u c t u r a l l y p e r f e c t and assumed t h a t each time a step on the c r y s t a l swept over the s u r f a c e , a new one had t o be nucleated on the f r e s h l y completed c r y s t a l l a y e r . A c r i t i c a l s u p e r s a t u r a t i o n was r e q u i r e d f o r continued growth because the c r e a t i o n of a step on the surface i n c r e a s e d t h e surface energy of the c r y s t a l . However, s i n c e i t was found t h a t c r y s t a l s grew at super-s a t u r a t i o n s which were immeasurably s m a l l , F r a n k ^ 1 concluded t h a t r e a l c r y s t a l s were not p e r f e c t , but .contained screw d i s l o c a t i o n s which were, de v e l -oped .during the e a r l y stages of t h e i r growth. I t was these screws which provided the c r y s t a l s w i t h permanent growth.steps. For the case of whisker-growth, i t was proposed by Sears-^ 2 t h a t whiskers contained a s i n g l e a x i a l screw d i s l o c a t i o n w i t h the l a t e r a l c r y s t a l surfaces bounded by surfaces which are a t o m i c a l l y smooth. As has been p r e v i o u s l y mentioned, whiskers were grown by the hydrogen r e d u c t i o n of CuClg. A c t u a l l y t h i s w i l l f i r s t be-reduced .to CuCl •3-3 which then disproportionates-'-' t o g i v e Cu and CuCLp^ Without going i n t o the thermodynamics of i t , the reduction;, p o t e n t i a l of the hydrogen w i l l be lowered by the presence of any HgO. However, i n t h i s case, the p o t e n t i a l i s s t i l l s u f f i c i e n t enough t h a t a l l the-CuCl^ i s reduced t o copper. - 58 -At present, because,of experimental results and thermodynamic 3 4 3 5 considerations ' , . so far only one theory of the growth of whiskers by hydrogen reduction is acceptable. This theory states that the liquid halide is transported up the walls of the whisker. This transport is then followed by catalytic decomposition at the whisker t ip. It has been observed by -Shetty that there is a growth step for each face at the tip of the whisker. HgO molecules are-strong dipoles with a.dipole moment of 1.7(10"-'-^) 3 7 esu. Sarakhov • studied the adsorption and .desorption of water vapour on gold fo i l at l8 ° C . The desorption never proceeded to zero concentration of RVJO at l8°C even after ten days evacuation and required 30-40 hours evacuation at 450°C. Allan and Webb-38 observed that when a boat containing CuCl was pushed into the furnace, growth occurred only after the boat became coated with a film of copper. Hence, i t is not unlikely that the whisker growth sites, would be poisoned by the presence.of any H2O dipoles. This process would therefore prohibit the growth of the whiskers. Also, even when a whisker had started growing, these dipoles could inhibit the growth by adsorbing on one or more of the faces at the whisker t ip. This would account for the...decrease in the quality of the-whiskers that are produced. 2. Comparison of Results With Previous Work The results of tensile tests performed on copper by Brenner-*-5 and 28 Saimoto are listed in Appendix VI. Fig. 33 is a comparison of the resolved cr i t ical shear stress '^tj c r, against diameter for whiskers of orientations [lOO^ , [ l lO^ and [ i l l ] as found by the•author,.Brenner and ..Saimoto. The points obtained by the author and Saimoto show, despite a high scatter, that there is a dependence, of the-critical resolved shear stress on diameter. However, in the case of Brenner, no particular dependence is observed. This 90 59 TO 60 Moore O Brenner G3 Saimoto 50 OJ kQ bO 30 20 10 Kl SI O o o P o # 6 # • H # ° # V o o o o . o • o • o 12 Diameter d( p.) 16 20 2k Figure 33. C r i t i c a l Resolved Shear S t r e s s Against^- Diameter. - 60 -is most .likely a result of. the relatively small number .of points obtained, most of which were for whiskers with diameters greater than 8 p. for which- the dependence of "tj c r . on .diameter is not so pronounced. It has been found.that the yield stress does depend on,diameter. Now- If, in fact, the cr i t ical shear stress did not depend on diameter this would lead to the rather startling conclusion.that the yield.stress of a whisker would no longer depend on the amount of stress necessary to cause shear on the slip plane. Therefore the yield.stress would have to depend on some other factor. Figs. 3^>.35 and 36 are plots of the cr i t ica l shear stress against diameter for whiskers. of orientation ^lOO^J , j^ l i o j and |jLllJ respec-tively. The.curves drawn through these.points were obtained from the .equation which relates yield stress against diameter as follows: =1571 +.-2.8.Kg/mm2 (d in microns) d l.6 This equation was.then multiplied by the appropriate Schmid factor.to give the :correct curve for the various orientations. For the case of whiskers with a ,j_10o3 orientation (Fig. -3*0/ "the points l ie reasonably well along the .curve. For whiskers with either a [ll0~j (Fig. 35) or a [_ 111"] (Fig. 36) orientation, the few points lie-more or less on'the curve. As was previously mentioned in the introduction,.Eder and Meyer^ found that the cr i t ica l shear stress was proportional to d . Fig..37 compares the results obtained by.Eder and Meyer with those obtained by the author,.Brenner and.Saimoto. As can be seen,from this graph,, the values of e r observed by Eder and Meyer are, on the whole, much lower than those observed by the.author, Brenner and..Saimoto. In fact,,Eder and Meyer's - 6i -0 k 8 1 2 1 6 2 0 . 2k Diameter d ( u) Figure 3^ - . C r i t i c a l Resolved Shear St ress f o r \l00~\ Whiskers Against Diameter. - 6 3 -7 0 6 o 5 0 ko OJ bD. ^ 3 0 o 2 0 1 0 Moore O Brenner gj Saimoto o 1 6 8 1 2 Diameter d( jx) Figure 3 6 . Crit ical Resolved Shear Stress for [ i l l ] Whiskers Against Diameter. 100 2 3 4 6 8 10 20 30 4o d( p). Figure 37- Log Plot of the Crit ical Resolved Shear Stress Against Diameter. - 65 -v a l u e s o f t h e c r i t i c a l s h e a r s t r e s s a r e , o f t h e - s a m e . o r d e r a s t h e c r i t i c a l f l o w s t r e s s e s ^fj_ m e a s u r e d b y B r e n n e r a n d . S a i m o t o . F i g . 38 i s a . l o g . p l o t o f fj. a S $ i n s t d i a m e t e r a s o b s e r v e d b y E d e r a n d M e y e r . T h e s e p o i n t s a r e a l s o o f t h e s a m e . o r d e r . a s t h e i r v a l u e s o f H i . „ T , . H o w e v e r , t h e . r e l a t i o n s h i p b e t w e e n f 1 a n d t h e d i a m e t e r i s d i f f e r e n t t h a n i n t h e c a s e o f r t j c r . T h e c r i t i c a l f l o w s t r e s s i s p r o p o r t i o n a l t o d " l . T h e r a t i o s , o f ' t / . c r t o ^ f ] _ a s f o u n d b y B r e n n e r a n d S a i m o t o , v a r y a n y w h e r e f r o m a b o u t .90:1 t o a b o u t . 4:1, w i t h t h e a v e r a g e r a t i o > . b e i n g ; a r o u n d 20:1. H o w e v e r , i n t h e c a s e o f E d e r a n d " M e y e r ( F i g . 39); t h e r a t i o s v a r y b e t w e e n a b o u t 5:1 t o 1:1, w i t h m o s t o f t h e r a t i o s . b e i n g b e t w e e n 2:1 a n d 1:1.. I t w o u l d t h u s s e e m t o b e t h e c a s e t h a t E d e r a n d M e y e r d i d n o t o b s e r v e t r u e w h i s k e r b e h a v i o u r . J u s t w h y t h e s e w h i s k e r s o f E d e r a n d M e y e r a r e s o w e a k i s n o t c l e a r , p a r t i c u l a r l y s i n c e n o i n f o r m a t i o n w a s g i v e n w i t h r e g a r d t o t h e p e r f e c t i o n o f t h e w h i s k e r s t e s t e d , . t h e m e t h o d o f g r o w i n g , o r t h e m e t h o d o f m o u n t i n g . T h e . o n l y i n f o r m a t i o n g i v e n t h a t m i g h t h a v e a b e a r i n g o n t h e i r r e s u l t s i s t h e i r m e t h o d o f t e s t i n g . A p p a r e n t l y t h e l o a d w a s a p p l i e d c o n t i n u o u s l y t o t h e w h i s k e r r a t h e r t h a n i n s t e p s a s i n t h e c a s e . o f t e s t s p e r f o r m e d b y t h e a u t h o r , B r e n n e r a n d S a i m o t o . T h e f o r c e w a s r e c o r d e d b y t h e d r i v e o f a c h a r t p e n . I f t h e s t r a i n r a t e w a s t o o h i g h , i t w o u l d b e p o s s i b l e t o m i s s t h e t r u e y i e l d p o i n t . T h e a u t h o r t e s t e d a . f e w w h i s k e r s o f l a r g e (y> 20 ja) d i a m e t e r o n t h e I n s t r o n T e s t i n g M a c h i n e . B e c a u s e . o f g r i p p i n g p r o b l e m s , o n l y o n e s u c c e s s f u l t e s t w a s o b t a i n e d . I n t h e c a s e . o f t h i s o n e w h i s k e r - ( F i g . 40), a w h i s k e r ^ . t y p e o f s t r e s s - e l o n g a t i o n c u r v e w a s f o u n d , b u t i t a p p e a r e d t h a t t h e t r u e y i e l d s t r e s s w a s m i s s e d b e c a u s e . o f t h e r e l a t i v e l y f a s t s t r a i n r a t e (^> 0 . 0 l " / m i n . ) . T h e a p p a r e n t y i e l d s t r e s s w a s 12.8 Kg/mm^-w h i l e t h e f l o w s t r e s s w a s 8.6 K g / m m . T h e w h i s k e r f a i l e d a t o n e g r i p . S i n c e t h e o r i e n t a t i o n . o f t h e w h i s k e r w a s u n k n o w n , t h e c r i t i c a l r e s o l v e d s h e a r s t r e s s - 6 6 -Figure 38. Dependence of Flow Stress on Diameter. Reproduced from Reference 19. - 6 7 -3 0 2 0 1 0 k .3 1 . 0 .1+ . 3 O o o o o o o o IS IS IS o o Crit ical Shear-Stress Crit ical Flow.Stress o cr o i s o WO o 6 8 1 0 2 0 ' 3 0 1+0 Figure 3 9 . Comparison of-Critical Resolved Shear. Stress With Crit ical Resolved Flow .Stress. Rate >. 0 . 0 1 "/rnin a = 55-7 u. -3 y :'l 0 .01 0 .02 0 .03 Elongation (inches) ^T" = 1 2 . 8 , Kg/mm2 = 8 . 6 Kg / mm 0 .04 Figure k0. Stress-Elongation Curve .of A Whisker Tested on the Instron. - 69 -and flow stress could not be calculated. However, the ratio between the two stresses is the same in both cases. ';This ratio is about 1.5:1 which is of the same order as the ratios observed by Eder and Meyer.., 3• Diameter Dependence of Stress One of the problems in explaining the diameter dependence , of yield .stress for copper whiskers, is'that the dislocation•content of the whiskers is unknown. As was previously mentioned in the introduction, Pr ice 1 6 found that zinc whiskers could be in i t ia l ly free.of dislocations. Gorsuch also observed that the dislocation density of some iron whiskers was so small ( i c f i dislocations/cm2) that essentially the whiskers contained no disloca-tions. On;, the other hand, . Pearson et al felt that the dislocation content of their silicon whiskers was not very different from ;that of bulk silicon. No. experimental data is available concerning the dislocation density of copper whiskers. Therefore the two possibilities of either dislocation free whiskers or dislocation containing whiskers wil l have to be considered. Assume, aside from the presence of axial screw dislocations which w i l l contribute nothing to the deformation, that the whiskers are in i t ia l ly free of dislocations. The problem that must now be considered is one of the nucleation of slip in a perfect .crystal. Becker39 was the first to•consider the.possibility that slip could take place.at some applied stress less than the .critical stress by the thermal fluctuations of the lattice helping the applied .stress to form a nucleus of slip on the slip plane. Cottrell 1'; .defines a nucleus of slip as "the smallest region of slip that can he made to grow by the action ,of the applied stress, alone.- Any.region smaller than this wil l slip back into its original and perfect configuration once the thermal fluctuation that caused the region to slip has passed." The smallest nucleus wi l l be a disk-shaped element in the slip plane bounded by a dislocation loop. Cottrell has shown that the radius of the loop has a cr i t ica l value such that.if the loop is of this size i t .can then grow by the action of the applied stress alone. If" the radius of the loop is less than this cr i t ica l value, i t wi l l collapse. This value .of the.critical radius r c is given by r c = jib log r c\. + 1 r 0 : (6) where ja-= shear modulus, ^T>= applied stress, r Q = equilibrium distance between atoms. Further, Cottrell has; shown that the activation energy for nucleating slip in a perfect lattice is of the order 1-2 eV. In effect, this means that slip cannot be nucleated unless the-applied stress 3^^ = J ^ J J . This is of the same.order as the theoretical shear stress which is about j = ^ Q : For copper, ji = h (10 ) dynes/cm which gives a value of applied stress^Xof } 2 around ho Kg/mm . This is of the order ,of that observed experimentally in whiskers. In.turn this means that the.critical diameter of a loop is about 10"° cm. The question now arises as to the maxiumu number.of sites that could produce loops of this cr i t ical size in any given area on the slip plane. . Since i t . i s not possible to calculate the number of sites from any fundamental considerations, some model wi l l have to be postulated in,order to give an estimation of this number. Consider unit area.of slip plane.containing at - 7 1 -each possible site a square loop with the length of one side equal to the cr i t ical diameter d c . Let each loop be a distance d from the next.loop as shown in Fig. kl. The -.number, of these sites wil l be given by d.+ d r where d = 10"°cm. Although no exact value of.d can be given, a reasonable Figure - kl. Model for -Estimating. the Maxiumu .Number. of ..'Loop • Sites. value is d .= ^ /k. This results in a site density of about I O 1 2 sites/cm 2. It should.be remembered however,.that the actual number of loops 1 2 / 2 of cr i t ical size wi l l be less than lCr- loops/cm since i t . i s highly unlikely that every possible site for creating a loop wi l l produce one of cr i t ica l - 72 -size. - Assuming a success factor of about .10 , the actual density of the loops wi l l be about 10 1 (\oops/cm 2. In view of the.above, i t would seem that the cr i t ica l shear stress P would be inversely proportional to d . However, i t has been found in.this investigation that the primary shear stress is inversely proportional to d 1 ' 6 . In order to explain this dependence,.the interactions between the loops wi l l have to be taken into - account. Consider a whisker.of diameter d containing many,loops of cr i t ical size as shown.in Fig. h2. The loops in the circle of diameter.d wi l l interact o c y o o (> p 1 ° /O o o d / o 1 | o o '.5o o \° \o o o 'do 2 \ Q oxo o 'O o 9-Figure h2. Diagram of a Whisker Containing Dislocation -Loops. - 7 3 -with each other and so they wi l l not he free to contribute to the deformation. Only those loops contained in the annular ring.of width w, wi l l take part in the deformation. The width.of this annular ring must be small, probably of the order.of 1000-A° which corresponds to the .possibility of having one or two loops in this width. For the general case of an inverse dependence of shear stress on d n , this means that the area A of the annular ring that contributes to the deformation must vary as d n . This implies that d Q = f(d) such that A = Kd n . The area of the annular ring.is given by A = TT l * 2 - * 2 o l This gives that TT ^-d2' - d 2 ^ = ^ -,2 n or d^ - d„ = kd From this dQ = d j . l - kd n 2 ' ( 7 ) and therefore k <f 1 (8) This means that for a whisker of diameter d, a.range.of values of dQ corresponding to values of k that satisfy equation (8) can be calculated. Figs, k-3, hh and h^ are plots of dQ against k for d'equal to 2, 10, and 20 jx respectively with n = 1 . 6 . These values of diameter were chosen.since they cover the range tested and the one used as a standard in previous calculations. i Besides the restrictions placed on the value of k by equation ( 8 ) , two other restrictions must be considered. The f irst one is that.since o<- d"n and therefore ^ 1 = ^2. d 2 cll n , then obviously the ratios of the ,. - Ik -areas of the annular rings hence once k is set for a particular•diameter, i t must be the same .for any other diameter. The other restriction on k is imposed by the necessity of the width of the annular ring being small. .k For the case being considered where n = 1.6 and taking k.= 0.05 ji , the comparisons of dQ and w for various diameters are given below. d •2ji. 10 JJL 20 ji do I.96 JX •9.92)i 19.. 8 5 p. w •200 A° koo A°: 750 A°. During the primary test on the whisker, nucleation of loops wi l l occur on many slip planes.; along the length of the whisker. Probably because of some localized condition, slip occurs in a small region. This region is then eliminated from the whisker and the whisker is retested. The whisker wi l l now contain a residual dislocation network because of the interactions of the loops formed during the primary test on the other slip 'planes. This dislocation network can be considered equivalent to a region of "dislocation pressure". Under the action of an.applied stress a .similar process wi l l occur as before, except that this dislocation pressure wi l l now aid the applied stress by helping to promote the action of loops that otherwise would not contribute to the deformation. This means that the width of the annular ring in which loops can act to produce deformation is now increased. 2^ n . This is true.only i f L = k and •T~ 1 2 - 78 -Experimentally.it was found.that the secondary yield stress varied inversely as d 2 ,5. Figs. 46, h7 and 48 are plots of. dQ against,k f or n = 2.5 for whiskers of diameter.-2, 10 and.20 ^ ..respectively. These values for dQ and .k were found from.equations (7) and-(8). As before, the restriction that.k must be the same for whiskers of various diameters s t i l l holds,, but the restriction concerning the width of the annular ring is no longer valid. -In order to make an estimation .of the new width, i t .is necessary,to assume that for whiskers of small diameter, w-does not vary greatly. This assumption;is reasonable as the difference -between values of stress for n.= 1.6 and n.= -2.5 for a whisker of small diameter-is small compared to the.difference•in-stress for whiskers of large,.diameter. For the actual case under consideration, for a whisker of 2 ^ - i n diameter,,w remains about the same,,ie. 200 A 0 . Therefore dQ is the.same as before and hence:the value.of k corresponding.to this value can:be calculated from equation (7). -Using this value of k which.is 0.028 '^,.the following results were obtained'. d 2p 10 )x 20 p. do 1.96 )X •9-55 / i 18.72p w 200 A° 2250 A° 6400 A° For a whisker with d = 10 jx, the width, of the annular ring was increased by about 5 l/2 times while for a whisker with ,d = 20p., by about 8 l/2 times. - 82 -It should be recalled that the whole of .'the above discussion was based on the assumption that the whiskers were in i t ia l ly free of dislocations If the assumption is now made that whiskers are not dislocation free, a different dislocation mechanism wil l have to be postulated to explain the observed diameter dependence of stress. However, at the present time, no adequate theories have been devised to explain this dependence. h. Variations of Young's Modulus. Aside from an apparent decrease in the Young's Moduli of about 30/o for whiskers subjected to a secondary test,. as was observed in this investigation, a significant change in the values of Young's Moduli from the accepted values has been observed by,several other investigators. ho Risebrough performed tensile tests on zone refined aluminium which was alloyed with 0.02, 0.1 and 0.2 percent magnesium. He found that the slope of the stress-strain-curve for these alloys increased as the amount of magnesium present in the. aluminium increased. -The average value of the Young's Modulus increased by a factor of about .2. Investigations into the mechanical properties of thin gold films hi h2 have been performed by Catlin.and Walker .and by Neugebauer . Catlin and Walker performed bend tests on single-crystal films varying in thickness between 1000 and 3000 A°. These films were grown by-vacuum .deposition on heated (375°C) rocksalt substrates which had been cleaved to expose ^ 100^ planes.. The orientations of the films produced were -completely 100 ^  . Fig. H-9 shows the variation of Young's Modulus with film thickness. The dashed line is their calculated value .0.785(l0 l2)dynes/cm2^J of Young's Modulus. For the thicker films there was close agreement between the cal-culated and measured values. However, as the. thickness of the films de-creased, .the value.of Young's Modulus increased by as much as 50$. - 83 -lOOO • 2000 Thickness in Angstrom Units 3C0 Figure ky. Variation of Young's Modulus With Film :Thickness. Reproduced from-.Reference kl. Tensile tests were -performed on gold films by Neugebauer. He prepared the films in approximately the same manner as Catlin and Walker. Completely, partially and randomly oriented films with respect to the ^1.00 planes of the rocksalt were produced. The type of orientation depended on the temperature of the substrate. In contradiction to Catlin and Walker, Neugebauer found no variation of Young's Modulus with film thickness. Coleman et al"^ have reported that the- Young's Moduli for zinc and cadmium whiskers were consistently lower than the accepted values by about .30$. •So far, no explanations for any of the various results mentioned above have been made. - 8k -SUMMARY AND CONCLUSIONS 1. Since the important results have been summarized in a previous section, they wi l l not be repeated here. 2 . The growth of whiskers by the hydrogen reduction .of cupric•chloride is greatly.inhibited by, the presence of any water in the cupric chloride. This is probably due to the poisoning of potential growth sites by the dipole action of the water molecules. 3. The results of tensile tests performed on-copper whiskers by the author, Brenner and Saimoto are reasonably consistent. However,.the results obtained by Eder and Meyers indicate that either their whiskers are abnormally weak or else their method of testing gives an apparent yield stress that.is much lower than the true yield stress. k. To explain the dependence.of yield stress on diameter and the change in dependence between primary and secondary tests, a dislocation mech-anism which assumed that the whiskers were in i t ia l ly free of dislocations was postulated. - 85 -RECOMMENDATIONS FOR FURTHER WORK The type of tensometer used in this investigation is very limited in its uses for these reasons: (i) the load is applied in steps, ( i i ) high temperature tests are impossible because of the low melting point of the gripping compound, ( i i i ) whiskers of large diameter ( >^ 20 u) cannot be tested. Therefore i t would be interesting to perform tensile tests of large diameter whiskers on the Instron Testing Machine. The effect of strain rate and temperature changes could be observed. Also, i t would be of interest to compare the type of stress-strain curve . obtained for the whiskers of large diameter to that obtained for whiskers of small diameter. - 86 -BIBLIOGRAPHY 1. Cottrel l , ,A.H. , "Dislocations and Plastic Flow in :Crystals, " Claredon,Press, Oxford 1953-2. • Mackenzie, J . JC., Thesis, University of Bristol,.I9I+9. 3- Bragg, W. L. and Lomer, W. M., Proc. Roy. Soc. (London),. Al 96, 171(191+9). 1+. Herring, C. and Gait, J . , Phys. Rev. 85, 1060 (1952). 5- Gyulai , ,Z. , Z . f . Physik, 138, 317 (195*0-6. Eisner, , R. L.Acta Met., 3, hih (1955). 7- Brenner,, S. S., J . Appl. Phys. 2J_, No. 12, 11+81+ (1936). 8. Fahrenhorst, W., and .Schmid, E . , . Z. f. Physik,.78, 383 (1932). 9.. Schmid, E . , and Boas, W., "Plasticity of Crystals, " F. A. Hughes and Company,,Ltd.,,London. 10. Coleman, R. V . , Price, P. B. , and Cabrera,. N. J . , J . -Appl . Phys. 28, .1360 (1957). _ 11. Evans, C. C . , , Marsh, D. M., and .Gordon, J . E . , Cambridge Conference on 1 Strength of Whiskers and Thin Films (1958)• 12. Pearson, G. L . , Read,, W. T . , and Feldman, W. L. , .Acta Met., 5 , , l 8 l (1957). 13- Cabrera, N. , and Price, P.B., "Growth and Perfection of Crystals" edited by,Doremus et a l , John Wiley and.Sons,.Inc.,. New York (1958)-lU. Brenner,, S. S., Ibid, p 170. 15. Brenner, S. S., J . Appl. Phys., 28, 1023 (1957)-16. Price, P. B. , Phil . Mag., 5,.No. 57, .Sept.(i960). 17. Taylor, G. F . , Phys. Rev., 23, (192U). 18. ASM Metals Handbook H98, (1961) . 19- Eder, F. X . , and Meyer, V . , Naturwis sens chaff en 1+7, 352 (i960). 20. Evans, C. C , and Marsh, ,D. M. , "Growth and Perfection of Crystals", edited by Doremus et a l , John Wiley and Sons, Inc., New "York .(1958). 21. Eisner, R. S.,, Acta Met. 3 ^ 19 (1955). 22. Gorsuch, P.D., J . Appl. Phys. 30, No. 6, 837(1959). 23- Costanzo, R. A . , Private Communication. - 87 -24. S h l i c h t a , P . J . , " G r o w t h a n d P e r f e c t i o n . o f C r y s t a l s / ' e d . h y D o r e m u s e t a l , . . J o h n W i l e y a n d . . S o n s , : I n c . , . N e w Y o r k (1958). 25. F l e i s c h e r , R . L . , a n d C h a l m e r s , . B . , J . M e c h . a n d P h y s o f . S o l i d s , 6, 307.(1958). 26. R o s c o e , . R . , P h i l . M a g . , 21 (7), 399 (1936). 27. C o t t r e l l , . A . H . , a n d G i b b o n s , D . F . , . N a t u r e , l62, 488 (1948). 28. S a i m o t o , . S . , M A S c T h e s i s , 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 , , i960. 29. B r e n n e r , . S . S . , - A c t a M e t . , 4, 62 (1956). 30. O v e r t o n , , W . C a n d G a f f n e y , J . , P h y s . R e v . , 98,,969 (1955) . 31- F r a n k , , F . C , D i s c u s s i o n s F a r a d a y , S o c . , . N o . 5, 67 (1949). 32. S e a r s , G . W . , , A c t a M e t . , 1, 4-58 (1953). 33- S a m i s , C . S . , P r i v a t e C o m m u n i c a t i o n . 34. B r e n n e r , S . S . , T h e s i s , R e n s s e l a e r P o l y t e c h n i c I n s t . , 1957. 35- G o r s u c h , P . , G . E . - R e s e a r c h R e p r i n t #57-RL1840 (1955). 36. S h e t t y , , N . , P r i v a t e C o r a m u n i c a t i o n . 37- S a r a k h o v , , A . I . , P r o c . A c a d . S c i . U S S R , P h y s C h e m . S e c t . , , 1 1 2 (1957) 38. A l l a n , W . J . , a n d W e b b , W . W . , A c t a M e t . J_, 646 (1959). 39- B e c k e r , R . , P h y s . Z e i t . , . 26, 919 (1925). 40. • R i s e b r o u g h , . N . , M A S c . T h e s i s , . U n i v e r s i t y o f . T o r o n t o , 1961. 41. C a t l i n , A . , a n d W a l k e r , P . , J . A p p l . P h y s . , 31, N o . 12, 2135 (i960). 42. N e u g e b a u e r , C A . , J . A p p l . P h y s . , 31, N o . 6, IO96 (i960). 43- F r e u n d , J . , a n d W i l l i a m s , F . , M o d e r n B u s i n e s s S t a t i s t i c s , P r e n t i c e H a l l , New J e r s e y . 44. S i m o n , L . E . , " A n E n g i n e e r ' s M a n u a l o f S t a t i s t i c a l M e t h o d s J o h n W i l e y a n d . S o n s , I n c . , . N e w Y o r k . - 88 -APPENDIX I - 8 9 -A.. Procedure for Calibrating the Tensomenter 1. Two helical springs were calibrated, giving a force-extension curve. 2. The restoring force was then measured by glueing.one of the springs onto the grips of the tensometer and then positioning grip B at various dis-tances from null . This gave a distance-from-null-extension relations from which a.distance-from-null-restoring force relation was determined. 3- The solenoid was calibrated with both springs mounted as above. The current-extension relations was determined by•increasing the current by increments of either l.Oma or 0.5ma. After compensating for the restoring force of the suspended rod, the current-force relationship was determined. - 90 -B. Calibration Curves and Tables TABLE IX Calibration of Impedance Transducer Null at 9.3^5 mm Bridge'Reading M i cr ome ter Reading 30u Scale lOu Scale 9-aO _ _ 9-22 9.65 ' 9-55 9.24 9.60 9.50 : : 9.26 9.60 9.60 9.28 9.80 9.20 9-30 9.50 9.4o 9-32 10.75 9.50 9.34 10.45 9.20 9.36 9.55 9.20 9.38 9-20 9.70 9.40 9.05 8.65 9.42 9.40 9.10 9-44 8.85 8.55 9-46 9.15 8.80 9.48 9.30 9-25 9-50 8.95 9.05 9.52 9.4o 9.00 9.5^ 9.05 9-75 9.56 9-4o 8.40 9.58 9.10 9-35 9.60 9-70 9.20 9.62 9.^5 . 9.40 9-64 9.45 9.25 9.66 9.25 8.90 9.68 9-75 9-35 9.70 8.50 9-25 9-72 10-20 ' 9 r25 9.7^ 9.30 9-35 9.76 9.75 9.20 9.78 9.90 9.45 9.80 9.30 9.10 9.82 9.50 9-35 9.84 9-75 9-75 9.86 9.30 8,65 9.88 9«io 9.80 9.90 8.90 9.20 0.1 - one small division 0.2 = one small division Sensitivity-: 2.12p./div. Sensitivity-: 0.432u/div. Calibration of Impedance Transducer Micrometer ' Bridge Micrometer Bridge Reading 3 /1 Scale Reading 3 /x Scale 9-30 - • 9.56 4.75 9-31 k.GO 9-57 U.15 9-32 3-85 9-58 •IT. 50 9-33 k.ko 9-59 k.-JQ 9-3^ 3-90 9 . 6 0 . 5.10 9-35 k.Qo 9.61 4 . 8 0 9-36 k.6o 9.62 4 . 5 0 9-37 k.oo 9-63 5.30 9-38 3 . 8 0 9.6k. 5.00 9-39 5-75 9 .65 4-35 9 . ko 3-95 9.66 U.90 9.ki 5.10 9-67 U.90 9.U2 U.15 9.68 ^5.00 9-^3 k.ko 9.69 4 .50 9.kk k.QO 9 . 7 0 . U.90 9M U.70 9.71 . U.70 . 9-k6 k.20 9.72 IT.60 9.U7 5 .00 9-73 k.90 9-kQ ^•55 9 .7^ 3-80 9.U9 k.20 9-75 U.90 9 . 5 0 k.80 9.76 1+.50 9.51 U.50 9-77 k.^O 9.52 5.25 9 .78 3-80 9-5.3 k.QO 9-79 5-00 9-5U k.90 9 . 8 0 k.10 9-55 U.70 0 . 1 = one small division Sensitivity-: 0.219 u/div. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Extension (mm) Figure 50- Calibrations of Large Helical Spring. Force-Extension Plot - 93 -0.0 0.0k 0.0Q 0.12 0.16 0.20 0.24 Extension (mm) Figure 51 • Calibration of Small Helical Spring. Force - Extension Plot 210 o 50 IOO 150 . 200 250 300 350 koo Force (mg-wt) Figure 53- Current-Force Relationship at Low Current. The Small Spring was Used. - 97 -APPENDIX II TABLE IV Comparisons Between-Measured and Calculated; Values of Area 2 Measured.Area in ii Whisker; Optical Calculated A 167 177 Bl 48 62 B 2 A8 71 C : •37 5^ D l 92 82 D 2 92 72 - D3 92 83 :E 77 k9 F l 85 102 F 2 85 ll+6 G, 77 71 H 37 1+1 I 37 •31 J 95 95 K 12 12 L 21 25 • M 111 90 N 11 ' 9 0 22 18 P 6.6 •8.0 Q 11 9.6 R :.i8 38 TABLE TV Comparisons Between:Measured and Calculated Values of Area. Whisker.: Measured 2 Area in u Optical Calculated . V lk 16 s 2 lk 18 s 3 ih 20 . sh ih 19 T l 72 92 T2 72 86 U 96 90 V 6h 65 w 55 72 x l 38 h9 x 2 38 h9 Y 62 5h Z 109 62 AA 115 102 BB 133 117 CC 102 119 DD 121 77 EE l238 227 FF 15 15 GG 50 58 -HH .. 58 35 TABLE V Primary Tests Whisker 4( ? ) L(mm) ^m(Kg/mm2) t £lm E(loNcg/mm2) ^ c ( Kg/mm2). ^(Kg/mm 2 ) A 15.0 1.415 34.0 0.50 0.68 23.4 6l.5 8.9 • 1.842 93.5 . 1.35 0.68 50.4 78.5 B 2 9-5 1.829 86.4 1.27 0.68 45.6 80.2 C 8.3 1.388 40.5 0.58 0,68 56.O 30.6 D l 10.2 2.4i6 4 l . l 0.575 0.68 41.0 42.4 D 2 9.6 1.602 32.6 0.47 0.68 45.1 30.6 D3 10.3 1.498 58.5 0 .84 0.68 4o.4 6l.2 E 7.9 1,549 44.9 0.63 0.68 60.3 31.5 F l 11.4 2.510 16.2 0.22 0.68 34.8 19.7 F 2 • 13-7 2.537 28.6 0.4l 0.68 26.6 45.5 G 9.5 1.629 37.4 0-53 0.68 45.6 34.7 H " 7-2 l . 0 4 l 92.5 1.325 0.68 69.6 56.2 I 6.3 0.534 106.8 1 .49 0.68 85.6 52.8 J ' 11.0 2.670 53-2 0.39. 1.34 36.7. 61.3 K . 4.o I.869 312.2 2.19 1.34 173.7 76.0 L 5.6 1.549 132.7 0.95 1.34 102.7 54.6 M 10.7 1.682 55.-9 O.285 1.96 38.3 61.7 TABLE V Primary Tests Whisker L(mm) Kg/mm2) £ m E(lO^Kg/mm2) O^C Kg/mm2) O 3 (Kg/mm2) n IT 3-4 1.255 157.8" 0.74 I.96 . 225.0 29.6 G 1.736 97.0 0.47 I.96 126.4 32.4 P 3-2 1.415 181.3 0.805 1.96 247.I 3-1.0 Q 3-5 1.428 110.7 0.445 I.96 214... 5 21.8 ' R 7-0 1.362 72.5 0.35 I.96 72.6 42.3 Si - 4.5 1.922 52.9 0.-22 I.96 144.5 15.5 S2 4.8 • 2.830 107.8 0.505 1.96 130.6 34.9 S3 5-0 1.922 155.8 0.74 I.96 122.4 53.8.• i.o4i 77.4 O.36 i . 9 6 126.4 25.9' T l 10.8 3.600 26.1 0.39 0.68 37-7 29.3 T2 10.5 5.280 41.5 0.6l 0.68 39-3 44.7 U. 10.7 1.295. 84.7 1.245 0.68 38.3 93-5 ' V 9-1 2.480 93-2 1.37 0.68. 48.7 81.0 w 9.6 2.456 15.O 0.21 0.68 45.1 14.1 7-0 2.360 60.5 O.87 0.68 72.6 •35.2 x2- - 7-9 2.456 74.5 1.095. 0.68 . 60.3 .' 52.2 TABLE V Primary • Tests Whisker d( p) L(mm) ^(Kg/mm 2 ) ^ m EClO^Kg/mm2) O^Kg/mm 2) Kg/mm2) Y 8.3 2.480 67.3 0.99 0.68 56.0 50.8 Z 8.9 1.228 35-0 0.515 0.68 50.4 29.4 AA 11.4 1.642 32.3 0.475 0.68 34.8- 39-2 BB 12.2 3.080^  88.1 1.295 0.68 31.5 ' 118.3 CC 12.3 5.120 12.1 0.178 0.68 31.3 55-2 - DD -' 9-9 1.202 45.4 0.668 0.68 42.9 44.8 EE 12.4 2.460 40.2 O.30 1.34 30.8 55.2 FF 4.4 1.695 98.0 0.57 1.34 149.8 27.7 GG 17.O 2.056 19:- 0 0.28 0.68 19.7 .40.8 HH 6,7 I.896 88.4 1.30 0.68 77.7 48.1 II 8.6 4.280 79-7 0.595 1.34 53.1 63.5 JJ T . l 2.590 89.8 O.665 1.34 71.1 53.4 * Whiskers used in Secondary tests. Whiskers used in annealing tests. TABLE VI Secondary . Tests Whisker d( p) L(mm) ^(Kg/mm2) 1 em E(loSvg/mm2) (Kg/mm2.) ^ f n ( Kg/mm2) E/Ei * GC. 12.3 5.120 12.1 0.178 0.68 • -; CC' * 2.123 27.3 - 0.637 0.35 47.3 30.6 0.51 CC' 1 1 I.896 14.4 0.284 0.41 47.3, 16.1 0.60 cc- 1 1 ' I..362 33-2 0.653 0.40 47.3 37.2 0.59 . T I - 10.8 3.600' 26.1 0.394 0.68 . , • _ 2.270 32.0 0.484 O.65 50.6 33-5 0.96 1.362 92-8 1.57 0.59 50.6 97.2 0.87 • # 10.5 5.280 41.5 0.6l 0.68 V 3.280 40.9 0.705 O.58 51,4 42.2 0.85 rp i 1 x2 2.189 58.8 • 1.062 O.56 51.4 • 60.6 0.82 * u - " 10.7 1.295 84.7 1.245 . 0.68 _ _ _ U' 0.881 60.6 1.13 0.54 50.8 63.2 0.79 TABLE VI Secondary Tests Whisker L(mm) O^Kg/mm 2) 4, €L m E(l0^Kg/mm2) <ry>c( Kg/mm2) 0^(Kg/mm2-) E/Ei * EE. 12. 4 2.560 , 40.2 0.30 1.3^ _ EE' . O.961 30.6 0.55 0.57 47.2 34.3 0.42 * V 9-1 2.480 93.2 1-37 ' 0.68 V 0.828 63.5 1.41 0.44 56.7 59.4 0.65 W 9... 6 2.456 15.0 0.21 0.68 w 1.028 33-7 0.495 0.62 ' 5^.5.. . 32.8 0.91 L 5-6 1.549 132.7 0.99 1.3^ — _ _ L' 0.748 130.2 0.13 1.07 98.7 69.9 0.80 * 7-0 2.360 60 »-5 0.87 0.68 _ Xi' 0.908 84.3 1.29 0.65 73.4 6 l . l 0.96 " TABLE VI Secondary. Tests Whisker d(^) L(min) ^ ( Kg/mm2) ' m E(loScg/mm2) Xfc (Kg/mm2) XJ>n{ Kg/mm2) E/Ei * X 2 7-9 2.456 7^-5 1.095 0.68 -V 0.721. 1+0.9 0.93 O.65 64.0 33-9 O.96 •# Y 8.3 2.480 67.3 0.99 0.68 _ Y l 1.121 11+9.6 3.32 0.45 61.0 130.0 0.66 *• DD 9-9 1.202 1+5.4 0.668- 0.68 .-. _ _ DD' 0.935 45.9 1.16 0.40 53-3 45.6 0.59 DD' ' 0.721 55.6 1.335 0.43 53.3 55-3 0.63 * -Z 8.9 1.228 3^.0 0.515 0.68 . _ _ _ z- 0.301 34.3 1.17 0.30 57-7 31.5 0.44 * AA l l A 1.61+2 32.3 0.475 0.68 AA' 1.068 34.6 0.565 O.58 49.1 37.3 O.85 TABLE VT Secondary Tests Whisker L(mm) C^(Kg/min 2) E(l0^Kg/mm2) 0-*c( Kg/mm2) O^ K g / m m 2 ) * FF 4.4 1.695 75-0 0-57 1.34 _ FF' :' 1..081 143.4 1.12 .1.31 ' 148.3 51.2 O.98 # BB 12.2 3.080 88.1 1.295 0.68 BB' 1.282 100.6 1.88 0-53 47.5 112.3 O.78 ** GG 17.0 2.056 19.0 0.28 0.68 -V r*rt l U J 0.587 15.9 0.95 0.17 47.5 19.7 0.25 HH " 6.7 I.896 88.4 1.30 0.68 - - ' HH' O.656 50.5 1.76 0.28 77-1 34.7 0-.4i I I - 8.6 4.280 79.7 0-595 1.34 • - - •-I I ' 1.469 65 . I " 0.73 0.90 59.5 58.O 0.67 TABLE VI  Secondary Tests . Whisker d( L(irun) ^(Kg/mm 2 ) E(loSv"g/min2) Kg/mm2) ^(Kg/mm2). E / E i *# JJ 7.1 2.590 89.8 O.665 1.3^ - - - -JJ* 0.908 75-2 0.505 . 1 .46 72;. 0 55 .4 1.09 * Ordinary Whiskers. ** Annealed Whiskers. APPENDIX III -: 109 -Sample Calculation of Young's Modulus for a Whisker With.the Axis l p ° Off L l O O ] Young's Modulus is given hy 1 = S n - 2(Sn - S 1 2 - S ^ / 2 ) ( t i tt + Y| i f + t i ^ ) E , where Sj_j = elastic compliances and ^1^2,3 = cosines of the angles formed by the axis.of the specimen .'with the three edges of the unit cube. The elastic compliances for copper^1-1 axe: _k o 511 =;1-^9 )111111 /Kg k p 5 1 2 = -0.63 (10 )mm /Kg Shk = 1.33 (l6U.)m^2/Kg For a whisker with a |[^ 10o"J axis, 2.. 2 Kr. 2 „ 2 . i 2 ,0 2. and E = 1 = 0.68 (l0 H) Kg/mm S l l Consider a whisker whose axis lies 15° off the [_100~j axis. The values ^ 1 2 3 c a n ^ e ^ o u n <3- using a stereographic projection. - 110 -[loo] Case 1. t . ^ 3 = cos 75° ^ x = c o s 1 5 ° o 2 = c o s 9° then ( ^ i 11 + t 2 K 3 + i I % 3) = 0.062 and.E = 0*76 (10^). Kg/mm2 Case 2. t 1 = cos 15 w ^ 2 = cos 79 „ cos 80 t h e n ( t ^ 2 + i 2 2 i 2 . + 0.062 >K __ , 2 and.E = 0.76 (10 ) Kg/mm* Case 3 ' Equivalent to Case 1. Therefore the error in assuming that a whisker has a \_ loo] orientation when, in fact, i t is off by 15 is $ error 0.76 --0.68 0768 100 = 11.7 $ Similar results are obtained for whiskers assumed to have orientations of Llio] or . - 111. -• APPENDIX' IV 112 A. The:Method of Least-Squares The curves in Figures 17, 19, 24 and 25 were obtained hy the method -of least squares^j^H-. . p o r a straight line y = a + bx where a and b can be determined by the following normal equations: . n n >".- y± = na + b > 1=1 1=1 n n > x i y i = a > Xi + b X 1=1 .1=1 1=1 . For results with a high scatter, i t is advisable to consider x as a function of y, ie. x = a 1 + b^y where a-*- and b"1" are found by similar normal equations as above. These two regression-lines wi l l coincide i f , and only i f , the correlation factor r equals 1. The best line.- lies between these two regression lines and a l l three lines wi l l pass through the point x^  and yj_. The•correlation. factor r is a measure of the "goodnes of f i t" and is given by r = x i y i - ^ j y i  x i yt where x±y± = the average of the products of the pairs. x i y i = ^ e average of the x|s times the average of the y^ s = the standard deviation of the x^s. • - 113 -= the standaifci. deviation.of the y. 's. y i 1 2 where = /> *± -.X^ * / i _ i y ± = — y | - y± i = l n If the f i t .is poor-., r wi l l be close to 0. However,-, i f the f i t is good, r w±ll:;be"'.:close to t 1, and there is a strong correlation. It can be shown that the correlation- between x and y is significant i f r V l.96 | n - l -r <" -i.96 4 n - l ' B- Application 1. Primary Tests For the plot of log O " ^ against log d, r = <r0 .Qk. The value.of -I.96 = -1-97 = -0-31 x/n-l" ^ J kl' Therefore the • correlation is significant. - 1 1 4 -2. Secondary Tests For the plot of log^J^ against log d, r •= -0.88. m The value.of -I.96 .= -I.96 = -0.50 Therefore the•correlation is significant. - 115. -APPENDIX V - 116 -A. Calculation of Theoretical Curves for the Volume and Surface Dependence  of Stress for Constant Diameter. 1. Volume Dependence of Stress Assume that the y i e l d stress < ^ i s , some function f('V) of the volume Where V = TT d L I t has been found that (1) 0 % A + B d n (2) However, t h i s equation i s only an.approximation since i t does not take i n t o account the effect of the length L on For-L = = constant, equation ( l ) can be written as 1/2 and d n = (3) Replacing d i n equation (2) gives A + B n/2 - hv -TT L i - 117 -and using equation (l) this can be written as d n n/2 , B W The value.of is chosen to'be about,the average•length of the whiskers tested. a. Primary Tests For primary tests on whiskers normalized to d = 10^u, equation (U) becomes 0^= 39.1 L l L ° - 8 + 2,8 (5) where L x = -2000 y. b. Secondary Tests For secondary tests on whiskers normalized to d =10 y, equation (k) becomes L 1 , 2 5 + 39-0 (6) where 1^ = 1500 p. The curves drawn in Figs. 22 and 27 were calculated from equations (5) and (6) respectively. 2. Surface Dependence of Stress Assume that the yield stress O"^is some function f(S) of the surface area where S = TT" dL (7) - 118 -It can be shown by similar arguments as in the above case that , ah + B (8) a. Primary Tests As before the whisker were normalized to 4 = 10 and equation (8) becomes 39-i: L 1.6 + 2 - 8 (9) where L. = 2000 b. Secondary Tests Similarly for d = 1 0 ^ , equation (8) becomes 2 - 5 + 3 9 . 0 (10) where Lj_ = 1500^1 The curves drawn in Figs. 22 and 27 were caluctlated from equations (9) and (10) respectively. B. Comparison of Expected Variation in Stresses Between Whiskers of Constant  Length L2 and L3 for Various Diameters. 1. Volume Dependence of Stress From the previous section A, i t has been shown'that 0"= A ,n L i L n/2 B (h) - 1 1 9 -C o n s i d e r t w o l e n g t h s L 2 = • L m i n a n d L3 = L ^ a x - I t w o u l d b e i n t e r e s t i n g t o k n o w w h a t v a r i a t i o n t h e t h e r a t l o i ^ l ^ c o u l d b e e x p e c t e d . f o r w h i s k e r s n o r m a l i z e d t o v a r i o u s d i a m e t e r s . T h e r a t i o b e t w e e n V J a n d i s g i v e n b y L3 L2 A ( L ] _ ) n / 2 + B ( L 3 ) n / 2 d n • L 2 n/2 (11) -a . P r i m a r y T e s t s F o r p r i m a r y t e s t s o n w h i s k e r s , Lj_ = 2000 p L2 = 1000 p L3 = 5000 y. a n d e q u a t i o n ( l l ) b e c o m e s 1000 = 5000 1.6 6865 + 7.63d 6865 +2 5.U8d1-6 3-62 (12) F i g - '55 ( a ) s h o w s a p l o t o f t h i s r a t i o f o r s e v e r a l v a l u e s o f d . b . S e c o n d a r y T e s t s F o r s e c o n d a r y t e s t s o h w h i s k e r s , = 1500 p. L2 = 500^ L3 = 3500 p a n d e q u a t i o n , ( l l ) b e c o m e s 500 = 3500 Ul3 + 0.922 d 2 ' ^ kl3 + 10.5 d 2 - 5 11.U (13) F i g - 56 (a) s h o w s a p l o t o f t h i s r a t i o f o r s e v e r a l v a l u e s . o f d . - 120 2 . Surface Dependence o f . S t r e s s [ From the previous s e c t i o n - A , i t has been shown that d 1 1 L l L n +B (8) In a s i m i l a r manner as b e f o r e , the r a t i o between i s g iven by T and L 2 b3 A ^ ) 1 1 + B ( L 2 ) n d n A ( L ! ) n + B ( L 3 ) n d n L 2 a . Primary Tests For same values of L-i_, L g , and as i n B - l ( a ) , equation ( l 4 ) becomes 1000 = 3004 + 1-77 d 1 , 6 3004 + 2 3 . 2 d 1 ' 6 13-1 (15) F i g . 55 (b) shows a p l o t of t h i s r a t i o f o r s e v e r a l values of d . ib. Secondary Tests For the same values of L-|_, Lp_, and as i n B - l ( b ) , equation ( l 4 ) becomes 500 0 * 3 5 0 0 3854 + 2.18 d 2 ' ^ 3854 + 283 d 2 - 5 129.6 ( l 6 ) F i g . 56 (b) shows a p l o t of t h i s r a t i o f o r s e v e r a l values of d . and 1,3=3500 )x. - 1 2 3 -APPENDIX VI TABLE VII Saimoto'a -Results Whisker Orientation ^ c r(Kg / m m 2 ) ^flCKg / m i r 2 ) ^ c r / ^ f l BI [ 100 ] - 3-3 50 .0 5 .15 9 . 7 B2 I 100 ] 2 . 6 1+7.0 - -B3 [ 100 ] 6-T 1+5.0 2.1+0 1 8 . 8 BIO [ 100 ] 6 . 6 2 7 . 0 -Bll [ 100 ] • -3.8 -3-5.8. - -B12 [ 100 ] 3-6 58 .5 - -B13 [ 100 ] 7 . 1 3 k - 2 2.1+5 1 3 - 9 BlU L loo ] 3-0 1+9.0 - -B20 [ 100 ] . ^ - 5 1 1 . 5 3 -00 . 3 -9 B21 [ 100 ] 3-9 39-6 3 . 7 10 .7 " BT [ 110 ] 9 - 9 1 6 . 7 1 .67 10 . 0 Bh [ 111 ] 6 - 9 21.1+ 2 . 5 2 8 . 5 B5 [ 111 ] 5-5 21+.0 3.02 7 - 9 B6 [ 111 ] h-3 1+8.5 1 . 9 1 2 5 . 3 B8 [ 111 ] 3-2 1 9 . 9 2 .91 6 . 8 B9 [ 111 ] h.Q 1+2.5 2 .86 1I+.9 B15 [ 111 ] . 5-0 2 6 . 0 3 .16 8 . 2 Bl6 [ 111 ] h.3 13-8 3 -29 1+.2 TABLE VIII Brenner's Results Whisker Orientation ^ c r ( Kg/mm2) Tj f ]{ Kg/mm2) ^ c r / ^ f l 1 [100] 6.3 15.2 1.52 10.0 2 [100] 8-3 32-0 1.74 18.4 3 [100] 8.9 . 20.5 1.48 •13-9 4 [100 ] 10.3 22.6 1.72 13.1 5 [100] 10.6 38.6 1.48 26.1 6 [100] •17.8 15.1 1.06 14.2 7 [100] 25.9 25.0 O.50 50.0 8 [100] ,9.6 ' 6...1 1.34 4.6 9 [100] 11.1 33-1 0.37 89.4 10. [no] •11.6 34,0 O.94 36.2 l l [no] 12.0 16.5 O.56 29-5 12 [no] 15.5 24.9 0.67 37.2-13 [110] 6.3 20.4 1.74 11-7 14 [no] 12.4 14.4 0.98 14.7 15 [111] 12.9 33-7 0.82 4 l . l 16 [111] 14.9 23.4 O.63 37-1 17 [111] 4.6 25.2 1.19 21.2 

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