THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THE EFFECTS OF TEMPERATURE AND HYDROSTATIC PRESSURE ON INTERFACIAL TENSIONS IN THE SYSTEM SOLID NICKEL-LIQUID LEAD Charles Ao Stickels A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Chemical and Metallurgical Engineering 1963 March, 1963 IP-610

Doctoral Committee: Professor Edward Eo Hucke Chairman Professor Wilbur C. Bigelow Professor Samuel Ko Clark Professor Maurice Jo Sinnott Professor Lawrence Ho Van Vlack

ACKNOWLEDGMENTS It is a pleasure for the author to acknowledge the assistance of many individuals and organizations during the course of this investigation. Professor Edward E. Hucke stimulated interest in the role of interfacial tensions in determining microstructures, and his advice was invaluable on many occasions. The interest and cooperation of the rest of the doctoral committee is also gratefully acknowledged. Dr. David V. Ragone, who served on the doctoral committee for more than a year, was a frequent source of counsel. The Universal-Cyclops Corporation, the University of Michigan Research Institute and the Muchnic Foundation provided fellowships for the author during his graduate study. The International Nickel Company supplied some of the materials used in this work and performed a chemical analysis on one alloy. Preparation of the dissertation was expedited by Janet Maloney, who did the typing, and the Industry Program of the College of Engineering, who prepared the figures and published the thesis in its final form. Finally, special thanks are due to Mro Robert W. Dunlap for a critical reading of many sections of the dissertation during its formative stages, and to the author's wife, Patricia F. Stickels, for her continued moral and financial support. ii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS.............. ii LIST OF TABLES........................ v LIST OF FIGURES.........o. o..... vi ABSTRACT...O.. a o.o. o o. o...... o viii CHAPTER I INTRODUCTION.. o o... o o o o....... o. 1 II THERMODYNAMICS OF INTERFACES............ 6 1. Definitions...... o o............. 6 2. The Gibbs Adsorption Equation,........... 15 3. Criteria for Stability of an Equilibrium State.. 18 4. Application of the Gibbs Adsorption Equation to Systems Involving Solids........... 20 5. Derivation of Useful Relations from the Gibbs Adsorption Equation............. 23 III LITERATURE REVIEW................ 33 1. Measurements of Ratios of Interfacial Tensions... 33 2. Measurement of Dihedral Angles......... 39 3. The Orientation Dependence of Grain Boundary Tension............ o......... 41 4. Changes in Surface Tension with Pressure...... 44 5. Brittle Fracture............... 55 6. Intergranular Fracture... o........ 59 7. Liquid Phase Sintering............... 62 8. Summary...................... 64 IV EXPERIMENTAL PROCEDURE. o.......... 66 1. Sample Preparation.......... 66 2. Hydrostatic Pressure Testing............ 71 3. Testing in Tension and Compression........ 76 4. Measurement of Dihedral Angles........... 79 V THE STATISTICS OF DIHEDRAL ANGLE MEASUREMENT...... 83 1. A Discussion of Errors Affecting the Distribution of Observed Angles................. 84 2. Effect of a Non-Unique Dihedral Angle on the Distribution of Observed Angles........ 87 3. The Contribution of Measuring Error........ 89 4. The Effect of Observed Particle Size....... 92 iii

TABLE OF CONTENTS CONT'D Page 5, Establishing a Confidence Interval for the Median Angle.o............... o.. 97 6o Summary.................... 100 7 Another Method for Obtaining Dihedral Angles,. 0 102 80 Statistical Tests for Significant Differences Between Samples, 0 00.000 0 104 VI EXPERIMENTAL RESULTS,, o, o o o, o o o,,,.o.o 106 lo The Nickel-Lead Phase Diagram... o.. o... 1 06 2, Temperature and Pressure Data, ooooooooo0 109 3. Tension and Compression Data...... 113 VII DISCUSSION OF RESULTS 1. 23 1l A Method for Obtaining Variations in Interfacial Tensions from Variations in the Dihedral Angle O o o 123 2, The Solid/Liquid Interface.......... 131 3. The Solid/Solid Interface,,,, o o o o o o o 135 4o Analysis of Tension and Compression Experiments, o o 139 5o Summary o, o o,, o, o o o o o o 0 0 146 VIII GENERAL SUMMARY, o o o o. o, o o, o o o o o 148 APPENDIX A DERIVATION OF THEORETICAL DISTRIBUTION FUNCTION FOR OBSERVED ANGLES o,.o........ o o o o. o, 150 B ESTIMATION OF NICKEL GRAIN.BOUNDARY TENSION IN THE PRESENCE OF LEAD.,,, o, o,o. o,,, o o., 157 C CORRELATION OF DIHEDRAL ANGLE VS TEMPERATURE DATA FROM THE LITERATURE, o o o o, o o, o,, 159 D PROCEDURE FOR FITTING: WEIGHTED LEAST SQUARES LINE TO DATA. o o..... o o o. o o o.. o 173 E ESTIMATION OF THERMODYNAMIC PROPERTIES OF NICKEL IN LIQUID LEADo o o. o o.,,,, o o o o,, 175 BIBLIOGRAPHY,,,o o o,,,,,, o,,. o o,.,. O O, 178 iv

LIST OF TABLES Table Pae I.Effects of Small Changes in Interfacial Tensions on the Dihedral Angle................ 38 II Spectrographic Analysis of Lead Used for Alloy..... 66 III Typical Analysis of Sherritt Gordon Grade "E" Nickel Powder.......... o............. o 66 IV Analysis of Heat "D"................ 67 V Effect of Non-Unique Dihedral Angle on Distribution of Observed Angles......... o.. 88 VI Effect of Measuring Error on Distribution of Observed Angles........... o 91 VII Effect of Ignoring Angles from Small Section Sizes on the Distribution of Observed Angles from Lenticular 96 Particles... o...... o...... o VIII Confidence Intervals for Various Sample Sizes..... 99 IX Comparison of Pairs of Samples for Significant Differences 105 X Summary of Temperature and Pressure Data....... 111 XI Summary of Tension and Compression Data........ 118 V

LIST OF FIGURES Figure Page 1 Vectorial Relations Between Interfacial Tensions.. 35 2 Thermodynamic Model for Creation of a Liquid/Vapor Interface................. o o o. 47 3 Density Profiles at a Liquid/Vapor Interface o.. 48 4 Tension and Compression Test Specimens.0,.... 69 5 Photomicrographs of Specimen DP8....... 72 6 Schematic Drawing of Hydrothermal Unit... e 75 7 Furnace Tube and Fittings for Tensile Testingo. o 77 8 Fitting for Compression Testing, 0 0 0 0.. 80 9 Frequency Plot - Theoretical and Observed.......85 10 Probability Plot Showing Regions Where Section Size of Lenticular Particles is Small. o 95 11 Nickel-Lead Phase Diagram o........... 107 12 Solublijty of Nickel in Liquid.Lead 0........ 108 13 Ratio of Interfacial Tensions as a Function of Pressure 114 14 Ratio of Interfacial Tensions as a Function of Temperature 0.00.00 0..0 115 15 Median Angle Observed in Tensile Testing as a. Function of Orientation with Respect to the Axis of Tension o 119 16 Comparison of Results of Testing in Tension and Compression...00 120 17 Comparison of Tensile Tests Run for 48 and 120 Hours o 120 18 R-Plot for Temperature Data at, 0 Pressure. o o. 126 19 R-Plot for Pressure Data at 371~C..,,. 0 129 20 R-Plot for Pressure Data at 593~C o o.,.., 130 21 Sketches of Thermodynamic Systems,.0......0 142 vi

LIST OF FIGURES CONT'D Figure Page 22 Vectors Used in Deriving Relation Between Observed Angle, Dihedral Angle and Orientation Parameters.... 151 23 Sketch Showing:Relationships Used to Obtain the Probability of Observing a Given Angle......... 153 24 Probability Plot After Harker and Parker(39)... 153 25 Theoretical Cumulative Frequency Ourves for Various Dihedral Angles................. 156 26 Copper-Lead Phase Diagram............ 160 27 Dihedral Angle vs. Temperature. Cu-Pb System..... 160 28 Aluminum-Tin Phase Diagram............... 162 29 Dihedral Angle vs. Temperature. Al-Sn System... 1 62 30 R-Plot for Al-Sn System................ 163 31 Zinc-Tin Phase Diagram.............. 164 52 Dihedral Angle vs. Temperature. Zn-Sn System.... 164 33 R-Plot for Zn-Sn System.............. 165 34 Tin-Tellurium Phase Diagram.............. 167 35 Dihedral Angle vso Temperature. Sn-Te System..... 167 36 R-Plot for Sn-Te System............... 168 37 Aluminum-Antimony Phase Diagram............ 169 38 Dihedral Angle vs. Temperature. Al-Sb System..... 169 39 R-Plot for Al-Sb System................ 170 vii

ABSTRACT A study was made of the effect of temperature and hydrostatic pressure on ratios of interfacial tensions in the binary: system nickellead at temperatures above the melting point of lead. Ratios of interfacial tensions were obtained by measurement of the dihedral angle made by the intersection of two nickel/lead interfaces with a nickel/nickel grain boundary. At zero pressure, the dihedral angle decreases from 52~ to 32~ as the temperature is increased from 371~C to 816~C. At 371~C, the dihedral angle decreases from 52~ to 39~ as hydrostatic pressures up to 50,000 psi are applied. At 593~C, the.dihedral angle decreases from:41 to 32~ with increasing:hydrostatic pressure up to 52,600 psi. The assumption is made, that, in the temperature and pressure range studied, the interfacial tensions in this system vary linearly with changes in temperature or pressure. It is shown that the experimental data is consistent with this assumption. Using this assumption, pressure and temperature coefficients are calculated for each interface. The values obtained for these coefficients are all positive. It is assumed that the Gibbs Adsorption Equation may be applied to solid/liquid and solid/solid interfaces to obtain expressions for the pressure and temperature coefficients of interfacial tension. Applying the expressions derived to the temperature and pressure coefficients of interfacial tension permits calculation of the excess entropy and the excess concentration of nickel at the nickel/lead interface. Although similar calculations cannot be carried out for the nickel/nickel grain boundary, it is shown that in all likelihood, lread segregates strongly at these boundaries. viii

Several specimens of leaded nickel were subjected to uniaxial tension and compression at 371~Go Significant changes in the dihedral angle were measured as a function of stress. It is felt, however, that the conditions which allow interpretation of the dihedral angle as a ratio of interfacial tensions are not satisfied under these conditions of loading. It is concluded that the changes observed in the dihedral angle in these experiments need not have been due to changes in interfacial tensions. ix

CHAPTER I INTRODUCTION The grain structures of well-annealed, single-phase polycrystalline metals have been shown to bear a. striking resemblance to the structure of soap bubbles in a. froth. This similarity arises because metal grain boundaries and soap films both possess a certain free energy associated with their interfaces. When certain conditions are satisfied, the most stable configuration of the boundaries in both cases is the one which minimizes the total interfacial free energy. The grain structures of well-annealed, multi-phase alloys are determined by the same condition. No soap froth analogy can be made here, however, because the interfaces between like and unlike phases in the alloy possess different energies. This fact has some interesting consequences. In a single phase metal, individual grains tend to the same shapes regardless of their interfacial free energy. In a metal with two or more phases, the minimization of total interfacial free energy produces configurations in which high energy boundaries tend to be replaced by low energy boundaries. Thus, the shapes of grains of different phases depend on the relative magnitudes of the interfacial energies of the various types of boundaries present. A simple type of multi-phase alloy, illustrating these principles, consists of a matrix phase in which a; small amount of a second-phase having a low melting point is dispersed. The second phase particles readily assume equilibrium shapes during annealing at temperatures above the melting point of the second phase. -1

-2The shapes vary depending on whether the particles occur in the interior of matrix grains, at grain boundaries, at grain edges, or at grain corners. A grain edge is the junction line where three matrix grain boundaries intersect; a grain corner is the point of intersection of four or more grain edges. When the particles are wholly included within a grain, they become spheroidal. When they are in contact with one or more of the matrix grain boundaries,their shapes are governed by their tendency to wet the grain boundary. The wetting is determined by the relative magnitudes of the interphasal and grain boundary energies. It is convenient to characterize the "wettability" of the second phase by its "dihedral angle," the angle formed by the apex of the second phase particle penetrating the grain boundary. Under certain conditions, the dihedral angle is a function of the ratio of grain boundary and interphasal energies. The dihedral angle serves as a "'boundary condition" (in the mathematical sense) on the shape of the second phase particle. Interfacial free energies or interfacial tensions (a distinction will be made later) are thermodynamic quantities whose values depend on the thermodynamic constraints on the system. If the temperature, pressure, or the composition is varied, the resulting changes in interfacial tension are reflected in changes in microstructure via changes in the dihedral angle. Sizable changes in interfacial tensions can occur when the composition of the system is changedo For example, a second phase consisting of a lead-bismuth alloy in copper can be made to produce dihedral angles from 15~ to 60~ by varying the ratio of lead to bismuth(5 Variations in temperature can also cause changes in interfacial tensionso The

-3dihedral angle of lead in copper decreases from 60~ to 0~ as the temperature is increased from the melting point of lead to the melting (51) point of coppero Similarly, changes in the dihedral angle are to be expected when the pressure on the system changes, because the surface tensions which determine the dihedral angle must change with pressure. The magnitude of this effect has not previously been established for systems involving metalso Changes in interfacial tensions with pressure have been observed in gas/liquid and liquid/liquid systems, however. The shapes of such second phase particles are important primarily because they determine the extent to which the second phase can influence the bulk properties of the alloyo Many phases with low melting points are weak or brittle When the bulk material is stressed, stress concentrations occur at these particles due to differences between phases in elastic or plastic behavior. The stress concentrations cause cracking and premature failureo These effects are most pronounced when the second phase can wet the grain boundaries and grain edges of the matrix, ioe,, when the dihedral angle is low. The detrimental effects of the second phase are minimized when the dihedral angle is high. For example, lead inclusions in nickel cause the alloy to be extremely.weak and brittle. This probably occurs because the dihedral angle of lead in nickel is low enough (about 50~) to allow unlimited spreading of lead along the edges between matrix grains. Lead inclusions in copper cause some decrease in strength and ductility of the alloy at room temperature, but the higher dihedral angle of lead in copper (about 65~) does not allow spreading along grain edges, hence the effect of the weaker phase is minimized,

-4Two other occasions may be mentioned where relative magnitudes of interfacial tensions play an important roleo Sintering is sometimes carried out with a liquid phase present, When this liquid wets the solid particles, sintering rates are much more rapid. This happens because the solid particles are held together by the surface tension of the liquid during the early stages of sintering and because high rates of mass transport are possible in the liquid. The effectiveness of the liquid depends on its wettability; this can be changed, as indicated above, by changing the temperature, pressure, or composition of the system. Secondly, it has been found that certain metals exhibit lowered fracture strengths and brittle fractures when stressed in certain liquid metal environments. Annealed copper has a fracture strength in liquid lead which is three times its fracture strength in liquid bismuth.7) Intermediate liquid compositions produce intermediate fracture strengths. It is generally felt that this indicates that the energy required for initiation and propagation of cracks at the solid surface is lowered when the crack is exposed to certain liquid metals. This energy is identified with various interfacial tensions in the system, For a given solid metal, only a few liquid metals, if any, produce a. marked decrease in fracture strength. Thus, in systems where this phenomenon is observed, rather special relations must exist between the various interfacial tensions. In an effort to obtain information on changes in wetting:with variations in the constraints on the system, experiments were undertaken to measure the effect of changes in temperature and pressure on the dihedral angle of lead in nickel. In addition, specimens were subjected to tensile and compressive stresses, and changes in the dihedral angle

-5were observed. These results are discussed in terms of changes taking place in the interfacial tensions of the solid nickel/liquid lead interface and the nickel grain boundary. In the next chapter the thermodynamic description of interfaces and interfacial tensions will be discussed as a backgrpupd for a review of the pertinent literature in Chapter IIIo

CHAPTER II THERMODYNAMICS OF INTERFACES In this chapter the thermqdynamics of interfaces will be reviewed. This is intended to serve as a background for the literature review in the next chapter, and as a basis for treatment of the experimental results in Chapter VIIo In this review the emphasis will be on the classical description of interfaces presented by Gibbs. No rigorous derivation of the Gibbs Adsorption Equation will be presented. Instead, the various parameters used by Gibbs and others to characterize an interface will be defined, and the conditions for equilibrium and stability in systems involving interfaces will be discussed. The assumptions needed to extend the application of the Gibbs Ads.orption Equation to solid surfaces will be reviewed. In the final section, the Gibbs Adsorption Equation will be written in appropriate forms for describing the dependence of interfacial tension on temperature and pressure. 1o Fundamental Definitions It seems particularly necessary when discussing surface phenomena involving solids to clearly define the meanings of the various quantities considered, There is little consistency in the literature, partly because great quantities of work have been done in special areas in solid surface chemistry where attention has been focused on the adsorbing substance and the properties of adsorbed films. Another major source of inconsistency has been the carry-over of concepts used to describe liquid-liquid and liquid-gaseous interfaces to interfaces involving solids. For example, -6

-7(65) McLean uses Guggenheim's expression for the dependency of the surface tension of liquids on temperature and composition to. describe temperature and composition dependencies in metal grain boundarieso But Guggenheim says that in solids "there is no quantity analogous to surface tension"(35) (at least the way he defines it) which suggests that McLean's application deserves justification which he does not provideo The Second International Congress on Surface Activity in 1959 included presentation of nine papers under the category of the "Solid/,1(112) Liquid Interfaceo The heated discussion which followed their presentation indicated a basic lack of agreement on the significance of the terms surface energy and surface tension when applied to solids, or for that matter, when applied to liquids in,-some caseso In the discussions to follow, the meanings of surface free energy, surface tension, and surface stress will be those given by (90) (52) Herring3 Shuttleworth,(9 and Johnson,(5 all of which are based on the original work of Gibbs (29,2) Dividing Surface and-Excess Quantities Gibbs defined the various properties of a surface in terms of excess quantities which are referred to a. dividing.surface separating.phases. Consider a system of bulk phases a and 3 separated by an interface So The dividing surface, according to Gibbs, is a geometrical surface "sensibly coincident with the physical surface of discontinuity" but having a "precisely determined positiono" Let a dividing surface of some sort be chosen to separate phases a and 3 so that it lies between surfaces C and CD as shown schematically belowo The portion of the dividing surface under consideration is sufficiently small to be considered planeo

-8C., —--- S The dividing surface a' ----... C The specific properties of the material in phase a exist unchanged up to C, and the specific properties in ~ exist unchanged up to Co Regions a' and P' lie, respectively, between the surfaces C and S, and Ca and S. Let T C and TV be the total quantities of some extensive variable (eogo, the maS's of component i, the free energy, etco) lying within a volume of a' and P, respectively, which is circumscribed by normal lines to a closed curve enclosing an area A T' T" in So Then -- may be considered to be the surface density of the variable To Now let T and TP be the quantity of the extensive variable T in the interiors of phases a and P, respectively, existing in the same volume as T and T o Then (S) i(~). + T ^ ) - (Ta + T') (2-1) -where T is the surface excess of the particular quantity T per unit area of dividing surface. In other words, T = TaV + V + T()A (2-2) T a 6 where T is the total quantity of T in the system, T and T the quantities of T per unit volume, V and V the total volumes of phases a and P, where the phases extend up to the dividing surface, and A the area of the dividing surface,

-9This analysis may be used to define excess internal energy, Helmholtz free energy, or mass of the ith component. Note, however, that since volume has been used to define the excess quantities, that (s) Oie VT V = 0, i.eo. V = V + VO One could however, define a non-zero excess volume per mass (using a mass rather than volume basis for defining excess quantities) which is related to the surface excess of mass on the volume basis used aboveo Guggenheim and Adam(3 discuss alternative conventions for describing the properties of the surface. Movement of the dividing surface in the direction of its normal affects the magnitude of the excess properties, except when the phases ca, p are identical (which is the case for grain boundary between like crystals in a solid)o The position of the dividing surface when a and a are not identical is usually chosen for convenience, eogo, to make the surface excess vanish for the major constituent of one of the phases. When the phases ca D are identical. the dividing surface cannot be chosen so that one of the surface excess quantities vanisheso Surface or Interfacial Free Energy The surface free energy f(s) is the excess Helmholtz free energy per unit area of dividing surface determined in the manner described above. Shuttleworth() points out that for a one component solid in equilibrium with its vapor in a constant volume enclosure at constant temperature, the amount of reversible work required to form a unit area of surface by a process of division is f'(s) The subscript i is 1 (s) needed for crystalline substances because f( ) varies according to the crystallographic plane of divisiono (For liquids and glasses f(s) is not dependent on orientation )

-10Surface or Interfacial Tension (29) The surface tension y is given by Gibbs( as Y f(s) - 2 i.r (2-5) i 1 where the subscripts i refer to various components in the system. The chemical potentials p.. are defined by the usual relations in either bulk phase: i ( ai TV N ( a, )i (2-4) (45) Herring ) points out that for solids, shear strains must be held constant as well as volume in the first expression, and constant stress should replace constant pressure in the second expressiono The quantities [ are the specific surface excess quantities of mass of component i. If N. and N are moles of the ith 1 i1 component, (corresponding to T and T above) then - + N + N) By 1 ( -- - )- _ ( — -— ) (2-5) Note that since Shuttleworth(90) considers only a one component solid in equilibrium with its vapor, he may choose his dividing surface S in such a way that the surface excess of his one component is zero. -Whenever this is so, the surface free energy and surface tension are equal. For a two component system however, one cannot in general choose a dividing surface so as to make-both surface excess quantities vanisho Hence, in this case, surface tension and surface free energy are not equal,

-11(45) Herring(3) has defined surface tension in precisely the same manner as Gibbso It must be realized, however, that when Herring defines f(S) as the Helmholtz free energy per unit area of surface it is the same as the surface excess free energy (surface free energy) defined above. Similarly, Herring defines i = i (S)/A where, in terms i i of quantities previously defined,. u(s)= (N +N ) - (Ni N. (2-6) 1 1 1 1 / vi\i/ (26 For a system consisting of more than one component, y measures the energy required to create unit area of solid surface where, as before, y will vary with the crystallographic orientation of the new surface createdo Surface Stress Gibbs 29) says that for solids "'we may regard (7) as expressing the work spent in forming a unit of the surface discontinuity,.but it cannot properly be regarded as expressing the tension of the surface. The latter quantity depends on the work spent in stretching the surface, while the quantity (7) depends on the work spent in forming the surface " In the case of liquids, the work spent in stretching or in forming a surface is the same, but for solids capable of existing in different states of strain, the quantities must be distinguishedo (45) Following Herring, consider an interface S separating two bulk phases a and A at least one of which is a solid. Then consider a plane P erected perpendicular to S9 having a normal n lying in So If we imagine P to be a cut across the bulk phases and the interface, then in order that atoms in the plane of the cut retain

-12the positions they had prior to cutting, external forces must be applied acting across and in the plane -P In general, a normal force acting in the direction of n and two perpendicular shearing forces in the plane P will be required. The contributions to these forces resulting from the presence of the surface is defined as the surface stress, One can imagine that this contribution.is arrived at by a process of dividing the total forces into bulk and surface contributions as was done in defining excess quantities earlier. When P is rotated about an axis normal to the interface the surface stress acting across the interface is given by 2 gpv'v where the surface stress tensor g v has components for the two values of v going with directions in the plane of the interface, and components for the three values of ip representing perpendicular directions in S and normal to So By considering infinitesimal deformations of the surface which may change y, Herring arrives at the following expression for gv = 9v + = + -;, 5 (2-7) gl-v ^ ^ ~v =1, 2 where E is the strain tensor of the deformation and 6 is the [I-V I-tV Kronecker Delta. Shuttleworth s treatment() of the same problem is simplified because he considers ohly surface stresses in the external surface of a one component solid in equilibrium with its vapor. As mentioned (s) before, this allows one to equate y and f() by choosing the dividing surface so that r = 0, For this case, if one chooses x and y axes in.the plane of the interface, the surface stress may be expressed in general by, three

-13components, g gyy and g When principal axes are chosen, g vanishes so the surface stress for an external surface can be xy characterized by two numbers. Shuttleworth considers an infinitesimal deformation of a rectangle in the surface, oriented with principal surface stresses perpendicular to the sides of the rectangle (no shearing stresses)o For a reversible, constant temperature deformation, the surface contribution to the work is g dA + g dA d(Af(s)) which, for the ~xx x ~y y P isotropic case gx = g g reduces to xx yy g = rf() + A ( /dA) (2-8) a relation analogous to the definition above, but clearly more restricted. (Shuttleworth calls f(s) the surface energy and g.. the surface tensiono This is logical nomenclature for a one component system, but it leaves no name for 7 when a multi-component system is considered. Note that Shuttleworth's symbol y is the same as g used here ) Alternative Treatments Guggenheim(35) has presented treatments of interfaces not involving solids which are not based directly on Gibbs: notion of a dividing surface between phaseso Guggenheim cites Van der Waals and Baeer( ) as the principal source for his development Consider a plane interface between bulk phases a and P and pass two planes C and C through the bulk phases a and a near the interfaceo C and Co are placed so that the properties of the matter at or near C and CD are the properties of the matter in the interior of the bulk phases a and 3, respectivelyo (It is possible

-14to show that such relations as are derived for the surface are independent of the precise location of C and C as long as the above conditions are true ) The distance between CC and Co is T, the thickness 6p of the interface, which should be less than 10 cm, but is probably smaller. The interface S is treated as a bulk phase which differs from phases a and P only in the property that when the system is under hydrostatic pressure, the force across any unit area of S is different on planes parallel and perpendicular to C and Ca, the difference being the surface tension. Let Ts be the total quantity of some extensive variable contained in a volume of interfacial material lying between C and CG, and circumscribed by normal lines to a closed curve in, say, C enclosing an area A in C o Then the interfacial density of T is simply T = T /A Note that this is not an excess quantity, Guggenheim's expression for the surface tension is T = Fr + PT - i C nr. (2-9) where F = FS/A is the surface density of Helmholtz free energy, n = ni/A is the surface density of component i, P is the pressure, Vs.i the chemical potential of the ith component, and T = - the volume of the surface phase corresponding to unit interfacial area. This expression may be reduced to Gibbs' definition of surface tension as follows:

-15s F2 n. = F?" + F.N 1 1 1 TA = V" + V V V= In terms of excess quantities: aC (s) F_ + FP "i = a i s = A Substituting in Guggenheim s expression: Y = f(s) 2- 3 l +F A (' + PVn ) i I i A 1 4 (FP + - +Z ^1^) The two terms in parentheses are zero because Fa + PV = G =. 1 a Hence the surface tensions defined by Gibbs and Guggenheim are entirely equivalent Guggenheim's skepticism as to the reality of surface tension in solids is undoubtedly based on the identification of y with a real force in the surfaceo The distinction that Herring has made between surface tension and surface stress g i should remove this objectiono 20 The Gibbs Adsorption Equation In this section a derivation of the Gibbs Adsorption Equation applicable to a system of two fluid phases separated by an interface will (29) be outlined following Gibbst In the next section, the criteriafor

-16stability of an equilibrium state with respect to motion of the interfaces will be discussed and applied to systems which involve solids as well as fluids o Finally, the applicability of the Gibbs Adsorption Equation to systems involving solids undergoing.changes in temperature and hydrostatic pressure will be discussedo Gibbs() shows that Equation (2-10) describes the dependence of the excess internal energy of the surface U, on the excess entropy of the surface S, the area of the surface, A, and the excess number of moles of component i present in the surface, m.o U; S and m. are excess quantities for an area A of interface; they are not specific excess quantitieso dU = TdS + dA + t[idm, (2-10) Equation (2-10) is a fundamental relation giving the internal energy as a homogeneous function of the first degree in the variables S A, m, oo mi. Hence, the temperature, T, interfacial tension y, and chemical potential pi are homogeneous functions of zero degree in the variables S, A, mi, o.o, mn given by the relations = T i = 1,. o.,n I A.m. m. 1 ~Ua)='? i = 1, o, n (2-11) A/ S ma -- -S = M 1 ^i/^-[ Sa, y7 m. J

-17Since UJ is of the first degree in its variables, then by the Euler relation U = TSc + yA + So.m 0 (2-12) i m 1 Differentiating Equation (2-12) we have dU& = TdS + S dT + ydA + Ady + idm d, (2-13) i m i i Setting Equations (2-10) and (2-13) equal to one another gives SdT + Ady + m, = 0 (2-14) i 1 1 Dividing (2-14) by the area A and rearranging, we have d = -S dT - d (2-15) i di where I(s) mo - P ^ is ) S and i - A — and Equation (2-15) is the Gibbs Adsorption Equation o Note that it is obtained (up through Equation (2-14)) by the same procedure which is used to obtain the Gibbs-Duhem relation for either bulk phase. Note also that in Equation (2-15) the interfacial tension depends only on the temperature and the chemical potentials of the components presento Gibbs shows that at equilibrium the temperatures of all phases and the interface are equal,9and the chemical potentials of all components are the same everywhere in the system. Thus, the interfacial tension, 7, is completely determined by intensive properties of the bulk phaseso For fluid phases separated by a curved interface, Gibbs shows that the condition for mechanical equilibrium in the system is

-18p - Pi = y (C1 + C2) (2-16) where p, p are the pressures of phases a and a on either side of the interface, and C1 and C2 are the principal curvatures of the interfaceo When the interface is plane, this reduces to p = p which is the condition for mechanical equilibrium between bulk phases when interfaces are neglectedo 35 Criteria for Stability of an Equilibrium State The equilibrium conditions for a system involving an interface, ioeo uniform temperature, equal chemical potentials in all phases, etc,, are derived by Gibbs from the condition that at equilibrium the energy of the system must be an extremum, when the total entropy and volume of the system, and the masses of all components are held constanto For stable equilibrium, the energy of the system must be at a minimum. Gibbs shows that when the temperature and the chemical potentials of all components are uniform throughout the system and the surface tensions are constant over each interface, a condition for stability is that "in a stable system each surface of tension must be a surface of minimum area for constant values of the volumes which it divides, when the other surfaces bounding these volumes and the perimeter of the surface of tension are regarded as fixedo" He then derives a condition of stability with reference to motion of interfaces which is sufficient, but not necessaryo This condition is the basis for several methods of measuring interfacial tensions. Consider a system composed of several bulk phases separated by interfaces of various kinds.Let the system be in equilibrium; thus the temperature and the chemical potentials of all components are every

-19where constant. The total entropy and the total quantities of all components remain fixedo Gibbs shows (29. p. 252) that under these constraints, a sufficient condition for the stability of an equilibrium state with respect to motion of interfaces is' the sum of the products of the areas of the interfaces by their tensions, less the sum of the products of the volumes of the phases by their pressures, must be a minimumo This condition is expressed in Equation (2-17), where the notation indicates that the function in brackets is a minimum in the stable stateo min,iAi - Spjdv. (2-17) L i j~L J JJ j Now in many situations Equation (2-17) can be reduced to a simpler formo Consider a system of two bulk phases separated by an interface, the whole system being in thermal, chemical and mechanical equilibriumo The bulk phases may now be either solids or fluidso Let the volumes of the two bulk phases be held constant and let us consider changes in the mechanical work done on the system when the interface is allowed to move. Changes must in general occur in the terms \ idAi because the interfacial areas are-being variedo If the state of stress on the bulk phases is substantially unaltered when the interfaces move*, then the terms for the bulk phases wi:Ll be unchanged because the volumes of the bulk phases are constanto In particular, when all bulk phases are under a uniform hydrostatic pressure, then the terms in Ipjdvj do (45) * Herring points out that there must always be an inhomogeneous stress in a solid near an interface which is needed to offset the surface stress. This inhomogeneous stress will in general change as the interface changes positiono Herring shows that this effect is negligibleo

-20not change when the position of the interface is changedo Under these conditions Equation (2-17) reduces to min i i dAi (2-18) If the interfacial tensions are independent of crystallographic orientation (if any of the phases is a solid), or if all phases are fluids, application of condition (2-18) to intersecting sets of interfaces produces the familiar "vectorial balance" of surface tensionso Herring3) has applied the condition (2-18) to systems in which the interfacial tensions do depend on crystallographic orientation and obtains a more general type of relation. These equations will be discussed more fully in the next chapter. It must be emphasized here that all equations of this type depend on the reduction of condition (2-17) to condition (2-18)o If for some reason this reduction is not possible, then none of the equations expressing the balance of tensions about a point are valid. Furthermore, the condition (2-17) is sufficient for stability, but not necessary; hence, stable equilibrium may be possible without satisfying either (2-17) or (2-18)o 4o Application of the Gibbs Adsorption Equation to Systems Involving Solids (43) Herring4) notes some conditions in which the Gibbs Adsorption Equation (2-15) does not give the total change in interfacial tension for a solid surface for all types of changes in the system. These circumstances are: (1) when a displacement of an interface changes its radius of curvature, (2) when the orientation of the crystalline surface changes, andy (3) when the state of strain of a crystalline surface is altered (othero th e dilation of the bulk phase normal to the interface)

-21Situation (1) above can generally be disregarded because the interfacial tension is substantially independent of curvature, except when the radii of curvature are on the order of atomic dimensionso Neither of the second two situations can be generally disregardedo The dependence of the interfacial tension on crystallographic orientation varies considerably from system to system. Studies of small angle grain boundaries between crystals of the same phase have shown a negligible dependence of the tension on boundary orientation, if the crystals across the boundary have a fixed orientation with respect to one anothero The appearance of generally spherical particles of low melting phases within solid grains indicates that the solid/ liquid interfacial tension does not have a strong dependence on the crystallographic plane exposedo On the otherhand, the development of facets on crystals at high temperatures may be related to significant variations in the surface tension on various crystallographic planes, as may the -ccurence of some plate-like or rod-like precipitates in metalso (These effects, however, may also involve preferred directions for growtho) If the change in surface tension with a change in the constraints on the system is measured on a large, random collection of interfaces of all possible orientations, then the average surface tension of all these interfaces should be independent of crystallographic changes occurring at specific interfaceso The Gibbs Adsorption Equation can then be applied to this average tension to describe its changes when the constraints on the system are changedo The final condition noted in the first paragraph seems to seriously restrict application of the Gibbs Adsorption Equation to

-22solid surfaces. Gibbs says (29, p. 528) that Equation (2-15) may be applied to solid surfaces subject to the limitation "that the state of strain of the surface of the solid remains the sameo" He goes on to say that "this limitation may in most cases be neglected," but "if the quantity (y) represented the true tension of the surface, as in the case of a surface between fluids, the limitation would be wholly unnecessary," It is unfortunate that Gibbs did not expand upon this subject, because it is not at all clear in which cases changes in the state of (45) strain of the surface may be neglected. Using Herring"s expression for the surface stress tensor as a function of surface tension, ~7y g. = yb& + * gij = Z ij + c.. 1j the condition for applicability would seem to be either that the change in the state of strain is very small or that the differential coefficient is small compared to 7, It must be assumed that Gibbs had in ij mind the latter caseo For example, a decrease in temperature of 50~C causes strains on the order of -7104 in/in in nickel. (About the same amount of strain is caused by the application of 60,000 psi hydrostatic pressureo) These strains certainly result in a change in the "state of strain" of the solid surfaceo It seems quite unlikely that Gibbs would make the statement that Equation (2-15) was applicable to changes in solid surface tensions "in most cases," if changes in the state of strain of this order of magnitude could not be tolerated, It will be assumed in all subsequent chapters that the Gibbs Adsorption Equation can be used to express the change in interfacial

tension with temperature and pressure for solid/solid and solid/liquid interfaces. This implies that the surface tension and surface stress (45) do not differ significantly. Herring3) notes that several atomistic calculations of the surface stress for homopolar crystals, alkali halide crystals and rare gas crystals indicate that the surface stress and surface tension should be significantly different. On the other hand, the conditions for equality of surface stress and surface tension in solid metals appear to be satisfied as much as 200~C below the melting point (cf. measurements of surface tension by the Udin-Shaler-Wulff method( 054)). McLean (65 feels that the surface tension and surface stress of ordinary large angle grain boundaries should be "nearly equal...at high temperature." No experimental measurements of the difference between surface stress and surface tension in solids have been made. To summarize, the stability conditions presented in the previous section insure that interfaces in systems involving solids under hydrostatic pressure assume a configuration such that, neglecting the influence of solid crystallography, a vectorial.balance of interfacia.l tensions occurs. As will be shown la.ter, this guarantees that ratios of interfacial tensions can be measured. However, expressing the dependence of these interfacial tensions on temperature and hydrostatic pressure by means of the Gibbs Adsorption Equation involves the assumption of equality of surface tension and surface stress. 5. Derivation of Useful Relations from the Gibbs Adsorption Equation The dependence of surface tension on temperature, pressure and composition of a system is given by the Gibbs Adsorption Equation:

-24.n dy = -S(S) dT - n C dpi (2-19) w (s) where S(s) is the surface excess entropy, and the sum is taken over all components of the systemo The discussion to follow considers a two-component system consisting of two phases, a and D, and an interface separating them~ For solids whose surface tension is dependent on crystallographic orientation, the following derivation is applicable for a particular set of orientation coordinateso Let (1) and (2) designate the components of the systemo The Gibbs Adsorption Equation is dy = -ss)dT - rdl - 2 dl (2-20) The chemical potential of each component is constant throughout the, (s a (s (s) system at equilibrium [1 = p= hence d = dp = dp o Let x, x be the mole fractions of component (1) in phases a and o Then (1 - x )9 (1 - xP) are the mole fractions of component (2). The chemical potentials in Equation (2-20) may be written: la 1a Otla a aX =t-l dT dP + -- ox (2-21) for phase a, and. d.1 = - dT + - dP + dx ft R R ~(2-22) fo2 r dT + p —s dP + d(xfor phase o.

-25Equations (2-21) and (2-22) may be simplified using the following relations~ T / jpx = -VS (2-23) P/ Tx i (2-24) - j = -. J j - ( - xj) = (2-25) xj )xji where Sa and V. are, respectively, the partial molar entropy and 1 1 the partial molar volume of component i in phase j, and Equation (2-25) is the Gibbs-Duhem relation between chemical potentials in phase j. Inserting Equations (2-23), (2-24), and (2-25) in (2-21) and (2-22), the differentials of chemical potential in each phase are: dL- = S dT + 1 dP + cdx dx Ca (2-26) d =- -S2 dT + 2dP - X dx 1 - x 6x and dL = - dT + V dP + - dx (2-27) l2 = -S2 dT + V dP _- x- dx ~2,~ 2 2 l-~x xx.Either Equations (2-26) or (2-27) may now be substituted into Equation (2-20)o Substituting (2-26) we have: [rd s r 2 dx (2-28) b,=cd

-26In some cases, Equation (2-28) may be considerably simplified. For example, if phase a and a have different compositions, then the dividing surface between phases can be chosen so that surface excess of one component vanishes, e.g. = Oo Then at constant temperature and pressure, Equation (2-28) reduces to: = F1() dx (2-29) where /1(2) is the surface excess for this choice of dividing surface. Since 1 - p{1n = RT n a RT in f x where a is the activity and f the activity coefficient of component (1), then a~ ET a in f~ RT._1 + RT (2-30) E a a Fo x oox For solutions in which either Henry's Law or Raoult's law.is valid for component (1), the second term on the right hand side of (2-30) vanishes. Under these conditions, Equation. (2-29) becomes dy= - (2) RT d In x (2-31) which is a commonly encountered form of the Gibbs Adsorption Equation. Simplifications of Equation (2-28) will now be derived for the solid/liquid and grain boundary, surface tensions. Solid-Liquid Interfaces Consider the interface between a solid and a liquid, where the solid consists almost wholly of component (1), while the liquid is

-27rich in component (2) with very limited solubility for component (l)o The dividing surface will be chosen so that F = 0. Then, with chemical potentials referred to the liquid phase, Equation (2-28) may be written: (2) - (2) s T - 1(2) 1 (2) x d -2) At constant pressure, the variation in surface tension with temperature may now be written: _7 S fg(s) r S1" r atl dx dT - (2) (2) 1 1(2) ax dTx (235) If the activity of component (1) in the liquid is proportional to its mole fraction x, then,- ~ k R 1_( RT 2-34) dx x and y _ (s) f 1 d in x dyF - ~> - - RT (2-35) dT- = S(2) 1 2)s\_ - 1(2)R- (235) Without making an assumption as to the relative magnitude of the quantities in (2-35)9 further simplification is not possibleo If the solubility as a function of temperature and the partial molal entropy (s) of component (1) in the liquid are known, the quantities S(2) and 11(2) could be determined if at least one was assumed constant with temperature and other was an assumed function of temperature, However, this would require great precision in measurements of y vso To At constant temperature, the variation of surface tension with pressure is:

-28-'1t7(2)Lvl + i-^ dx-J (2-36) dP 1(2) + - -- dP Substitution of Equation (2-34) gives: dyb - r& < R d ln x1 d-P - T(2) [v1 + RT d (2-37) If the solubility as a function of pressure and the partial molal volume of component (1) in the liquid are known, 1(2) may be determined from measurements of y vso Po In mrany cases, the thermodynamic properties of the liquid solution are not known. One can obtain simple expressions for dY/dT and d7/dP replacing Equations (2-35) and (2-37)9 if the solid can be assumed to be pure component (1). Then the differential of chemical potential referred to the solid is: dps K= -sdP V 5dP at constant T., s sd'l = -s SS dT = - SldT at constant P. Inserting these results in Equation (2-19), taking r = 0, one has d.y/d.T = -S(s)+ E s dy/dT = -S(2) + (2) SlS at constant P (2-38) dy/dP = - r2 V1 at constant To (2-39) S Q Since values for the molar entropy, Si and the molar volume, Vi, of the pure solid are generally available, measurement of the temperature and pressure coefficients of interfacial tension permitscalculation of (2 and S(2)

-29Grain Boundaries For grain boundaries between phases of the same density and composition it is not possible to choose a dividing surface so that the interfacial excess concentration of one of the components vanisheso In addition, the interfacial excess quantities are independent of the choice of dividing surfaceo Thus, formal simplification of Equation (2-28) is not possible by the methods previously usedo At constant temperatures and pressure, Equation (2-28) simplifies to: dy = _ -- a d ] ar a dxa (2-40) [- K! - Xk 2 (14) Cahn and Hilliard ) have suggested that the quantity in brackets be designated f o)o r 2)is the difference between the actual excess concentration of component (1) at the interface, and the excess concentration of (1) which would be required just to maintain the bulk composition in the boundary, given the excess concentration of component (2). Thus, (2) indicates the direction in which the composition of the boundary deviates from the bulk compositiono In the case of strongly segregating component (2) in the grain boundaries of almost pure (1), r would be negativeo One could, in principle, determine (2 from measurements 1(2) of the variation in 7 with x using a Udin-Shaler-Wulff type experiment (5) The precision required is probably unattainable, however In a more general situation, this approach provides no simplifications. One can, however, obtain more usable equations if a few assumptions are made.

-30At constant temperature,. Equation (2-28) applied to the grain boundary gives the pressure coefficient of grain boundary tension: L lvi + le (v1 - Lr - ~ r (241) dP 1 1 2 1 1 - 2 ax P In this equation, the surface excess quantities refer, of course, to the grain boundary, but the differentials of chemical potential occurring in Equation (2-20) are now referred to the liquid phase. If the system is in equilibrium, the chemical potentials are the same in all phases; hence this procedure is permissible. 1 dxr An expression for - - is easily obtained if one assumes dx that the solid is pure component (1) and the liquid is a mixture of the two components Then d-i V'. dP in the solid a- ~ (2-42) --'1 ~ du = V1 dP + - dx in the liquid. Equating these two expressions and rearranging gives: a s b- -- X1 - -v1 s where V is the molar volume of the solid and V is the partial molal -l 1 volume of component (1) in the liquid. Substituting (2-43) in (2-41), rearranging and using the relation V = x V + (l - ) where VT is the molar volume of the liquid, one has y = (Vv (2-44) dP 1-1 - ( 1 -44) 1 - x

-31Now if one could assume that the surface excess quantities are independent of temperature, measurement of dy/dP at two temperatures would allow calculation of the two surface excess quantities rI rIo But2 1 r2 are most likely to be constant with temperature if there is little change in the liquidus composition with temperature. Since molar volume changes in the solid and liquid in this case would then be, essentially, only due to thermal expansion, the change expected in dy/dP would be very small. This small difference would make the computation of I very ina..ccurate For the case of a grain boundary in a one component solid, the pressure coefficient of surface tension is given by dy/dP - r rv where V. is the molar volume of the solido If the grain boundary is 10% less dense, on the average, than the grain interior and if the density of the bulk material is p and the boundary is 10-7 cm thick, then -8 2 = -10 p gms/cm and V. = 1/p cm3/gm hence dy/dP = 10 cm So a pressure change of 1000 atmospheres will produce an increase in the surface tension of about 10 ergs/cm o It will be sihown later that this change is e enough o be endetected frommeasurements of ratios of surface tensionso

-32When a solute is present which segregates at the grain boundary one would expect r1 0 and r > 0o Referring to Equation (2-44) for the saturated solution, it is apparent that dy/dP could be either positive or negative depending on the extent of the segregation and the thermodynamic properties of the system. The temperature coefficient of the grain boundary tension may be expressed in an entirely analogous manner. Assuming the solid is pure component (1)9 then ~ P1 dx s dxP - S (2-45) d T 1 -l where S.' is the partial molal entropy of component (1) in the liquid, and S. is the molar entropy of the solid. The temperature coefficient is: -1 d _T s) + S1 - x S1 (t2-46) where S is the molar entropy of the liquid~ For a grain boundary in a one component solid, (2-46) reduces to d.y/dT = _ S ) + s Since < 0 one would expect the temperature coefficient of the grain boundary tension to be negative. In a two component system, it is obvious that the temperature coefficient may be either positive or negative depending on the degree of segregation at the boundary and the thermodynamic properties of the system.

CHAPTER III LITERATURE REVIEW The literature chosen for review pay be classified under three main headings~ (1) a survey of some measurements of ratios of interfacial tensions, obtained by measuring the dihedral angle, (2) a survey of measurements of changes in interfacial tension with pressure, and (3) a review of two areas9 brittle fracture and liquid phase sintering, in which interfacial tensions play an important roleo 1o Measurement of Ratios of Interfacial Tensions Many methods have been devised for measuring the interfacial tension between two fluid phases. The two most common are measurements of capillary rise and measurement of drop shapes. Adam's book describes these methods and their theoretical basiso The only means of measuring with any degree of reliability the surface tension between a solid and a fluid phase is the method of Udin, Shaler and Wulffo (0 This involves suspending small weights from fine wires, measuring rates of extension, and finding (either experimentally or by extrapolation) the weight which just compensates for the tendency of the wire to contract due to surface tensiono This method yields an absolute measure of the solid surface tension for polycrystalline wires if the ratio of surface and grain boundary energies is known, but is only useful to within 200~C of the melting pointo (41) Hayward and Greenough I have measured the surface tension of nickel by this methodo Shuttleworth(90) Herring,(3) McLean (5) Udin15) and others have discussed this methodo Herring concludes that surface tension and surface stress are equal in these experimentso -33

-34Measurements of ratios of surface tensions are easier to obtain. These are based on equations describing the surface tensions of three interfaces meeting at a line as a vectorial balance of forces. These equations are strictly true when all of the phases are fluids, but are not necessarily true when one or more of the phases is a solido Thus, if phases 1, 2, 3 meet at a line, the surface tensions Yl2 t 253 Y31 are given by the equations 712 723 (31) sin 3 sin e1 sin (See figure l(a)o If two of the phases are identical, the relation is: "ll = 2712 cos e/2 (3-2) See figure 1(b). If the (1) phases here are metal grains, then yi is the grain boundary tensiono If (2) is a gas phase or vacuum, then this expression describes the angle formed during thermal grooving at a grain boundary. If (2) is a liquid or solid phase, then e is termed the dihedral angle 93) The dihedral angle is important in determining the effect of a dispersed "second phase" on mechanical properties. When a liquid droplet is placed on a rigid, undeformable, insoluble solid, the vectorial balance of tensions only takes place in the plane of the solid, as shown in Figure l(c)o sv - s = v cos (3-3) where e is the contact angle of the liquid on the solid (Figure l(c))o

-35Y23 02 83 i " 0I'0 ( 12 3 (a) Y72 (b) V //LV s Ysv (c ) Figure 1. Vectorial Relations Between Interfacial Tensions.

-36All of these relations are consequences of the necessity for minimization of the sum of the products of interfacial tensions and areas at equilibrium. (See Section 3, Chapter IIo) From this principle, Gibbs9) and Johnson() present derivations of Equation (3-3)> Herring3) outlines a derivation leading to (3-1) or (3-2)o It is worth noting at this point that the notion of surface tension as a physical force acting in the surface of liquids leads directly to the proper vectorial relations between tensions on intersecting surfaces. In solids, on the other hand, the surface tension is not a force per unit length but an energy per unit area.o With a modification to be noted, however, the same relations between surface tensions hold for interphasal boundaries involving solids. The name "surface tension" is a misnomer for solids; it is retained because the quantity it describes has the same thermodynamic definition for all interfaces. (4[2) Herring ) has shown that the vectorial balance of interfacial tensions given above is not strictly true when one of the phases involved is a. solid whose interfacial tension depends on crystallographic orientation. By considering virtual displacements of three phase boundaries meeting at a line, the necessity for minimization of _5 (65)r'i l 7yidAi leads to the conditions(65) i=l J 1 l 253 + 3 12os )3 +e2 2 7135 72 cos a + Y12 cos 3+ s - sin = 0 (5-4) 712 + 725 cos 2 + 713 cos1 - sin2 a sine = 0

-37For derivatives with respect to e. clockwise rotation is positive (Figure l(a)). Only two of the above relations are independent, and they do not suffice to determine the interfacial tensions or their derivatives with respect to rotation. For the most general case of the intersection of three grain boundaries, the situation is further complicated because five parameters of orientation are needed to describe the position of the boundary with respect to one of the adjacent crystals, and the terms ay/l6 are dependent on these parameterso If the conditions on a system involving solids are such that the stability condition for equilibrium does not reduce to simply minimization of the total interfacial free energy, then none of the Equations (3-1) to (3-4) are true. Usually one assumes~that for anything but a small angle grain boundary or a twin boundary, the terms expressing orientation dependence in (3-4) are negligibleand Equations (3-1) or (3-2) are used, This point will be expanded upon in the section on the orientation dependence of interfacial tensionso In his classic paper on the role of interfacial energies in (95) determining microstructures, C. S. Smith93) noted that a vectorial balance of surface tensions at the junction of a twin boundary and a curved grain boundary would imply, in some cases, that the twin boundary had a negative surface tension. Since this is impossible, Smith thought that perhaps the twin boundary tension was so small as to be sensitive to slight variations in grain boundary tension as the grain boundary (28) curved past the twin. Fullman pointed out, however, that the behavior of coherent twin boundaries could be understood if one regarded them as "locked" in position due to the large amount of energy needed to reorient

-38them fro'm their position of Coherency. This, then, would only allow a vectorial balance of surface tensions in the plane of the twin boundary. This proposal leads to values-for coherent twin boundary tension of about (28) (72) 0035 of the grain boundary tension2) Mykura(7 has studied the thermal grooving of twins in nickel and used Herring's equations to determine the coefficients of orientation dependence of the surface tensiono These are on the order of 005 to 025 of the surface tension. Measurements of relative interfacial tensions are capable of detecting rather small changes in tensionso For example, the angle of penetration of a liquid between solid metal grains is given by the balance of surface tensions in Equation (5-2). If Q= 500 and Y = ss 500 ergs/cm, then changes in 7 and sl by 10 ergs/cm produce the changes in dihedral angle shown in Table Io TABLE I EFFECTS OF SMALL CHANGES IN INTERFACIAL TENSIONS ON THE DIHEDRAL ANGLE Ratio of Tensions Dihedral Angle 7ss /sl 500 7 + o10 450' sl Yss 58~ Ysl + 10 sY + 10 54 ~ 7sl + 10 7ss - 10 10.Ss — ~ 55~ Ysl s- 10 Ysl -10

-39For a dihedral angle less than 50~, the same change in the surface tensions will produce an even larger change in dihedral angleo 2. Measurement of Dihedral Angles The most straight-forward way to measure a dihedral angle is to measure the trace of the angle on a surface normal to the line of intersection of the surfaces forming the angle. This may be done in sheet specimens where grains are allowed to grow to a width several times the thickness of the sheeto To a very good approximation, all grain boundaries become normal to the surface of the sheet during a long anneal and true dihedral angles can be measured directlyo A second method requires the measurement of a large number of angles made by the traces of grain boundaries on an arbitrary plane of polisho If there is a single dihedral angle characterizing the structure, this will give rise to a particular distribution of observed angles. From the observed distribution, one can determine the dihedral angle which generated ito The details of this method, originated by Hlarker and Parker(39) are given in Appendix A. The first method has the disadvantage that the same crystallographic surfaces will not lie in the plane of the sheet for all grains; hence, these may influence the equilibrated angle obtainedo The second method has two disadvantages a) the model for the distribution of angles observed may be inexact, and b) sampling error limits the precision of the measurements. These last disadvantages will be considered at length later.

-40All of the work described in the following section on the orientation dependence of surface tension was done measuring the dihedral angle directlyo For some experiments, however, it is desirable to use the statistical method to determine the dihedral angle and assume that orientation differences exert a negligible influence on the distribution of measured angleso C93,9 94995) (51) Co. S Smith(939 995) and Ikeuye and Smith(5 measured the angle formed by particles of a low-melting second phase in the grain boundaries of a solid matrix phase, and studied the variation of this angle with temperature~ Ikeuye and Smith(51) noted that the dihedral angle observed was dependent on the composition of the liquid phase present. When the composition of the liquid was changed either by changing the temperature or by adding alloying elements to the system9 regular changes in the dihedral angle were observedo They concluded that major shifts in the dihedral angle with increasing temperature would only occur if the liquid composition varied markedly with temperatureo Van Vlack106) also employed this technique to establish interfacial energies in the Fe-Cu-Cu2S system. Both Smith93) and Van Vlack 6) showed conclusively that the relative interfacial tensions determined in this manner were internally consistent by comparing direct measurements and indirect calculations using other measurements to arrive at the relative tension for a particular interfaceo McLean, (6) Taylor (3 and Chalmers (16) have reviewed this worko More recently, Riegger( ) has correlated the change in dihedral angle made by a. liquid phase with changes in composition of the liquid phase using a semi-empirical relation suggested by Taylor )based on

-41the Gibbs Adsorption Equation for binary systems. If one assumes that the grain boundary tension is nearly independent of temperature and composition changes in the liquid phase, then changes in the dihedral angle reflect changes in the solid/liquid interfacial tension. A plot of the solid/liquid interfacial tension obtained in this manner, versus the log of the solubility of the minor constituent in the liquid, is found to produce a linear relationship. From the slope of this curve one can then, presumably, calculate a surface excess concentrationo Several other assumptions, beyond that of having the grain boundary tension constant with temperature, are also involvedo Referring to Equation (2-35) it is apparent that the entropy terms must be negligible and that the activity of the solute must be proportional to its mole fractiono A constant slope with temperature, then, indicates that the surface excess varies as 1/T. This correlation is not too helpful, because all these assumptions cannot reasonably be made without considerable experimental justification. For example, McLean's calculations (65, po 148) indicate that the change in grain boundary tension with temperature cannot be ignored. 35 Orientation Dependence of Grain Boundary Tension Five parameters are necessary in order to specify the orientation of a boundary between grains 1 and 2o Three determine the orientation of grain 2 with respect to axes in grain 1, and two determine the orientation of the boundary with respect to grain 1o It has proven difficult to establish orientation dependencies of grain boundary tensions because no absolute measure is available. Tensions must always be measured by means of a junction angle formed between the interface of interest and a reference interface which is assumed constant from sample to sampleo

-42Read and Shockley ) calculated the dependency of the energy associated with a grain boundary formed from dislocations on the degree of misorientation between the crystals on either side of the boundaryo They predicted that for angles of misorientation, up to about ~ = 15~0 the energy should be given by E = E f> [A - In ] where E0 is a. function of the macroscopic elastic constants and the orientation of the boundary, and A is related to the energy in the material immediately surrounding a dislocation and is also dependent on the orientation of the boundary This calculation was based on a. model of a simple tilt boundary in a simple cubic lattice. Since this reduces to a two dimensional problem, only one parameter, 4, is needed to specify the orientation of the grains, and another to specify the orientation of the boundaryo Dunn and his co-workers (202122) obtained the.first confirmation of this relationship. Dunn used oriented tricrystals of silicon ferrite, in sheet form, all having the same crystallographic plane in the plane of sheeto Variatios in tensions with the degree of misorientation between crystals were measured with reference to a standard interface between crystals with a fixed degree of mismatch. Although the orientation of the boundary could not be controlled in these experiments, the predicted relationship holds quite well. Some of the scatter in the data (20) is attributed to the variation in boundary orientation which occurred ) Since this work, Chalmers, et alo (1i 5664107) andAust( have determined this orientation dependence for Sn, Pb, Ge, Ag, and AgClo (27) Friedel, Cullity, and Crussard studied this effect in Al, and

-43Gjostein and Rhines(1) in Cu. Weinberg 108) Chamers16) Aust and ChaLmer(7) and McLean 5 have reviewed this work. Conclusions which may be drawn are: a) the grain boundary tension is a function of misorientation between crystals, most of the variation in energy occurring between 0 and 15~ for simple tilt boundaries, b) the grain boundary tension is not a strong function of the orientation of the boundary between grains of fixed orientation relative to one another, except perhaps for low angle boundaries, c) Cusps in the tension versus misorientalicn curve (which are predicted from the dislocation model) are not observed except, possibly, near twin orientationso It is important to know the frequency with which low tension boundaries will be encountered in well annealed polycrystalline materials. Read and Shockley 8) estimate that "energies less than 50% of the average will be obtained in only about 3% of the cases. Consequently, observations of grain boundaries for grains selected at random would show such a small percentage of low energy boundaries that their presence would have only a slight effect on the statistics." This point will be dealt with later in the discussion of errors in the distribution of angles observed due to the presence of a range of dihedral angles in the specimeno Several workers have determined the orientation dependence of the surface tension for metal vapor surfaceso The method involves the use of Herring s relation (Equation (3-4)) applied to thermally etched twins. By measuring various angles formed it is possible to determine the coefficients 1/y -e which express the change in surface

-44(72) tension with orientation. Mykura() showed that the surface tension changes about 6% with a rotation of about 300 from a (100) plane in nickel. Robertson and Shewmon ) found changes of the same order of magnitude in coppero In both cases, the rate of change of tension with orientation is greatest near low index planeso Before employing the method of statistical analysis to determine dihedral angles, it should be established that the large fluctuations in interfacial tension which occur, occur so infrequently as to be negligible, and that minor fluctuations in energy are too small to influence the distribution obtained. On the first point, reference can be made to Read and Shockley's statement above, and to similar statements by Co So. Smith93 (It is a simple matter to show, that two adjacent grains, misoriented by a combination of tilt and twist less than 15~9 occur randomly only 3.4% of the timeo) Experiments on solid/ vapor interfacial tensions lead one to expect that the orientation dependence of any solid/fluid interface is small, but that a few percent variation can be expectedo Hence, large fluctuations in grain boundary tension undoubtedly do not influence the distributiono On the second point, fluctuations of a few percent seem likely in a large percentage of the grain boundary and solid/fluid interfaces one might encountero These variations will lead to a broadening of the distribution of angles seen; a statistical analysis based on the ideal distribution (Appendix A) may be in error due to this broadeningo This point will be dealt with further in the section on errorso 4. Changes in Surface Tension with Pressure Comparatively few attempts have been made to determine the effect of pressure on surface tensiono Before the importance of

-45adsorption was appreciated, it was believed possible to determine the molecular volume of a liquid at an interface by measuring the pressure coefficient of its surface tensiono Several earlier experiments were done with this end in mindo More recent experiments have been performed on systems of interest to the petroleum industryo The motivation for the early work was provided by an easily derived thermodynamic expressiono Consider a system of n components in two bulk phases separated by an interfaceo For the entire system one may write~ dG = -SdT + VdP + ydA + E idn.i (3-5) Since dG is an exact differential, we may write V^/A TPa ni = ) T.An (3-6) That is, the change in surface tension with pressure is equal to the increase in volume of the system with an increase in interfacial area. Hence, in Bridgman's words,() the pressure coefficient of surface tension is a measure of the difference of effective volume of a molecule in the interior and at the surface of a liquido" Lewis and Randall (1923) (6) support this view. It was later realized ) that this interpretation implied an over-simplified picture of the interface, as will be shown below. For a two phases one component system, the quantity' a'y/P)T Ani cannot be determined because the pressure on the system cannot be varied at constant temperature and still retain two phases. If one has a binary system, howeVer, consisting of a pure liquid of one component and a pure gas of another, with no mutual solubility, then one can certainly determine the quantity 7y/aP)T Arnio To see the significance of the term aV/aA)T p ni the following illustration is helpftlo Let a container fitted with a moveable

-46piston P be divided into two sections A and B by a sliding partition C, (Figure 2)0 In section A is pure vapor of substance 1 at a given temperature and pressure., and in section B is pure liquid of substance 2 at the same temperature and pressure. Assume that the properties of the liquid and vapor exist unchanged up to the partition C. Let C be slowly withdrawn and the piston P continually adjusted to maintain a constant pressureo Figures 3(a) and 3(b) show the density relationships before and after the partition is withdrawno Note that in this picture there is an abrupt density change clearly defining a physical "surface" at ao Also there is an "interfacial region" from a to b where the density of material (all substance 2) approaches the value of the density in the bulk liquid phase. Then the volume change of the system may be entirely ascribed to the change in density of the liquid phase in the region abo Now given measured values of 6y/6P)TA~ni and estimating a thickness for the interfacial region_ ab one is able to calculate an average density for the interfaceo Statistical mechanical models for the interfacial transition region in the liquid can then be checked directly from experimental datao This is the physical situation that Bridgman had in mindo A somewhat more real physical situation might be imagined if adsorption of the gas were allowed at the interface, so that the density profile might look like Figure 3(c)o The change in volume of the system with creation of the interface is now the sum of volume changes Which occur in the gas and liquid phases at the interfaceo The quantity ) T,Pni cannot be broken down into OA /V)~r~n^

-47P V v ---- cL Fiur ~2. ermodynamicModelfor CreatI Figure 2. Thermodynamic Model for Creation of a Liquid/Vapor Interface.

-48DENSITY pv C DISTANCE (a ) I I',. I I I PL DENSITY i Pv I I a C DISTANCE (b).) A.e Ra PL I I DENSITY I Pv a C b DISTANCE (c ) Figure 3. Density Profiles at the Liquid/Vapor Interface. (a) Before Withdrawal of the Partition. (b) After Withdrawal. (c) A More Realistic Profile.

-49contributions from the gas and liquid phases without some information about the process occurring. Based on the above approach, statements are often made that since a liquid should be less dense near its surface than in its interior, the expression 6V/6A)Tppni will be positive and the surface tension should increase with increasing pressure. Clearly, however, with adsorption occurring, nothing can be said a priori as to whether surface tension should increase or decrease with pressure In any real case, where the liquid has a significant vapor pressure and the gas dissolves somewhat in the liquid, the transition in density across the interface is probably continuous (on a statistical basis, ignoring atomistics) and even less can be said about the physical meaning of the expression'V/A)rTPni Recent workers have dealt with the change in surface tension with pressure in terms of the Gibbs Adsorption Equation. This approach emphasizes the importance of adsorption at the interface. Recently, experiments have been performed on systems generally involving water and hydrocarbonso In oil reservoirs, recovery of oil in porous rock is affected by the displacement of oil by natural water which is in turn affected by the interfacial tension between the oil and watero But these interfacial tensions are established under conditions of very high pressure; hence it is important to know how interfacial tensions are affected by pressure, A number of experiments have been performed to determine the pressure effect on some important hydrocarbon systems Apparently the first measurements of the change in surface tension with pressure were performed by Kundt (60) in 1881o He measured

-50the change in vapor-liquid interfacial tension with pressures up to 212 atm in the systems CS —air, CHC1 -air9 C2H50H-air, C2H50H-H22 (C2H )20-air, and (C2H5)20-H2o In all cases he found the surface tension to decrease-with increasing pressure. In 1906, Lynde 3measured surface tension changes with pressure in the systems mercury-water, mercury-ether, water-ether, chloroform-water, and carbon bisulfide-water at pressures up to 5000 psi. He found increases in surface tension with pressure in the systems mercury-ether, mercury-water, and carbon bisulfide-water and decreases (ll) in the others. Bridgmani has commented on Lyndevs work in connection with determining the molecular volume at the interface by means of Equation (3-6). (83) Rice ) has discussed the effect of pressure on surface tensiono He points out that a one component liquid in equilibrium with its vapor is'almost certainly" less dense near the surface than in the bulk, hence aV/cA)p T n should be positive and the surface tension of a binary system involving this liquid should increase with pressureo Since KundtVs work shows that the surface tension decreases with increasing pressures Rice sought to meld the interpretation of surface energy changes with pressure on the basis of Equation (3-6) with the interpretation based on the Gibbs Adsorption Equation He considers two component vapor-liquid systems and breaks down i) PTn into two terms' a AV of the gas phase arising from adsorption of gas at the surface of the liquid, and a &V arising from a density decrease in the liquid near the interface. (Implicitly, his model demands that the dividing surface to which the excess quantities refer be placed. so that on one side of the surface the properties of the gas phase are uniformo)

-51The volume change due to adsorption of gas in the liquid surface may be written AV = -FvI, or for an ideal gas AV = kT where is the surface excess of the gaseous component at the interface,with the dividing surface indicated above, and P is the total pressure on the systemo So for an ideal gas _ _ - kT + V (3-7) P Cr 6' TTA,n~ Now Rice applies this equation to Kundt's data assuming AVe = 0 and is able to account for the fact that the surface tension decreases with increasing pressureo He obtains values for r as a function of pressure. The same equation that Rice obtains may be obtained immediately from the Gibbs Adsorption Equation, avoiding the awkwardness of Rice s approach. For an ideal gas (1) above a liquid (2) the Gibbs Adsorption Equation at constant temperature is - dy= Vl-jt + r2di2 If one chooses the dividing surface so as to make the surface excess of the liquid component vanish, i.e, r = 0 then - d7 = 1(2)dd l (3-8) where 11(2) is the surface excess of component 1 for this choice of dividing surface Now for the gaseous phase d = dP ~V dP V dP kT For a one component ideal gas, d1 = dP- therefore dy/dP = - (2) ( ) which is Rice's result, obtained in a more conventional mannero

-52Slowinski, Gates, and Waring(92) measured the effect of six pressurized gases above water and n-hexaneo The gases were C02, C2H6) CH4, N2, H2 and He. With all gases except He above either liquid, a decrease in surface tension with pressures to 100 atm was observedo Helium pressures to 100 atm produced a very slight increase in the surface tension for both liquids. These results tend to confirm Equation (3-7)2 which ignores any mutual solubility of the componentso From the shapes of the curves obtained it appears that over a fairly wide range of pressures that l(2) = KP; that is, the adsorbed excess is proportional to pressureo It is interesting to note that the order of increasing effect of a particular pressurized gas on the surface tension of water is the same as the order of increasing gas solubility in the water at atmospheric pressure. Gielessen and Schmatz3o) measured the pressure dependence of the liquid/vapor surface tension for the liquids n-hexane, heptane, octane, nonane, a C3 -fraction, and cyclohexane under nitrogen. In addition, they also considered n-hexane and nonane under pressures of helium and argon. In all systems except those involving helium, the surface tension was found to decrease with pressure. The data.in these systems were correlated with an empirical equation, with two constants, formally similar to the Langmuir Adsorption Isothermo (92) These authors confirm the results of Slowinski, et al(9) for systems involving helium; i.oe a small, linear increase in surface tension is found. Hassen, Nielsen, and Calhoun(40) measured the effects of pressures to 204 atm and temperatures from 26~ - 82~C on the liquid

-53liquid interfacial tensions of a series of hydrocarbons with water. The hydrocarbons used were benzene, propane, n-butane, n-pentane, n-hexane, n-octane, and i-octaneo In all cases increasing pressure lowered the surface tension, the magnitude of decrease varying with the liquid pair. Presumably, this data can be subjected to the same treatment outlined aboveo If we choose the dividing surface so that the surface excess of water vanishes, \r = 0 then - dy = r d. 1(2).* Assuming the hydrocarbon liquid does not dissolve water, then dt = -a dP = V1 dP = V dP So dY/dp = - r(2) V (3-10) where -V is the molar volume of the hydrocarbon liquid. Knowing V1 V1 (P) io.e. compressibility data for the hydrocarbon, surface excess quantities may readily be computed. Michaels and Hauser') measured the liquid/liquid interfacial tension in the systems n-decane —ater and benzene-water as a function of temperature and pressures to 700 atm. The interfacial tension decreased with pressure in the benzene-water system and increased with pressure in the n-decane-water systemo (48 49Y50) Hough and his co-workers 995) have measured surface tension changes with pressure in systems involving water and methane, helium, nitrogen or carbon dioxide at pressures up to 1000 atm. They find rather irregular behavior in some of these systems~ 1) in the watermethane system the surface tension at 100~F and 160~F decreases with

-54increasing pressure to about 5000 psi, then increases; while at 74~F it decreases continuously with pressure, 2) the surface tension in the water-nitrogen system decreases with pressure at temcreperatures from 80~F to 280~F, 3) the surface tension in the water-helium system increases with applied pressure by a rather small amount at temperatures from 80~F to 280~F9 4) the surface tension in the water-CO2 system decreases with increasing pressure, showing an initial sharp decrease to about 3000 psi followed by a more gradual decrease thereafter. Using the relation (3-7) Hough et al(495) calculate surface excess quantities as a function of pressure for these systems using AdV = Od Although their surface tension measurements show little scatter about a smooth curve, they are not sufficiently precise to allow calculation of the surface excess to much more than an order of magnitude. The plots of r vs P which they presented imply much better precision. The pressure dependence of interfacial tension has been measured for a number of systems involving crude oil, water and reservoir (47,539102) gasess i4753 02) The results are rather irregular, and because of the multitude of chemical species present (in unknown proportions), have little scientific.value. A certain amount of work has also been done by Russian investigators on surface tension changes with pressure in hydrocarbon systems important to the petroleum industry(88) This work appears to be fundamentally no different than the American work discussed aboveo One exception is that differences in the wetting of solids by liquids have been noted when the gas pressure surrounding the liquid-solid system is increased~

-55In summary, the above results indicate that changes in pressure can produce measurable changes in surface tension in liquid-vapor and liquid-liquid interfaces and possibly in liquid-solid interfaces. The change in surface tension with pressure can be described by the Gibbs Adsorption.Equation at constant temperature with the total differentials of chemical potential for each component expressed as functions of pressure and composition. In certain cases, especially for binary systems where one component is not present in both phases, the change in surface tension with pressure is given by the particularly simple relation - - r dP 1(2) 1l which is rigorously true for a plane interface, with the dividing surface chosen so that = 0 and where V is the molar volume of component 1 in the phase which is assumed to be pure component lo (2 2) is then the surface excess of component 1 for this choice of dividing surfaceo 5. Brittle Fracture In two papers in the early 1920 s Griffith proposed a theory of rupture for isotropic brittle materials which has formed the basis for much of the subsequent work done to explain brittle and nearlybrittle failure The principle behind the theory is quite straightforward. If one assumes the existence of some sort of microcracks in a material, then the minimum energy that must be supplied to create a unit area of new crack surface (propagate the crack) at constant temperature is the surface tension (as defined earlier) of the crack surfaceo Thus, the

-56energy stored in the surface of a very flat elliptical crack in a plate per unit plate thickness is W = 4yC where the length of the crack is 2Co s In addition to this energy, elastic energy is stored in the material due to the action of external forces. This energy may be divided into two parts, (a) the elastic energy W which would be stored if a the crack were absentand (b) a correction term which accounts for the additional strain energy due to the presence of the cracko This latter term will be dependent upon the length of the crack, and is given for (1 - v2) 2C22he the plane strain problem by W v2) t where = is the b E stress normal to the crack, v is Poisson's ratio and E is Young's modulus. For a fixed stress, the total decrease in the potential energy of the material and its surface tractions is equal to the strain energy stored inHence the materialen t potential energy of the system at equilibrium with reference to the unstressed, uncracked state is: Lw = w (c) - Wa - Wb(c). If the crack is to propagate the total energy must decrease as C increaseso The point at which the crack is in equilibrium with the applied stress is given by d.C = o0 When C exceeds the value given by this expression the crack may propagate without limit. Alternatively, one may also determine the stress above which a crack of length 2C will propagate. For a flat elliptical hole in a plate in a condition of plane stress the critical stress applied normal to the crack is c - Ey (3-11) C it,

-57For the plane strain problem the critical stress is given by 2 Ey (3-12) c itC (1 - v ) Sack ( has treated the corresponding three dimensional problem of a circular "penny-shaped" crack in a solid and finds that ac (5-15) c 2C (1 - v2) where a is the critical stress normal to the plane of the cracko Griffith's theory adequately explains the fracture behavior of glasses and some ionic solids;o In recent years, the Griffith theory has been modified to account for the plastic deformation associated with cleavage-type fractures in (77) metals which have some ductility. Thus, Orowan7 suggests that y in Griffith's expression be replaced by y + p, or simply p, where p represents the energy stored in plastic deformation per unit area of crack surface formed,, and is about 1000 times as large as yo The presence of deformed metal layers on fracture surfaces lends support to this suggestion 25) Griffith's theory does not explain how microcracks originate in metals, or how they grow to a critical size. The answer to this problem has been provided by the development of theories of crack nucleation based on dislocation models, Stroh has been a major contributor in this field, and has written an excellent review of many aspects of metal fracture (100) According to the model developed by Stroh, cracks are nucleated in polycrystalline metals by the pile-up of dislocations in a slip band at a grain boundary, The grain boundary is usually an effective obstacle

-58to the propagation of slip because the orientation of adjacent grains across the boundary is differento That is, slip in one grain cannot readily be extended to a neighboring grain because there is no slip system properly oriented to accomodate ito This situation produces a pile-up of dislocations in the slip band at the grain boundaryo When a sufficient amount of slip has taken place in a grain, the total stress field arising from the piled-up dislocations becomes large enough to create a cracko Stroh~s relation for the resolved shear (100) stress on the slip plane needed to produce a crack is( TyG (3-14) s 8(1 - v)L (-14 where G is the rigidity modulus and L is the length of piled-up dislocations in the slip planeO (L is usually taken to be of the order of the grain diameter-) As Stroh points out0, the above relation for crack nucleation does not contain a dependence on the crack length, thus the crack may grow with an overall decrease in energy until it has consumed the dislocations which produced ito Once this occurs further growth of the crack is dependent upon the normal stress across the crack and the ability of the material to yield plastically to reduce the stress concentration at the crack tipo For an ideally brittle material, Griffith's criteria now determines whether or not the crack will propagate; for a material with some ductility Orowan's modification may be appropriateo The theories outlined above explain many aspects of fracture behavioro Fracture, however, is a very complicated subject Reference (9) to a recent symposium on fracture shows that many modifications and refinements of the foregoing notions are needed to explain the many phenomena observed,

-59It is apparent from the preceding discussion that the surface tension of the crack surface is of paramount importance in fracture theory because it occurs in every expression (3-11) to (3-14) for the critical stress for crack nucleaticn or growtho Changes in the state of the system which'change this surface tension can produce changes in the fracture behavior of a metalo But not all changes in surface tension produce identifiable changes in fracture strengtho For example, changes in temperature and pressure (or stress) produce changes in the surface tension, but they have a much more pronounced effect on the generation and movement of dislocations; hence any effect of surface tension changes from these sources is masked by changes in the modes of deformation. Changes in composition, on the other hand, change the surface tension and may not radically change the mechanism of crack nucleation and growtho Petch' has developed this last idea to explain the hydrogen embrittlement of iron. Using Stroh s criteria for the nucleation of a cleavage crack due to dislocation pile-up, Petch corrects the surface energy term for adsorption of hydrogen on the crack surfaceso He employs the Gibbs Adsorption Equation. written for the adsorption of an ideal gas, and substitutes the Langmuir.Asorption Iotherm to express the surface excess concentration as a function of pressure.Although sufficient data is not available to allow a precise calculation to be made, it is clear that the lowering of surface tension due to adsorption should produce a drop in strength which is of the same order of magnitude as that which is observedo 60 Intergranular Fracture The dislocation model of crack nucleation proposes that the

-60nucleating crack is transgranular, but there are many occasions where fractures are observed to proceed along grain boundaries rather than across grainso The minimum energy required to form a unit area of grain boundary crack at constant temperature is 2Ys - ygb and the energy s gb required to form a transcrystalline crack is 2ys where y is the surface tension of the crack surface formed and ygb is the surface (65) tension of the grain boundary. One usually assumes that a.crack propagates so as to expend the least amount of energy; this suggests that variations in the quantity 2y are important in determining s - Ygb the fracture path. While it is always true that 2- 1 Ys7 2s s is generally much larger than 7gbo This ratio is on the order of 0o90 so other factors probably determine whether the fracture will proceed intergranularly or transgranularly, or show a mixture of the two. But if -.-is caused to decrease, eogo. by changes in composition, 27 s " gb then the ratio 2y is smaller and;preference for the intergranular path is now marked. One.of. the ways in which 7 can be significantly lowered has been mentioned above in the case of hydrogen embrittlement. And indeed. intergranular fracture is observed in this caseo This is at besthowever, a qualitative correlation; no theory has yet explained all that is involved in the choice of fracture path. The work of Eborall and Gregory2) and Rostoker(73'74'75l17'87) on embrittlement by liquid metals lends itself to interpretation on the basis of changes in interfbdal tension, Eborall and Gregory showed that small amounts of lead present in P-brass and tin bronze produced a large drop in the hot ductility of these alloyso (Experiments were conducted above the melting point of leado ) They pointed out that

-61for intergranular cracking, the surface energy that must be supplied to propagate a Griffith crack in the presence of a liquid phase is 2Ysl - 7.gb where s is the surface tension of the solid-liquid sl- Y gb sl interface. As a rule, s > Is(l) > io.e the surface tension of the pure metal surface is greater than its surface tension in the presence of the liquid's vapor, and the latter is in turn greater than the solid-liquid interfacial tension. Hence, if the liquid fills the crack tip, a Griffith crack can propagate at a much lower stress. It is likely, in this case, that the lead particles existing in the grain boundaries of this material acted as pre-existing Griffith crackso Rostoker and his associates have published several papers (73,74,75 7) and a book(87) on the embrittling effect of liquid metals. The features they have observed are characteristic of many types of embrittlement; for example: (1) a ductile-brittle transition temperature, (2) delayed failure behavior, (3) dependence of embrittlement on grain size, and (4) intergranular, rather than transgranular crackingo They have observed that cracking commences at the surface of a metal stressed in a liquid metal environment. Hence, the environment probably influences not only crack propagation but nucleation as well. This supposition has been confirmed^, and by use of the Stroh-Petch relation between grain size and fracture stress, the effective surface tensions for the nucleation and propagation of cracks have been estimated, While the interpretation of the results in terms of surface tensions depends on the model chosen for the initiating crack, the data suggest that the initiating and propagating steps are characterized by different surface tensions.

-62Brittle, intergranular failures are also observed -when a casting fails by'hot-tearingo" Hot tears may occur whenever stresses are produced in the casting due to either thermal gradients or constraints on the cooling metal from complicated mold shapes. In either case, severe hot tearing is usually associated with the presence of a low melting phase, eogo an iron sulfide-iron eutectic in steels. In the absence of this liquid. tearing seldom occurs because the metal plastically deforms to reduce the magnitude of the stresses presento With some. liquid present, these stresses are reduced by crackingo It seems likely that the mechanism Eborall and Gregory propose to explain the detrimental effect of lead in brass and bronze is also applicable here 7o Liquid Phase Sintering Liquid phase sintering is a process in which a liquid phase, permeating the interstices between solid particles, accelerates the densification of the compacto As in all sintering, interfacial tension plays a, dominant role in this process, and the lowering of the total free energy of the system by the motion of interfaces (and resulting change in particle shape)provides the driv'ing force for densification. Although liquid phase sintering has been a successful industrial technique for many years, the theoretical development of the subject has laggedo Recently, however, Kingery5 575) has proposed a densification mechanism for liquid phase sintering which conforms to experimental observations of the importance of major process variableso Liquid phase sintering begins, typicallyT, with a pressed compact of low and high melting materialso When the compact is raised

-63to a temperature sufficient to melt the low melting phasef an initial densification occurs due to the rearrangement of the solid particles into a more closely packed configurationo In order to obtain the accelerated densification characteristic of liquid phase sintering, several criteria must be satisfied: (1) there must be sufficient liquid present to fill most of the interstices of the solid particles, (2) the solid must be appreciably soluble in the liquid, and (3) the liquid must wet the solid completelyo The structure of a compact which has been liquid-phase sintered generally shows rounded solid particles outlined by the totally wetting liquid phaseo (If the solid is non-metallic, eogo, a carbide, euhedral grain shapes will often be preserved,) If insufficient liquid is present or if the liquid does not wet the solid completely, bridges may form between the solid particles and the sintering rate slows to that characteristic of conventional solid state sinteringo The solid particles change shape by a process of solution and reprecipitation, mass transport taking place through the liquid phase; hence the requirement that the solid be soluble in the liquid phaseo After the initial readjustment when the liquid phase melts, the solid particles change shape so as to~ a) round off sharp edges, b) flatten points of contact (or points of nearest approach), and c) increase in sizeo Kingery(5 proposes that step (b) is most important in promoting densification, and derives an expression for the rate of densification due to this processo Several requirements on interfacial tensions may be inferred from the above requirements for successful liquid phase sintering.

-64(a)o The solid-liquid interfacial tension must be less than or equal to half the solid-solid grain boundary tension for the liquid phase to wet the solid particles. (b)o Following Kingery's model, it is necessary that the contact angle, formed by the balance of liquid-vapor, solid-vapor and solid-liquid interfacial tensions, be acute, which implies that the solid-vapor tension is greater than the solid-liquid tensiono Equilibrium measurements of relative interfacial tensions of a liquid-solid pair should indicate whether liquid phase sintering is possible for this system. In practice, however, if the liquid is not saturated initially with the solid (and vice versa) the wetting characteristics may deviate far enough from equilibrium behavior so that relative interfacial tensions determined at equilibrium are a poor guide -'58o discussion) n a.dt to the actual sinterability of the system. di) In addition, surface active impurities present in small amounts may also influence wettability. Since surface tensions vary with the pressure on the system, it is possible that some of the benefits of sintering under pressure (hot-pressing), with or without a liquid phase present, are due to pressure induced changes in surface tensionso 8. Summary The effects of temperature and composition on surface and interfacial tensions have been widely studied and are generally appreciated. Relatively few investigations have been made of the effect of pressure on interfacial tensions, mainly because pressures beyond the customary range of laboratory experimentation or commercial application must be

-65applied to produce a significant changeo This is particularly true for interfaces between condensed phases. The tension of a solid/liquid or solid/solid interface is difficult to measure absolutely. Ratios of these tensions, however, can be more easily determined. Complications which arise due to the crystallographic and elastic properties of the solid are frequently resolved by assuming that metals behave as liquids This assumption is warranted in many cases, but the departures from liquid-like behavior (eogo, the dependence of grain boundary tension on the misorientation of grains across the boundary., the behavior of doherent twin boundaries) have provided a most fruitful field for research. A study of changes in interfacial tensions in metals is profitable because many properties of the metal depend indirectly on interfacial tensionso For example, interfacial tensions are the major controlling factor determining the shapes of grains in annealed polycrystalline metals. The magnitudes of certain of these tensions and the relative magnitudes of tensions at various types of interfaces must also be considered in theories of brittle fracture and theories of sinteringo Changes in interfacial tensions arising.from changes in composition have been shown to significantly influence such properties as fracture strength and ductilityo It is likely that changes in interfacial tensions caused by changes in pressure (or state of stress) or temperature may also influence these properties, although the effect may be masked by changes in the mode of deformationo

CHAPTER IV EXPERIMENTAL PROCEDURE lo Sample Preparation All specimens of leaded nickel were made using high-purity (99~99+%) lead obtained from the American Smelting and Refining Company. Table II is the spectrographic analysis provided with this materialo TABLE II SPECTROGRAPHIC ANALYSIS OF LEAD USED FOR ALLOY Bi Oo0008 * Fe 0o0002 Cu 00003 Zn NoDo Sb NoDo Cd NoDo As NoDo Ag NoDo Sn NoDo Te NoDo Ni NoD. Pb 99o99+% by difference NoDo = None Detected -. Weight Percent Mond Nickel shot (99o95% Ni + Co), obtained from the International Nickel Company, was used in all but two meltso All specimens with the letters B, C, or D in their designation were from heats made using Mond Nickelo Specimens from heats E and F were made using Sherritt Gordon Grade "E" Nickel Powdero A typical analysis provided with the latter material is given in Table III TABLE III TYPICAL ANALYSIS OF SHERRITT GORDON GRADE "E" NICKEL POWDER Ni 99 g9x Fe 0O004 Co 0O03 S 0O002 Cu 001 C 0O03 * Weight Percent Heats B, C, D, consisting of about 4 lbso of Ni and Pb were induction melted in a LECO zirconia crucible under a protective atmosphere -66

-67of tank argon.; The system was.evacuated and flushed with argon before melting and while the system was being heated. From 8 - 12 psig argon pressure was kept above the melt to keep the lead from boiling offo Melts were allowed to solidify in the crucible under argon pressure. In all cases ingots showed no sign of oxidation during meltingo Heats E and F were prepared by compacting mixed lead and nickel powders and sintering at 15000F for about 18 hours in purified flowing hydrogeno The sintered compacts were then melted under a pressure of about 3 psig purif:ied flowing hydrogen in a small globar furnaceo The melt was contained in a small Morganite, recrystallized alumina boatO The hydrogen gas was purified by passage through a DeOxo unit, a Drierite dessicant, and a liquid nitrogen cold trapo Pressure was maintained on the system by bubbling the outlet gas through about 7 inches of mercuryo Analyses were performed on a sample of material from Heat D by Chicago Spectro Serv'ice Laboratory. The results are shown in Table IV TABLE IV ANALYSIS OF HEAT "D" Pb Lo98* C 0o02 Cu. 001 S O0o 013 Fe Oo002 0 0o004. ~Si OO1 * Weight Percent Since the analysis of Heat D showed a sulfur content which was not negligible, heats E and F were made to obtain specimens with a low sulfur level. A sulfur analysis on Heat F performed by the Research Laboratory of the International Nickel Company indicated 5 ppm sulfur^ which is considerably lower than the sulfur content of the Nickel used for the melto

-68Sufficient information as to the free energies of formation of the nickel sulfides is not available, Thus, the partial pressure of H2S in equilibrium with them cannot be calculated. However, the data that are available (24) indicate that the free energy of formation of these sulfides is not much different from the free energy of formation of H2S, per mole of sulfuro It is probable that the partial pressure of H2S in the flowing hydrogen stream was kept low enough to result in removal of sulfur from the nickel in heats E and Fo Test Specimens Test specimens were machined from ingots according to the dimensions shown in Figure 4. Specimens used in hydrostatic testing and for equilibration at various temperatures varied in dimensions. Most were small cylinders, 3/16 inches diamo, 1/2 inches long. Tensile specimens were copper plated before useo A flash coat of copper was applied using a cyanide bath, at low current density. Subsequently a heavier coat), about O0OOl inches thick, was applied from a sulfate bath at higher current density. Plating conditions (54) and solutions used are given by Kehl(5 Heat Treatment Specimens were equilibrated at temperatures from 371~C to 816~C at times varying from 24 hours to one weeko All specimens subjected to hydrostatics tensile, or compression loading at 371~C were annealed for one week at 371~C prior to testing. Specimens hydrostatically tested at 593~C were annealed for five days at 593~C prior to testing.

-691/4 5/16 "-24 NF THREAD __ I7/532 "__ / BOTH ENDS I~l~l l l llllF! 111'11 3" I" 2 4 t (a) BOTH ENDS L Z ~ GROUND FLAT' —-9/16 3/16" (b) Figure 4. Tension and Compression Test Specimens. (a) Tensile Specimen. (b) Compression Specimen.

-70The only exceptions were specimens from heats E and F which were annealed three days at 593~Co Annealing was done in a vycor tube which was fitted with a ground glass joint and a stopcocko Samples were positioned in the tube using ceramic pieces so that when the vycor tube was placed in a tube furnace, the specimens were at a determinable position in the hot zone, Previous calibration of the furnace insured that the annealing temperature was within +5~C of the desired temperature during the course of the run. The portion of the annealing tube not occupied by the specimen was filled with clean uranium turnings and cerium chipso Prior to placing the tube in the furnace, it was evacuated, flushed repeatedly with argon or helium, then evacuated againo The tube was placed in the furnace while still open to the vacuum system. After about 15 minutes of pumping while the tube was at the desired annealing temperature, the stopcock was closed and the vacuum system detachedo This proved to be a very satisfactory method for long time annealso The presence of the very reactive metals, U and Ce, insured that residual amounts of gases were removed. At the completion of the anneal, the vycor tube was quickly withdrawn from the furnace and quenched in watero This seemed to provide a satisfactory. quench for the temperature range covered. It probably would not give a rapid enough quench from temperatures much higher than 816~C, however. In every case, the specimens annealed in this manner were as bright as when they were placed in the furnace. The U and Ce present

-71wei usually covered tith a light coating of reaction productso This was easily removed by cleaning in nitric acid. The same uranium turnings were used for more than a dozen long time equilibrations. The average matrix grain diameter after annealing for heats B9 C, D was about 0.5 mm. Lead was present largely as spherical shapes within grains and as lenticular and elongated shapes in grain boundaries. Figure 5 is typical of the appearance of the lead in these alloys. Heats E and F were different. The average matrix grain diameter in F was about O02 mmo The lead was distributed as in the previous heats except that no elongated lead particles were observed in grain boundaries. No significant change in matrix or particle grain size occurred with heat treatment in the temperature range usedo Recrystallization of samples did not occur during heat treatment. If the specimens were sufficiently deformed to cause recrystallization, they were so badly cracked as to be unusable. Failure to have a recrystallized matrix structure did not appear to introduce any systematic sampling error in dihedral angle measurement 2. Hydrostatic Pressure Testing All tests under hydrostatic loading were carried out in a hydrothermal unit, model HR-lB, manufactured by Tem-Press Research, Inc. A schematic drawing of this apparatus is shown in Figure 6, Each reactor vessel, about 8 inches long, with 1/4 inch bore afid 3/8 inch thick walls, Was mounted so that a vertical tube furance could be raised into a position around it. After completing a run, the furnace could be quickly d.ropped, and a quenching bath placed about the reactor. Specimens were successfully quenched from 595~C and 50,000 psi by this method.

- T -:igp;ure 5j.?hotom'*.cropgrapihs of Spec..men D.!P8. 1000 X. Not Etched.

-73X.. Fl gutre 5 (cont' d). Photoim nierographs of Specimen DP., 0OO0 X. Not Etched.

-74The unit above is built to operate using water as a pressure mediumo The procedure used for a typical run was: (1) The specimen was placed in reactor vessel and the vessel was filled with water, assembled and installed in the unit. (2) The pressure was raised on the vessel using the water pump. (3) The furnace was raised over the reactor and the vessel brought to the desired test temperature, (4) Excess pressure was bled from the system during heating and when stabilized at the test temperatureo When the desired test temperature and pressure were obtained, the system was isolated by closing the appropriate valve. When the system was adequately pressure tight, week-long runs could be made with no significant pressure drop occurringo The temperature of the vessel was measured by a thermocouple inserted in a well on the outside of the reactor vessel opposite the specimen. A calibration showed that temperatures recorded at this point were about 2~C higher at 371~C and 593~C than temperatures inside the reactor vessel after the system had become stabilized at a particular temperature. The temperature of the furnace was regulated by a thermocouple inserted in the furnace windingso A West Model JP controller was usedo Pressures were followed during a run by means of the pressure gauges shown in Figure 6 o The two units with 0-50,000 psi gauges were used for runs of less than 30,000 psio If the pressure on the system was suddenly removed, the resulting jar to the gauge needle often changed the indicated zero point. Over the several months that these units were used, several zero point calibrations of the pressure gauges were necessaryo These were made by attaching a dead weight gauge tester directly to the system~ The calibration of the gauges was otherwise unaffectedo

-7 5A B C ID u^G E I U JU K J L M M M M Figure 6. Schematic Drawing of Hyydrothermal Unit. Key: A —Water Reservoir. B —High Pressure Air Driven Water Pump. Max Output Pressure 28,000 psi. C, D, E, F, G —High Pressure Valves. K, L —0-100,000 psi Pressure Gauges. M —Reactor Vessel.

-76It was found that no oxidation of specimens occurred in the course of a run, if all dissolved oxygen was removed from the water in the unit. This was accomplished by using freshly boiled distilled water in the reactor vessel and by placing a few uranium turnings or cerium chips in the reactor. These reactive metals prevented any specimen oxidation; any metal left after reaction with the dissolved oxygen (or nitrogen) was converted to oxide by reaction with the water. Two runs, (Specimens El and Fl) only one of which was completely successful, were made using argon as a pressure medium. To obtain sufficiently high argon pressures at 593~C, it was necessary to begin the run with the reactor vessel filled with liquid argon. This was readily done by placing the reactor vessel in liquid nitrogen, and running in tank argon until the vessel was filled. When the reactor was installed in the unit and allowed to warm to room temperature a pressure of 25-30,000 psi was obtained. Subsequent heating produced higher pressures. It was also necessary in these runs to add reactive metals to the reactor vessel to remove oxygen and nitrogen from the argon. Enough water was left in the line leading to the pressure gauge to insure a. proper pressure reading. 3. Testing in Tension and Compression All tests in tension and compression were done at 371~C (700~F) in a helium atmosphere. Direct dead weight loading was used. A sketch of the apparatus with a tensile specimen in place is shown in Figure 7 A split wound tube furnace, 1 1/4"1 ID, 12" long was placed around the tube. All threaded fittings inside the furnace tube were loose fitting; this provided the necessary flexibility to prevent bending of the specimens.

-77A B B c I / —-c Figure 7. Furnace Tube and Fittings for Tensile Testing. Key: A —Pull Rod to Framework. B —Conax Seals C —"O"-Ring Seals. D —Furnace Control Thermocouple. E —Thermocouple. F —Specimen. G —Threaded Fittings. H —Helium Supply. I —"O"-Ring Packing Gland. J —Packing Gland Nut. K —Pull Rod to Weight Pan.

-78Prior to starting a run, the furnace tube was evacuated and flushed several times with heliumo During the run a positive pressure of 1-2 psig helium was maintained in the furnace tube. This kept oxidation of the specimens to a minimum, but never eliminated it entirelyo The temperature of the furnace was controlled using the control thermocouple shown in Figure 7 o A "Celectray" controller, manufactured by C. Jo Tagliabue Manufacturing Company was used for temperature controlo Temperatures during runs were checked using a thermocouple which touched the specimeno The temperature gradient along the specimen was less than 2oCo Temperature variation during a run was about +2~C. The lower pull rod attached to the specimen passed through a packing gland which consisted of 3 teflon "0" ringso The packing gland nut was used only to keep the "0" rings in position; they were not compressedo A high polish was kept on the pull rodo It was found that the 1-2 psig helium pressure inside the furnace tube was sufficient to push the rod through the packing glando The load placed in the weight pan plus the weight of the pan and fittings -was taken to be the true load on the specimeno In other words, it was assumed that the helium pressure maintained inside the furnace tube just offset the frictional forces in the packing glandO At most, this could result in a 2% error in the effective load on the specimen. Before each run, three sets of scribe marks were placed on the specimen, The distance between marks was at least one inch.Measurement of the distance between these marks before and after each run revealed no indication that bending occurredo For runs made at the heavier loads, -4 about 210. permanent strain was indicated, but this is less than the

-79error with which the distances between scribe marks could be measured on successive trials. Hence, one can conclude that no sizable permanent elongation occurred during these runs. A compression run (DCl) was made in the same apparatus using fittings which permitted tensile loading to achieve compression of the specimeno Two interlocking "U"' shaped fittings, one of which is shown in Figure 8r were usedo One run in compression was carried outo Microscopic inspection after the run showed no signs of bending. 4. Measurement of Dihedral Angles Specimens were prepared for metallographic examination using conventional metallographic techniques applied with some care. Specimens were generally cylindrical in shape, 1/2 to 3/4"long, 5/32 to 7/52" diameter. Most specimens were copper plated using solutions (54) recommended by Kehl, prior to mounting in bakeliteo This prevented rounding of the specimen edges during polishing. After mounting, specimens were ground using 60C silicon carbide paper, so that the final plane of polish bisected the cylinder along its axiSo Some of the tensile specimens were also examined on transverse sectionso Following this, 240C and 600C grit papers were used; 6 micron diamond on a silk cloth was used to remove the scratches from the 600 papero Typically, at this point it appeared that the nickel matrix had been smeared to such an extent that only the largest lead particles were visible. Since no etchants were used, (because of preferential attack of either the lead phase or the interfaces in the system) this flowed metal had to:be removed by careful polishing with a Linde A

-805/16" PULL ROD I ] a —', -1/4" DOWEL PIN SPECIMEN Fir 80.005" SOFT COPPER PLATE Figure 8. Fitting for Compression Testing.

-81Al20 suspension, followed by two or more hours of mechanical polishing using a Syntron vibrating polisher with Linde B Al 203 suspended in water. In both of the final polishing steps, well-worn Buehler Microcloths were usedo The high nap on a new cloth tended to remove lead from the structure. This polishing treatment did not produce."relief" polishing in the sense that adjacent nickel grains were at different levelso It did, however, cause all grain boundaries to be lower than the matrix grains. It is not felt that this in any. way distorted the shape of the lead particles presento On the contrary, unless this situation obtained, the lead particles were often partially obscured by smeared nickel, Angles were measured on a Baush and Lomb Research Metallograph equipped with a rotating stage. By means of a vernier attached to the stage, angles could be read to the nearest 0ol~o All angles were recorded to the nearest 0ol~ even though the probable accuracy of measurement varied from +0.5 to +50 from one angle to anothero An 80X, oil immersion objective was used for all measurements with a,15X filar eyepieceo For angles measured on tension and compression specimens (and on sample D, a blank), the distance from the edge of the specimen was recorded for each angleo In addition, the angle which the grain boundary made with the axis of tension (or compression) was measured. The accuracy of the latter measurement was probably about +-50 The distance from the specimen edge was also recorded for all specimens equilibrated at various temperatures or subjected to hydrostatic loading with the exception of specimens from heats E and Fo The specimens used in these cases were irregular in shape which made such a measurement difficulto

-82In every case, specimens were repolished at least once so that angle measurements were made on more than one polished section. This was done to minimize any systematic error due to the quality of polish. As experience was gained, good quality polishes could be produced consistantly.

CHAPTER V STATISTICS OF DIHEDRAL ANGLe MEASUREMENTS The statistical problem involved in the determination of a dihedral angle from a sampling of observed angles should be a problem A in determining a best estimator, 3, for the dihedral angle i which is the single parameter in the theoretical cumulative frequency function (Appendix A) which describes the population from which the sample is drawno Straightforward solution cf the p-oblemn is difficult because of the complexity of the analytical form of the distribution function. More important, however, is the fact that the d:istribution function does not, in all likelihood, accurately characterize the population from which the sample is drawno For these reasons, a less direct approach is requiredo It has been shown(8) that the median angle of the theoretical distribution always lies within 1~ of the dihedral angle, o Given a value of the median angle one can apply a correction which will produce the dihedral angle. One must then consider the factors which are likely to cause the actual distribution to depart from the theoretical distribution and determine their effect on the median angle. If the effect on the median angle is predictable, then it is still possible to determine the dihedral angle from the median angle by applying several corrections, Since the actual form of the distribution function for the population is unknown or exceedingly complex, the most satisfactory way to determine the probabhle errors in the varlue of the median angle obtained is to apply methods of non-parametric statistics to the sample, -83

-84This will allow one to establish a confidence interval for the median of the population. Other non-parametric methods will allow comparisons for significance to be made between different samples. The sections to follow elaborate on these points and discuss the calculations made. lo A Discussion of Errors Affecting the Distribution of Observed Angles The statistical analysis of observed angles to determine a true dihedral angle was first employed by Harker and Parker(39) in 19455 (93) (51) A few years later, Smith() and Ikeuye and Smith used this analysis to determine the dihedral angle of dispersed second phase particles lying in grain boundaries ca matrix phase, eogo, lead in coppero These authors, and most subsequent workers who have employed this technique, have observed that in general the distribution of angles observed departs considerably from the expected distribution, in that the peaks observed at the modal angle are blunter and broader than one would expecto Figure 9 illustrates this point with data from the Nickel-Lead system, (Specimen D)o Application of a Chi-Squared test to this data indicates there is virtually no chance that the sample of angles obtained came from a population with the theoretical distribution shown. Application of the Chi-Squared Test to other data in the literature dtows some data which fits the theoretical distribution better than the data shown, but no cases of good fits. Several reasons have been advanced to explain this discrepancy. Harker and Parker( suggested that a) the presence of a range of dihedral angles in a specimen rather than a unique dihedral angle would cause a spreading of the distribution function, b) a non-random selection

-8528 1,-, - 26 - EXPERIMENTAL- 500 MEASUREMENTS 2 4 ---- THEORETICAL - =52. 1~ BEST FIT TO DATA 22 - 20 () I 18 r 16 Vn I r 14 0 0 I 00 Lm:LI a. 8~ I I 4 -1 1 2 - 00 I I I i 1 1 1 1 1 1 II 0 1' 20 30 40 50 60 70 80 90 100 110 120 OBSERVED ANGLE IN DEGREES Figure 9. Frequency Plot - Theoretical and Observed.

-86 - of orientations with respect to the plane of polish would distort the distrubution function, eogo., arising from a columnar grain structureo This is more likely to be a problem when measuring the angle at which matrix grains meet, as in Harker and Parker s experiment, c) "bad luck" in samplingo The last cause can be ruled out because the effect is invariably observed. Smith(95) and Ikeuye and Smith(51) suggest that failure to observe the expected peak frequencies is due to either a) differences in grain boundary energy due to orientation effects, or b) failure to obtain equilibriumo For well annealed specimens. such as they used, Ikeuye and Smith do not consider the latter effect to be importanto They ascribe the bulk of the peak broadening to the first cause even though "there is only a small probability that two grains will meet i (95) each other at an angle that gives a boundary of low energy'o Another possible source of this discrepancy is the contribution of measuring erroro If the errors in measurement are sizable9 then even if they occur randomly they will change the effective distribution function for the population of observed angles. Increasing the sample size only insures that the sample distribution approaches the effective distribution of the population, not the "true" distributiono In measurements performed on dispersed second phase particles9 another type of error can arise from an improper selection of particleso It is easier to measure the angles from a particle whose section size in the plane of polish is largeo This same particle can be sectioned to produce a variety of section sizes, and it can be shown that if the difficult measurements on small section sizes are omitted, then a systematic error in the sample distribution can resulto

-87The following three sections show the effects of three sources of error on the effective distribution function of observed angles The calculations were performed with aid of an IBM 7090 computero 20 Effect of a Non-Unique Dihedral Angle on the Distribution of Observed Angles Blunting of the distribution peak has usually been ascribed to variations in grain boundary tensions arising from the dependence of grain boundary tension on orientation parameterso It is undoubtedly true that the few orientations which produce very low grain boundary tensions contribute negligibly to a sample of observed angleso However, minor fluctuations in composition due to segregation (influencing adsorption at the boundary) might very well produce a range of dihedral (95) angles. Smith93 points out that the general occurrence of particles with nearly spherical, symmetry within matrix grains argues against any significant dependence of solid-liquid tension on interface orientationo This may be so, but variations in interface tension by a few percent might not noticeably affect the sphericity of these particles, but would have significant effect on the dihedral angle. For the moment we will assume that fluctuations in tensions only occur for grain boundarieso If the grain boundary tension is uniformly distributed over a range e ~ 0o01579, then if the dihedral angle corresponding to y is 50~9 the range in dihedral angles will vary from about 46~ to 54~o If the range is y ~ Oo03y the dihedral angles will vary from 42~ to 58~~ The spread in the frequency function peak for each of these cases is indicated in Table Vo Comparison with the sample presented in Figure 9 shows that a uniform distribution of dihedral angles from at least 42~ to 58~ would be needed to explain the peak depression.

-88TABLE V EFFECT OF NON-UNIQUE DIHEDRAL ANGLE ON THE DISTRIBUTION OF OBSERVED ANGLES Observed Theoretical Cumulative Cumulative Cumulative Angle Cumulative Frequency Frequency Frequency Frequency for =50+2 for 9=50~4 for =50+8 for 9 = 500 5 0,29 0.29 0.29 0.30 10 1.18 118 119 1.22 15 2070 2.o71 273 280 20 4.94 4.95 4.98 5.12 25 8.00 8.02 8.08 8.32 30 12.09 12.12 12.22 12.62 35 17.51 17.57 1772 18.38 40 24.82 24.92 25.19 26.42 45 35.23 35.44 36.10 39.31 48 44.42 45.04 46.67 48.63 49 48.60 49.74 50o71 51.75 50 54.73 54~75 54~78 54.84 51 60.75 59.68 58.77 57.88 52 64.69 64.13 62.62 60.86 55 72.80 72.62 72.03 69.24 60 80.94 80.85 80.61 79-55 65 85.95 85.90 85.76 85.17 70 89.33 89.30 89.20 88.81 75 91.73 91.70 91.63 91.36 80 93548 93.47 93541 93.21 85 94.81 94.80 94.76 94.60 90 95.83 95.82 95.79 95.67 95 96.63 96.63 96.60 96.51 100 97.27 97.27 97.25 97.17 105 97.79 97.78 97.77 97.71 110 98.20 98.20 98.19 98.14 115 98.55 98.54 98.54 98.50 120 98.83 98.83 98.82 98.79 Median Angle 49.3 49.1 48.8 48.4 Relative frequencies near Peak of Frequency Plot 40-45 10o41 10.53 10o91 12.89 45-50 19.49 19,31 18.67 15.53 50-55 18.07 17.87 17.25 14.40 55-60 8.14 8.24 8.58 10.32

-89If this factor is important, then no unique dihedral angle existso If the number obtained for the median angle of the population is corrected to obtain the dihedral angle, the latter can only be regarded as a number representing the central tendency of an existing range of dihedral angles, 3o The Contribution of Measuring Error According to the theoretical distribution (Appendix A), if one has a dihedral angle of 50~, then 20o3% of the angles observed should lie- between 48~ and 52~ But if the reproducibility of measurement is on the order of 1.~ then even if the sampling is perfect, errors in measurement would not allow one to observe such a large percentage of angles in such a small range. The following discussion presents an approximate method for determination of the amount of peak broadening due to errors in measurement. Let Iy designate the variable which is measured experimentally. Then, when errors in measurement exist 4v * = - + y (5-1) where J is the value of the variable which would be observed in the absence of errors, and y is the contribution of erroro Let y be normally distributed with mean 0 and standard deviation a, and let be distributed according to Harker and Parker's expressiono Then, what is required is an expression for the distribution function of U o The distribution function for { * may be calculated according to the following scheme. Let the fraction of angles less than or equal to 4 be called C4, where C, is determined from the theoretical distribution equationo Let the fraction of angles between y and L -A,tbe designated f + __ where

-90f w r = cW + ^- - C ~ (5-2) Errors in measurement cause the fraction f + At to be normally distributed over all o. (Since the range of ) is finite, the normal distribution is a good approximation except for ) near 0 or 180o ) The fraction of f + which is greater than 4 + l + x or less than Y + -i- - x is 2 0 1 1 ~~ -t22 f^ ~ Y —--- e dt j Hence the cumulative frequency corrected for measuring error is C* where K \ n r I =* C + -f( /2)AI f -(n / 1 \ e-t2/2 (5-3) *2i' \ (n 1+ _/2)A_~ The summation is carried out up to n = K_, where K is high enough so that the value of the error integral is very small. This wil1 of course, depend on the value chosen for a and for A o For the calculations that were made, At and a were adjusted so that K = 40, which meant that the smallest value of the error integral used was 0.00004o The results of this calculation for a = 2 and a = 5 are shown in Table VI. Computed values are shown only for the central portion of the distribution, since the assumed normal distribution of error cannot hold for very large or very small angles. Although a pronounced blunting of the distribution peak occurs for a = 5, neither the modal angle nor the median angle is much affectedo While both of these criteria remain relatively unbiased for this type of error, neither the sample

-91TABLE VI EFFECT OF MEASURING ERROR ON THE DISTRIBUTION OF OBSERVED ANGLES Observed Theoretical Cumulative Cumulative Angle Cumulative Frequency Frequency Frequency for 2=2 for a=5 ~ =50~ =50~ =50~ 20 4.94 5.00 25 8.00 8.09 8.53 30 12.09 12.20 12.78 35 17o51 17.66 18.52 40 24.82 25.06 26.60 45 35.23 358.1 35848 48 44.42 45.81 47.63 49 48.60 50 o6 50.88 50 54.73 54.74 54.15 51 60.75 59.03 57.38 52 64.69 63.12 60.53 55 72.80 72.17 69.04 60 80.94 80.70 79.10 65 85.95 85.83 85.05 70 89.33 89.25 88.81 75 91.73 91.68 91.39 80 93.48 93.45 93526 85 94.81 94.79 94.65 90 95.83 95.82 95.72 95 96.63 96.62 96,55 100 97.27 97.26 97.21 105 97.79 97.78 97.74 110 98.20 98.20 98.17 115 98.55 98.54 98.52 120 98o85 98.85 98.81 Median Angle 49.3 49.0 48.8 Relative Frequencies near Peak of Frequency Plot 40-45 10.41 10.75 11.89 45-50 19.49 18.93 15.66 50-55 18.07 17.43 14.90 55-60 8.14 8.53 o1.06

-92mode nor median will be as reliable as calculations based on the theoretical distribution would suggest. The standard deviation a of the error in measurement depends on the magnitude of the angle, the size of the particle being measured relative to the magnification, and the quality of the metallographic preparation. Thus, while a single value for a really does not exist, the assumption of a fixed value is valid for estimating the magnitude of the effect. For most samples measured in this work, a conservative estimate of a would be 2,0~o From the results of the above calculations one can see this would have s small effect on the shape of the distribution, and a negligible effect on the median or modal angles. 4. The Effect of Observed Particle Size The two types of errors previously discussed probably account for most of the departure observed from the theoretical distribution of angleso A third type of error may on occasion arise due to biased sampling which occurs because of differences in the section size of dispersed particleso For example, if the dispersed second phase consists mainly of lenticular shapes lying in matrix grain boundaries, measurement of only the largest sections seen on a plane of polish introduces a bias toward a higher median angle. This occurs because sections through a lens which are of a smaller apparent size have angles which are on the average, smaller. To see this, consider a single such particle in the form of a (93) lens with surfaces which are portions of spheres meeting' in a dihedral

-93angle 0 and having a diameter d in the central planeo Then, passing planes of random orientation through the particle gives rise to observed grains whose apparent length 1 is such that.~ - do If one is measuring angles observed on a plane of polish at a particular magnification, the ease of measurement is dependent upon the particle sizeo For particles whose diameters. lie between d and d + Ad, there will always be a length Io below which it is not possible to make an accurate measurement. However, disregarding particles which are too small to measure gives rise to a systematic error in the distribution of angles obtained, whose net effect is to raise the median angle of the distribution. It is worthwhile to determine the value of Qo/d which will allow one to disregard particles having ~/d < ~o/d without appreciably affecting the distribution of angles obtained. This in turn gives one a qualitative guide to the magnification needed to satisfactorily measure dispersed particles having a certain size rangeo Let N be the normal to the plane intersecting a.lens having its central plane lying in the XZ planeo The normal has direction -cosines (cos. since sin3 sin ac cos a) where ca, are the usual spherical coordinates. Thus, the equation of the interspcting plane is cos s sin a x + sin. sin a y + cos a z = 0, and the trace T of this plane on the XZ plane is the line cos. sin a x +- cos ca z = 0o The equation of the circle C in the central plane of the lens is (x - r) 2 2 + z = r o Solving the latter two equations for x, and expressing the length of the chord I in terms of x results in the following expression

-94/d = = (5-4) 1 +' cos p tan a Rearranging (5-4) and using the trigonometric substitution 2 sin2a tan a = - -sn2 1 - sin a one has 2 %K sin = K + cos (5-5) where K = (d/)2 - 1 (39) Harker and Parker() have shown that the probability of observing an angle less than or equal to ( for a given dihedral angle 6 is given by the areas to the left of curves of constant ) on a graph of sin a vso. o Superposition of curves of constant ~o/d from (5-5) on the curves of constant Y will indicate the portion of the distribution not obtained when particles having i/d ( ~o/d are ignoredo For = 500, the probability plot of sin a vs a is given in Figure 10 with the curves for three ratios of ~o/d superimposed. The fraction of the population of observed angles having ~o/d < 0ol is about 0o5%, for io/d < 0o5 about 4.6% and for ~o/d < 0,5 about 13o4o% The data for the frequency plots thus obtained are given in Table VII and the median angle for each case is indicatedo Note that the modal angle is about the same for each case. Clearly, if all particles with e/d > Ool can be measured, no significant error in the distribution will result. On the other hand, if only 85% of the particles present are large enough to measure, the distribution of angles obtained. will be in erroro Two considerations prohibit the application of these results in any quantitative manner. First, a distribution of particle sizes

35 0 ^45 0 10 20 30 40 50 60 70 80 /3 IN DEGREES 0.80 5 0.85 20 Sin' a 0.90 15 75 0.9 5 10 I.00 0 10 20 30 40 50 60 70 80 IN DEGREES Figure 10. Probability Plot Showing Regions Where Section Size of Lenticular Particles is Small. Dihearal Angle is 50~. Curve A: lo/d = 0.5. Curve B: c/da = 0.3. Curve C: ~o/a = 0.1.

-96TABLE VII EFFECT OF IdNORING ANGLES FROM SMALL SECTION SIZES ON TBE DISTRIBUTION OF OBSERVED ANGLES FROM LENTICULAR PARTICLES Observed Theoretical Cumulative Cumulative Cumulative Angle Cumulative Frequency Frequency Frequency Frequency, /d=O.l /d=0o 3 ~/d=0o5 =500 ~ = 500~ 5 =50~ 10 1.18 0~73 0 0 15 2.70 2,24 0 0 20 4.94 4.48 1.25 0 25 8.00 7.56 4.16 0 30 12.09 11o67 8.30 195 35 17.51 17.12 13590 7.18 40 24.82 24.46 2151 15o06 45 355 23 3493 32.39 26.73 48 44.42 44.16 42.01 37 19 49 48.60 48 36 46.38 41. 96 50 54~73 54.52 52.81 49.00 51 60~75 60.57 59.12 55.92 52 64.69 64.53 63524 60.43 55 72.80 72.68 71.73 69.70 60 80.94 80.86 80o25 78.96 65 85.95 85.90 85.49 84.65 70 89.33 89530 89.02 88.46 75 91.73 91o70 91.52 91.15 80 93548 93547 9 35 93.o12 85 948.1 94.80 94~73 94.60 90 95.83 95.83 95.80 95.73 95 96.63 96.63 96.63 96.61 100 97.27 97.28 97 29 97 31 105 97~79 97.79 97.82 97.87 110 98.20 982.1 98.25 98531 115 98.55 98.56 98.61 98.68 120 98.83 98.84 98.90 98.97 Median Angle 49.3 4953 49.6 50o. Relative Frequencies Near Peak of Frequency Plot 40-4.5 0o41 10o.46 lo.88 11.67 45-50 19049 19 59 20.41 22.27 50-55 18 07 18.16 1893 20.70 55-60 8o14 8.18 8o52 9~26

-97is always presento This means that for relatively large particles, nearly all orientations of the particle with respect to the plane of polish produce sections large enough to measure, while for particle sizes at the other end of the spectrum, a large proportion will produce sections too small to measureo Without knowing the distribution of particle sizes, no correction can be applied. It is probably true, however, that if the magnification used is adequate to resolve all but a few percent of the particles present, no significant error should resulto When a fine dispersion of particles is present and the best..magnification available cannot resolve a large proportion of them, the distribution of angles obtained is sure to be in error. An approximation to the true dihedral angle can, however, still be obtained from the modal angle. The median angle of the measured distribution will be too higho Th.the alloys used in this work, lenticular shapes were predominant. However, this is often not the case. When the matrix grain size is small relative to the dispersed particle size (as in powder compacts sintered with a partially-wetting liquid phase present), more particles will lie in grain edges and in corners where four grains meet than when the matrix grain size is relatively largeo Obviously, the same considerations that have been dealt with above for lenticular shapes apply here, too, but the more complex geometry of these particle shapes would make a similar calculation difficulto 5. Establishing a Confidence Interval for the Median Angle Two criteria have been proposed to determine the dihedral angle from a sample distribution of angles- a) the modal angle, and b) the median angleo Smith(93 pointed out that the modal angle in the theoretical

-98frequency function is the dihedral angleo If a range of dihedral angles exists, the mode then tends to a central value in the distribution of dihedral angles Unfortunately, the mode is a clumsy tool; the sample size must be large and the value of the modal angle depends on the classification of the data. The value of the modal angle is insensitive to many types of error, but the frequency of the mode is not. (84) Riegger and Van Vlack proposed use of the median angle to characterize the distribution. They show that the median angle of the theoretical distribution is always within 1~ of the true dihedral angle, (see Figure 25)o The median angle is obviously determined more easily than the mode, and with a greater precision. Riegger and Van Vlack estimate the sampling error in using the median angle of a Sample (drawn from a population having the theoretical distribution) to obtain the median angle of the populationo They assume that the theoretical distribution function of observed angles approximates a normal distributionO Then, the standard deviation of the median angle is given as an asymptotic approximation for large n by the relation a -ed 1o255 -3med ei p where ac is the standard deviation of the population. They then apply this relation to show that 25-50 angle samples characterized by means of the median angle give the same accuracy in determining the dihedral angle as much larger samples characterized by the modeo These last results may be criticized on three points~ (1) the theoretical distribution deviates from a normal distribution at small and large values of the dihedral angle; the useful range of this assumption is probably, 20~< 1 60~; (2) the approximation used to obtain the standard deviation of the median

-99angle is valid for large samples. It has not been shown that a 25-50 angle sample is sufficiently "large"; (3) the population from which the sample is drawn is almost certainly not characterized by the theoretical frequency function (or a normal distribution with the same mean and standard deviation); the standard deviation of the actual population is largero A better approach is to use a non-parametric method to establish a confidence interval for the median. Such a method has the virtue of being independent of the distribution function assumed for the population. S, S. Wilks (110) derives the following probability relationships. A sample of size n from a given population may be ordered according to the magnitude of the elements in the sample. Thus, the sample may be represented (x(1) X(2)9 ~ X(n)) where X(k) ( X(k + l) Then a confidence interval for the median of the population x is given by' P (x ) < < X(k + k2)) = Io.5 (k i n - kL t 1) (5-6) - 15 (k1 + k2 n - (kl + k2) + 1) where I (v, v ) is the Incomplete Beta Function. Tabulated values p` 1'v2 of this fimction have been prepared by Pearson') for p = 001 to l00, and for v -'V2 = 0005 to 50~ For large samples, an asymptotic approximation using a normal distribution permits a similar calculation. Table VIII shows 90% and 95% confidence intervals for samples of five different sizeso TABLE VIII CONIFIDEN'CE INTERVALS FOR VARIOUS SAMPLE SIZES Sample Size 90% Confidence Interval 95% Confidence Interval 25 8- 17 7- 18 50 19 - 32 18 - 33 75 30 - 45 29 - 46 250 112 - 158 110 - 140 500 252 - 268 228 - 272

-100Using this method, one can then say, for example, that for a sample size of 500, there is only one chance in twenty that the median of the population has a value lying outside the values of the 228th and 272nd members. Thus, without making any assumptions as to the distribution function of the population, the median of the population can be established to within fairly close limits. Although this method gives an interval in which the median of the population is likely to occur, it gives no best estimate for the median of the population. For purposes of further calculation, the median of the population is taken to be the median of the sample. An equally valid choice for the median of the population would be the mean of the limits of the confidence intexivalo For large sample sizes, there is no significant difference between these two figures. To obtain the dihedral angle from the median, the factor obtained by Riegger and Van Vlack4) is added to the median angleo From the calculations of the previous sections one could also apply a correction for measuring erroro A value of a = 2 is a conservative estimate for the measuring error of any specimen; this would mean an additional factor of about 035 should be added to the median angleo This last correction was not applied because it is small and computations show it is independent of the size of the median angle in the range of, interest. Furthermore, the measuring error was probably not the same for all specimenso 6. Summary Three types of errors have been considered which will change the effective population of observed angles andthereby the distribution function for that populations (1) Existence of a range of dihedral angles,

-101distributed in an unknown manner, (2) errors in measurement, and (5) a range of apparent particle sizes, some of which are too small to measureo One should be able to avoid the third type of error by using sufficiently high magnification on a sufficiently coarse distribution of particleso This type of error is most likely to show itself when comparing results of samples taken from two different dispersionso Both of the first two effects must always be present, however, to some extent. The resulting samples with flattened peaks represent a superposition of both effects. A reasonable estimate of the error in measurement, a = 29 cannot account for all of the peak broadening. One must conclude., therefore, that a range of dihedral angles existsin the specimen. If one assumes that this range of dihedral angles is rectangularly distributed, then the above calculations suggest a spread of dihedral angles from 8~ to 16~ wide is present. The median angle of the population of observed angles can be determined. To relate this to a dihedral angle one must retain the convenient fiction that a single dihedral angle exists. This assumption leads to the calculation of ratios of interfacial tensions which are some sort of averaged values. While it is important to realize that all interfaces may not have the same tensions, conclusions made as to how the averaged tensions vary with the constraints on the system must also be true for the individual tensionso In the following section, a brief description is given of another method for determining a dihedral angle from a sample of observed angleso In the second section, the results of non-parametric tests for statistical differences between certain pairs of samples are giveno

-1027. Another Method for Obtaining Dihedral Angles In a previous section, a non-parametric method for establishing a confidence interval for the median angle of the population was discussedo For large samples, the 95% confidence interval is small enough so that the probable error involved in identifying the median angle of the sample with the median angle of the population is tolerable. In addition, however, one would like to know how well the samples fit the theoretical distribution equation. An alternative method for choosing a value of dihedral angle likely to produce the observed sample was devised which is applicable to samples of any size. This method is empirical and does not give an indication as to the accuracy of the values obtained, but it does give an indication as to how well the sample fits the distribution equationo Given a sample (Il oo,^ V n) ordered so that 4)i - i+ ~ one can construct a cumulative distribution function for the sample 109 y < 0 *(t) = ( 2 - 0 ~<t< iK; i =,1 0, n (5-7) ( 2n ^ n If F(y; e) is the distribution function given in Appendix A which describes the population, then one can define a parameter J given by: n J = F (Yi;G) - F* (yi)] 2 (5-8) i=l Thus, a value of J for a sample can be computed for any choice of the parameter 0 o The value of G which gives the smallest value to J, thus giving the "best" fit to the data, is taken to be the dihedral angle. The value of J for this "best" fit, is an index as to how well the sample fits the theoretical distributiono

-103This empirical method for choosing the parameter & is based on the Cramer-von Mises test for "goodness-of-fit" (8) According to this test, if the theoretical distribution function is known, the "goodness-of2 fit" of a data sample can be established by means of the parameter ncu n -L~n ~ C 2 2n + F(x ) -_ -Y 2 1 + 77 F2jFx - 2 12n-' 2j=n j where F (x) is the distribution function of the population and 2j - 1 2n is the distribution function for the sample. Tables of the probability that'2 a sample has rxn <_ Z for various values of Z are given by Anderson and Darlingo(4 While there is no assurance that the above method of minimizing J produces an estimate of e which is any way better than a value'of 6 determined via the median angle, one does obtain in the minimum value of J an indication as to how well the data can be made to fit a theoretical distributiono As was pointed out in the discussion of errors, the distribution function of Appendix A does not satisfactorily describe the population distribution, a fact which may be confirmed by the Cramervon Mises test. Nevertheless, the minimum values of J obtained allow judgments to be made as to the relative reliability of various sampleso For purposes of comparison, an index of fit, I, is defined as follows -o I = =J[ =; EF(i ( ) - FX (W)i)J 2/n (5-9) where F ((Li; G) and F* (Y) ) are expressed as percentages. This index is not dependent on sample size, as is J. Typically, "good data" gives values of I less than 5.

-i04The values of the dihedral angle found by this method are up to 10 higher (and never lower) than those obtained by using the corrected median angle. One would expect this result, because (1) measuring error has been shown to decrease the median angle, and (2) a range in dihedral angles probably decreases the median angle. In subsequent calculations, the dihedral angle used was the ore obtained by correcting the median angle. No significant difference in any of the results is obtained by using the dihedral angles which fit the data best according to the above criteria o 8, Statistical Tests for Differences Between Samples Since the form of the distribution function for the population of "observed angles" is unknown, it is desirable to use non-parametric methods to establish whether or not certain samples could have come from the same populationO The methods used are described in referemce (91), Two types of "two-sample" tests are suitable ~ (1) the median test9 and (2) the Kolmogorov-Smirnov test. In the median test the hypothesis being tested (null hypothesis!) is that the samples come from populations with the same mediano Used as a one-tailed test, rejection of this hypothesis implies that the medians are different in the manner predictedo In the Kolmogorov-Smirnov test, the null hypothesis is that for any X, F (1) (X)= (2) (x), where F (X) is the distribution function for the population from which the sample is drawno Subscripts refer to the two samples being testedo Used as a two-tailed test, rejection of the null hypothesis implies that the distribution functions are different in central tendency, dispersion, or skewnesso Used as a one-tailed test, rejection implies that F(1) (X) >F(2) (X) for X near the center of the distributiono

-105The one-tailed test is most sensitive to differences in the central tendency of the distribution. The Kolmogorov-Smirnov test is more sensitive to differences, (91) and is the preferred test(9) Some results of these comparisons are presented in Table IX. The level of significance at which the null hypothesis is rejected is the probability of rejecting the null hypothesis when it is in fact true. In each case, the lowest level of significance at which the null hypothesis can be rejected is indicatedo The median test applied to the same data does not allow rejection of the null hypothesis with as high a degree of certainty. TABLE IX C.,OMPARISON OF PAIRS OF SAMPLES FOR SIGNIFICANT DIFFERENCES KOLMOGOROV-SMIRNOV TWO-SAMPLE TEST Level of Significance at which.Null Hypothesis is rejected Sample Pair Two-Tailed Test One Tailed Test Dll-7 1 0.10 0.05 Dll-8O 2 D 1 Not rejcted Not rejected C2A 1 3 Ooe025 oo0.01 C2A C4 C o.0 O 00 1 Cl 3 5 Dl 1 o5 o05 Dl 3 Reference to Tables X and XI in the next chapter will show that all the samples in each pair in Table IX, except the second pair, should have come from different populationso The Kolmogorov-Smirnov test indicates that it is likely that the samples in each pair, except the second, do come from different.: populationso For example, the results of the twotailed test for pair (1) show that there is only one chance in ten that the samples are drawn from the same population~

CHAPTER VI EXPERIMENTAL RESULTS 1. The Nickel-Lead Phase Diagram Figure 11, the phase diagram for the Nickel-Lead System, is taken from Miller and Elliot(9) It is based on Hansen(37) with corrections from the more recent data of Fleisher and Elliott,() and (2) Alden, Stevenson, and.Wulffo) Figure 12 shows the solubility of Ni in liquid lead from reference (2)o It is not known whether a eutectic or peritectic reaction occurs near the melting point of pure leado No information is available as to howr the phases present change composition with changes in pressure. A rough calculation can be made to indicate the order of magnitude of the change of nickel solubility in liquid leado Assume that the solid is pure nickel and the liquid is a dilute solution of nickel in leado Then since d di (6-1) N=i = Ni and di s = dP (6-2) - Ni ~I Ni VNi dP + dxNi T constant axi one has ndi N Pi = VNi -i T constant (6-3) ~ dP -Ni Ni aXni where MNi is the chemical potential of nickel, Vi is the molar -106

-1071600 1452 I 400 \ 1340 1200 C) 1 000 800 I1 4C. S 800 - 600 400 - (328) (324)? 327 327 200 1 0 20 40 60 80 100 WEIGHT PERCENT LEAD Figure 11. The Nickel-Lead Phase Diagram.

-108r 000 900 800 / 0U 0 O LUi 500 4 00 300 0 1 2 3 4 5 6 7 8 9 10 ATOMIC PERCENT NICKEL Figure 12. Solubility of Nickel in Liquid Lead. Reference (2).

-109volume of nickel in the solid, V the partial molar volume of nickel VNi in the liquid. and XNi the mole fraction of nickel in the liquid, If the activity of nickel in liquid lead is proportional to its mole fraction, then a'Ni RT ~ I (6-4) ax.ii Xnix aNi Ni So d ln xi i - i.dP. i -Ni i (6-5) dP RT From (66, 96) the molar volume of solid nickel at 600~C is 6066 cc/molo From (98) the partial molal volume of nickel in liquid lead at 600~C in in the range 352 to 5o5 cc/molo Taking the value which will produce the greatest change in solubility one haso ~ d In x-i d n XNi 48 10- atm (6-6) dP The solubility of nickel in lead at 600~C is 1o85 atomic percento From Equation (6-6)9 5000 atm (44,000 psi) pressure increases the solubility to 2o14 atomic percent. From the data available, this must be regarded as an upper limit to the solubility change which occurs. One can conclude that pressures in the range used in these experiments exert very little influence on the phase diagram. 2o Temperature and Pressure Data Specimens of leaded nickel were equilibrated at temperatures from 371~C to 816~C (700~F to 1500~F) in a vacuum. Other specimens were equilibrated at pressures up to 50,000 psi at 371~C (700~F) and 595~C

-110(1100~F)o The results of 500 angles measured on each specimen are shown in Table X. All of the specimens with designation D were cut from the same ingot, and had about the same grain size, impurity content, etc, All of these specimens were hydrostatically tested in watero The only inconsistent values from this data are from DP2 and DP35 The first gave an abnormally low value for the dihedral angle and the second an abnormally high value. (Compare with specimens DP5 and DP9, respectively ) Re-measurement of these specimens produced virtually the same dihedral angles. These two specimens were among the first ones run in the hydrothermal unit and neither was quenched after the run as were all other specimens. DP2 was cooled in an air blast, and DP3 was cooled in an air blast until the pressure dropped to about 25.000 psi, then quenched in watero It is not clear why failure to give a rapid quench should produce a low angle in one case and a high angle in another. This seems to be the only difference, however, in the way these two specimens were treated, so one must assume that the cause of the unusual values is the quenching rate. Neither of these two specimens will be considered in the analysis of the results. As was mentioned in Chapter IV, heat D proved to have an undesirably high sulfur level. Sulfur is known to be surface active i(6) in liquid iron() and there is indirect evidence which suggests it might segregate to nickel grain boundarieso(7 Oxygen is also surface active in liquid nickel(55) For these reasons, heats E and F were prepared in such a manner as to insure low sulfur and oxygen contentso

TABLE X SUMMARY OF TEMPERATURE AND PRESSURE DATA 500 ANGLES MEASUIRED ON ALL SPECIMENS Sample Pressure Temp Time Dih. 4 Median 95% Confo Dih. 2. Index psi ~F Hours from Angle Limits on Median from of Median Computer Fit D 0 700 186 51o9 51.0 49,1 52.2 52.1 3521 D900 0 900 168 46.5 45~ 5 44.1 -470 470 53 95 D10 1100 1-68 41.1 40.2 38.8 41.9 42.2 5.82 D1300 0 1300 120 36.8 36.0 34.8 37.6 37.8 5 54 D1500 0 1500 120 32.3 31.6 306 33.53 3356 6.94 DP1 10,600 706 168 47e7 46,7 45.3 48.4 48.2 3566 DP2 24,000 716 161 40o1 39.2 37.9 40.7 41,0 5.00 DP3 335500 708 144 448 4358 42.1 4444 44.8 375 DP5 19,700 709 167 45,0 44,0 42.7 45.7 45.5 4.76 DP6 41,000 712 168 40,9 40,0 50 4195 4417 5o74 DP8 50,200 7035 168- 38,6 577 36 6 38.8 392 4.73 DP9 31,000 705 83 42.6 41,6 39,8 4351 43.0 4o39 Dll-l 23,200 1096 124 36.1 35~ 3 34,3 36.8 37.3 3o85 D11-2 11,500 1098 120 38,9 38.0 36,7 39.1 9.3 4.14 D11-6 33,100 1096 120 34.7 3359 32.5 34 9 35.5 4.38 D11-7 42,000 1097 120 33.6 3528 31.o3 339 3543 5.52 D11-8 52,600 1096 52 32,1 31i4 30.1 32,6 32.8 7.81 E 0 1100 72 435 420545 4 3 438 44.5 5 27 E1 23,500 1098 28 39.5 38.6 37.5 39.8 40.4 4.39 F 0 1100 72 42.1 41o1 40.2 43-0 43 6 5 28 Fl 30,200 1097 30 3558 3550 3357 36.3 36.5 4.17 F1A 0 1100 36 42.1 41.1 40o2 42.5 42.6 3579

-112Due to the melting practice used, heat E contained only a few percent lead which was very finely dispersed throughout the nickel matrix. The median length of lenticular particles was about 34pi. Heat F contained 8-10% lead with a median particle length of about 5o4po Particles in heat D were larger; a median particle length of about 711 was measured. Any particle less than about 2p. in length is too small to measure with any degree of accuracy at the magnification used (l200X)o Particles from 2-4[t in length can be measured, but the measuring error is probably high. The measurements from specimens E, F, and D1100 show decreasing median angles with increasing particle sizeo In the light of the errors discussed in the previous chapter, it is felt that a high median angle was obtained from E because of biased sampling due to the omission of measurements on small particleso The same tendency may have been present in F; but the results from F and D1100 are not sufficiently different (nor are the particle sizes sufficiently different) to warrapt such a conclusion. Specimen El was hydrostatically tested using argon as a pressure medium. However, a slow leak in the system allowed the argon pressure to drop about 6000 psi during the course of the runo Hence, the measurements made on El have little significance. Specimen Fl was also hydrostatically tested in argono The resulting change in median angle from F to F: checks well the change in median angle from D1100 to D11-6. These results allow one to conclude that, (1) composition differences (particularly sulfur content) do not significantly influence the results, at least at these impurity

-113 - levels, (2) hydrostatically testing in water rather than an inert medium does not influence the results, (3) the length of time allowed to obtain equilibration at 1100~F for all specimens from heat D was more *than adequateo The reversibility of the change in dihedral angle is shown by specimens F. Fl, and FlAo Specimens F and Fl were heat treated at 1100~F for'72 hours. Angles on F were then measuredo Specimen Fl was then subjected to hydrostatic pressure at 1100~F for 30 hourso After angle measurements were completed on Fl, it was reannealed at 1100~F for 36 hourso The angles then measured are designated by FIAo It is clear that there is no significant difference between F and FlA. The results of the temperature and pressure measurements are summarized in Figures 13 and 14. The ratio of surface tensions is related to the dihedral angle by: gb//Ysl = 2 cos 9/2, where e is the dihedral angle. Points designated by squares in Figure 14 are picked off the smooth curves of Figure 135 35 Tension and Compression Data The angle measurements made on stressed specimens were treated:in the same manner as similar measurements made on specimens equilibrated at different combinations of temperature and. pressure. For purposes of characterizing a sample of angles obtained from these specimens, the median angle will be usedo The problem of relating this to a dihedral angle expressi.ble as a ratio of interfacial tensions will be left to the next chapter. (59) Preliminary experiments on leaded copper suggested that the median angle decreased. when a tensile stress was appliedo The exact

-1141.94... I.92 1500~F -.J,1i.9o- 0_ 1 I 90 1300~ F (1 1.88 - zX E- ^11000F z x 1.82 LU. 0 1o0o0 20,000 30,000 4000 50,000 60,00 PRESSURE ( PSI) Figure 13. Ratio of Interfacial Tensions as a Function of Pressure. 95% Confidence Intervals Are Shown for Two Data Points. 0 0 / I.82 I.80 0 0 F o10,000 20,0 0, 000 000 4qo,00 50,000 60o000 PRESSURE (.PSI) Figure 13. Ratio of Interfacial Tensions as a Function of Pressure. 95~o Confidence Intervals Are Shown for Two Data Points.

RATIO OF INTERFACIAL TENSIONS KGB'/ SL 001 OBB ~00 C(00(0(ODD CD 0 r 0) 0B 01) 0) I I I I I III HH H.0 0 - CD 0 N H N -f-I C\ t~~~~~~~~~~~~~~~ H 0 o \^s \ \ s ^ ~ ^-\ N' \N \\ 0 0 N o - " x <,\~ H m5 H H n N. H?.~~~~~~~~~~~~~~ N \ \ \ -4 N C N H NN I - o (. o 0 (D ci-0 P.~II

-116 - nature of the variation was not established by the preliminary work, so these experiments were repeated using leaded nickel. All tension and compression testing was done at 371~C in an inert atmosphere for 48 hours, with the exception of one specimen run for 120 hourso Two specimens were run at stresses of 1000 and 1500 psio Both showed a lower median angle than an unstressed specimen treated at the same temperature, but the decrease in both cases was about the sameo The angle measurements were then grouped on the basis of the distance from the edge of the specimen at which the angle was observedo It was found -that the median angle was lower(by 5-6~) in the center two thirds of each specimen than it was in the outer third of the specimen. This confirmed the observations made in reference (59)o The median angle in the outer third of the specimen was almost the same as the median angle of the unstressed alloyo Observations of the polished section showed some cracks occurred. These began at the specimen surface and followed grain boundaries into the interior of the specimen. On the order of 30 cracks per inch along the section edge were observed, the longest running 0,015-0020"T into the specimen. The portion of the specimen showing unchanged median angles was about Oo035" thicko It was postulated that the applied stress was indeed affecting the median angle observed, but that cracks occurring at the surface relieved the stress in the outer portion of the specimeno This produced a median angle in the outer portion of the specimen characteristic of the unstressed material. Since observations of crack depth can only be made in the plane of polish, it was considered likely that some cracks were at least 0o035" deepo

-117To test this hypothesis, subsequent specimens were plated with about 0.001" thick copper plate prior to tensile testing. This soft plate was applied to eliminate crack initation at the surface. In every subsequent tensile test with plated specimens, no dependence of the median angle on distance from the surface was noted. On the basis of these results, the suggestion put forward in reference (59) that diffusion of oxygen from the atmosphere along grain boundaries was responsible for this behavior must be rejectedo In a tensile test on this sort of material, the liquid lead particles probably behave very much like stress-free holes in a solid matrix. Stress concentrations occur at the particles; particles lying in planes normal to the axis of tension must have a concentrated normal stress at the apex of the particle tending to cause crackingo It was considered possible that lead filled cracks originating at these particles might have been measured and were responsible for the overall decrease in median angle observed in stressed specimens. To check this possibility, the angle that each lead particle made with the axis of tension was recorded along with the apex angles. The data was classified into three groups on the basis of angle off the axis of tension. This was done for all tensile specimens. A summary of all the data on plated tension specimens,9 classified in this manner, is presented in Table XI and in Figures 15 to 17. The data show that for all but one tensile test the median angles taken from particles lying normal to the axis of tension are larger than the median angle from particles lying along the axis of tension, Non-parametric tests for statistical significance indicate

TABLE XI SUMIMAY OF TENSION AND COMPRESSION DATA TEMPERATURE 700~F Run Stress Angle Time Median No.Angles 95% Conf, Dihedral Index psi Off Axis Hours Angle Measured Interval Angle From of For Median Computer Fit Tension D 1 0 0-29.9 186 51.1 145 53.8 48.4 52.4 5.12 2 50-59.9 5l1 165 55.8 47.9 52.1 4.67 5 60+ 50.9 191 52.4 48.7 51.8 2.50 Dl 1 2010 0-29.9 48 58.6 23355 41.9 54.9 40o,8 7,88 2 50-59.9 58.8 299 41.2 56.5 4l,o 8.27 5 60+ 42.9 452 44.8 40.7 44,5 8.75 D2 1 975 0-29.9 120 42.9 205 44.8 40o,8 44.6 4.56 2 50-59.9 46.5 206 48,5 45,0 47.5 4.88 5 60+ 47.8 191 50.1 44.9 49.2 4.21 H Cl 1 1500 0-29.9 48 40.6 251 42.6 59.4 42.6 4.12 2 50-59.9 42.5 175 45.2 40,8 44,4 4,55 6o0+ t. 187 46,-7 45.5' 45.8 24 C2 1 995 0-29,9 48 43.8 182 46,4 41.9 45.5 5.19 2 50-59.9 46.3 147 48.2 42.6 47.4 4,78 3 60+ 47.4 167 49.9 45.1 49,2 5,06 Bl 1 2550 0-29.9 48 57.9 150 4o.5 34.8 59.7 7,90 2 50-59.9 57.-1 145 58,8 55.2 58.0 7.86 3 60+ 51.7 141 4.8 26,7 3355.6 9,64 Compression DCl 1 1500 0-29.9 48 45.7 182 47,6 42,1 46,6 4.62 2 50-59.9 42,3 206 44.6 59.2 43.8 5.00 5 60+ 59.6 212 43.4 56.0 41.6 7.68

-119o 0 PSI A 995 PSI 55 x 1500 PSI TENSION - X 1500 PSI DATA + 2010 PSI 3 2350 PSI 50 45 w x d XX / z w 40 o_ 35 30 -I I 0-29.9 30- 59.9 60- 90 ANGLE OFF THE TENSION AXIS Figure 15. Msdian Angle Observed in Tensile Testing as a Function of Orientation with Respect to the Axis of Tension.

-12048 0 1500 PSI TENSION 46- A 1500 PSI COMPRESSION j 44 z 42 w 40 38 0- 29.9 30-59.9 60 - 90 ANGLE OFF AXIS OF TENSION (COMPRESSION) Figure 16. Comparison of Results of Testing in Tension and Compression 50C O 995 PSI - 48 HOURS 48- A 975 PSI-120 HOURS -JJ Z 46 z c 44 42 0-29.9 30-59.9 60-90 ANGLE OFF AXIS OF TENSION Figure 17. Comparison of Tensile Tests Run for 48 and 120 hours.

-121that the differences indicated perpendicular to and along the axis of tension are significant. The only specimen which shows a smaller median angle perpendicular to the axis of tension is Bl which was run at the highest stresso Attempts made to run specimens at still higher stresses all resulted in fracture soon after the load wasappliedo (In fact, some specimens loaded at a lower stress than Bl also fractured.) One must conclude from these results that the lowered median angle observed in tension is not influenced by crack formation at the lead particles at stresses below the observed fracture stress of these alloys Specimen D2 was run for a longer time to determine whether the median angles changed with time. Comparison with specimen C2 run at the same stress shows no significant change in the median angle between the two specimens. One specimen (DCl) was subjected to compressive loading under the same conditions that were used in tensile testingo The specimen was not plated. The median angles measured for this specimen were also lower than for the unstressed material, but the lowest angle -as now observed normal to the axis of compression. The changes observed in compression ere not much different than the changes occurring at the same tensile stress (specimen Cl)- but the dependence of the median angle on the angle off the axis of tension is reversedo Comparison of the data from Table XI and the data from hydrostatic testing at 371~C in Table X shows that a pressure of 50,000 psi is required in hydrostatic loading to produce the same drop in median angle as was observed with a 2000 psi tensile stresso

-122It is probable that the variation in median angle observed in these specimens depends on the orientation with respect to the axis of tension of the plane of the grain boundary in which the lead particle occurs. The angle off the axis of tension measured in the plane of polish, I), is related to the spherical co-ordinates (8 ) of the normal to the grain boundary by the relation cos sin sin (6-7) l1- sin2 cos 2 The orientation parameter ~ represents the angle between the normal to the grain boundary and axis of tensiono When P is allowed to range from 0 to 30~0 Q can take on values from 60 to 90~o For 30~ < t < 60~, then 30~ < < < 90~9 and for 60~ 4 L < 90~9 then 0 < G < 90~~ Thus the grouping of angles occurring 60-90~ off the axis of tension can arise from particles occurring in planes with any inclination from 0 to 90~ with the tension axis. The majority, however, will have inclinations from 0 to 30~0 so the variations observed are still. qualitatively correct. The significance of this data will be discussed in the next chapter

CHAPTER VII DISCUSSION OF RESULTS 1. A Method for Obtaining Variations in Interfacial Tensions from Variations in the Dihedral Angle Changes in the dihedral angle with changes in temperature and pressure are the net results of changes in interfacial tensions at the solid/solid and solid/liquid interfaces. If one can obtain, however, the change in interfacial tension occurring at each interface, then the thermodynamic equations developed in Chapter II from the Gibbs Adsorption Equation can be applied and some properties of the interfaces may be calculatedo In order to do this, one must make an assumption as to the dependence of dy/dT on temperature and of dy/dP on pressureo The simplest assumption one can make is that for both interfaces dy/dT is independent of T, and dy/dP is independent of P, Since one cannot show, in general, that the thermodynamic quantities which determine these coefficients are independent of temperature or pressure, the worth of this assumption can only be measured by how well the data can be correlated on this basiso Assume that the tension of each interface changes according to the expression: 7 = a+ Ld J =y + X (7-1) where X represents either temperature or pressure, AX= X - Xo y~ is the interfacial tension at X6 and dy/dX = y? is independent of % o Then at any value of'X one may write-123

-124Y _ gb + 7gb (70 (7-2) sl ~ s7Y's + a where subscripts gb, sl refer to the solid/solid and solid/liquid interfaces respectivelyo Subtracting ~ yb/Y~ from both sides, dividing y gb/ sl by 2 and rearranging gives: 7gb ygbb + 7 gb7 b sl + gb7s A. 2yL 27y - 270y s(70 + 7 sl a si si ^si sl si or gb gb ( gb sl - gb sl) s (7-3) 27sl 27~s 27 slosi + 2Ysi~ i AX taking the reciprocal, assuming 7 gb/ y0sl egb' /Ys Sgb s r 4 ) - r o = sl + 2 s l y sl ( I A 27 gb ( 7 0gbY sl ( gb 7 7 sl- 7 gb7Y slgb S l 2sl 2 sl" Now when the ratio of interfacial tensions is such that 0 <(Ygb/2 sl< 1 then a dihedral angle 0 exists, 0< e < 180~o So the ratios of interfacial tensions on the lefthand side of (7-4) may be represented in terms of the dihedral angleo The first term on the righthand side of (7-4) is independent of X and the coefficient of 1/eX in the second term is independent of X o Thus Equation (7-4) may be written~ (cos e/2 - cos`o/2)" = b + m (- ) (7-5) If the assumption that dy/daX is independent of X is true, then a plot of (cos /2 - cos 0~/2)-l vs (AX)o1 will yield a straight line with slope m and intercept bo If m and b are determined from such a plot, then they may be used to find Y gb and

-1257s in terms of either y~ or ~b and cos 0~/2o This is done by solving the equations for m and b simultaneously and using the relation: 2 cos e~/2 Y= gb/7~sl The results are: _ b 7gb sl m sl 2m cos ~/2 (-6) 70l T ~(7. L' - m (1 + b cos 0o/2) - g ( 1o +b /gb m m cos 0/2 Thus dys,/dX and d7gb/d)X are determined in terms of sl gb the surface tension of either the solid/solid or solid/liquid interface and the dihedral angle e at X O0 Using the above technique the coefficients d.y/dP and dy/dT can be determined for each interface from measurements of the dihedral angle as a function of pressure and temperature, respectively. In. Appendix C. the change in dihedral angle with temperature is correlated on this basis for data obtained from the literature. The dihedral angle at zero pressure was measured at five temperatures from 371~ to 816~Co This data is presented in Table X. Using a reference temperature of 3710~C a plot of (cos &/2 - cos ~/2)- vs (AT):' is shown in Figure 18. The line drawn through these points is a weightedleast squares lineo The reasons for using a weighted least squares fit are discussed in Appendix Do The method for fitting weighted least square lines is described in reference (68). It is interesting to note that this plot extrapolates to a zero dihedral angle at the moiotectic temperature (1340~C), If one takes the monotectic temperature as a reference and constructs a

-12655 i0 50 40 0o Cos -e Cos/ 2 SLOPE 5.15 o3 0C / INTERCEPT 4.71 i i/ 20 - 100K 0 20 -! 0 o 4 6 8 10 I/T (C I x 103 Figure 18. R-plot for Temperature Data at 0 Pressure. The Line Shown is a Weighted Least Squares Line. The Line Extrapolates to a 0~ Dihedral Angle at 1340~C, the Monotectic Temperature, Indicated on the Abscissa.

-127(cos 6/2 - cos ~0/2) 1 vs (T)1 plot, the data is linear and the calculated temperature coefficients of surface tension agree to within one standard deviation with the temperature coefficients obtained using 371~C as a reference. The variance of the coefficients obtained is about the same in both cases. An attempt to quench a specimen equilibrated at the monotectic temperature produced a specimen with a small but finite dihedral angle. Angles appeared to be smallest at the surface of the specimen, indicating that the quenching rate is an important variable. Ikeuye and Smith() show the dihedral angle going to zero at the monotectic temperature in the Cu-Pb system, and it is possible that this occurs in the Ni-Pb system, too. A slope and intercept may be determined from Figure 18 and used to compute the temperature coefficients of the surface tension at each interface. The results are: dysl dT^ 5.09 (+~276) 104 7(3710) ergs/cm-C d7gb =11o2 ( 5.06) -104 y( 71~) ergs/cm2-~C dT where y(3710~) is the grain boundary tension at 371~C and 0 pressure. The standard deviation of the coefficient is indicated in parentheses (+X). For purposes of comparison, all data will be expressed in terms of the grain boundary tension at 593~C. From Equation (7-1) 7(593~) (7 + 113 32 10 7(3710) (593 - 371) 7(593~) = 1.252 y(37.0~) Hence, the temperature coefficients become:

-128I = 4o.0 (+2o20) 0- 4 7(5953) ergs/cm2- C aT (7-7) rgb= 9o04 (4.05) 104 y(5935) ergs/cm2-oC Note that both surface tensions are increasing with increasing temperatureo For pure liquids, surface tensions invariably decrease with increasing temperature, but, as McLean points out 6) positive or negative temperature coefficients are possible in two component systems. Reference to Equation (2-28) clearly indicates that the temperature coefficient is dependent not only on adsorption at the interfacesbut also on the thermodynamic properties of the two component phases presento The assumption is often made in treating changes in dihedral angles with temperature, that the grain boundary tension is constant with changing temperature. If this were so in this system, then the intercept of the above plot should equal - sec e~/2 or -1oo1 It is clear from Figure 18, that such an intercept is improbable. Changes in the dihedral angle with pressure were measured at 371~ and 593~C0 This data is presented in Table X and plotted as (cos 9/2 - cos ~o/2) vs P.1 in Figures 19 and 20. The reference points are the zero pressure values of the dihedral angle at these temperatures. The lines drawn are weighted least squareslineso From the slope and intercept of these plots, pressure coefficients of interfacial tension are determined: dysl dP- = 2.00 (~0.82) 110 (53710) cm at 371~C (7-8) dygb -10 p = 5 90 (~1.55) 10 Y(3710) cm ~ c=3o0(53 1

-12965 60 50 C - 0O -;2 Cos 0/2-Cos0/2 40 - SLOPE 3.62 100~ dynes /cm2 30 / INTERCEPT: 13.01 ~20~0 20 - 15 ----- - 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 /p x 10~1 (cm./dyne) Figure 19. R-Plot for Pressure Data at 371~C. The Line Shown is a Weighted Least Squares Line.

-1300 140 SLOPE 9.48 x I ODYNES/Cm 2 INTERCEPT: 14.37 120 100 0 _ <S 80 - 60 <S /O 40 _r 20 0 | | | I I I l 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 I/P x100( Cm2/dyne) Figure 20. R-Plot for Pressure Data at 593~C. The Line Shown is a Weighted Least Squares Line.

-131dy sl - = 0.81 (0.o63) 101 7(5935) cm at 593~C (7-8) db - 1.63 ( l.32) *10 7(59) cm dP Referring the measurement at 371~C to the grain boundary tension at 593~C. as before, one has sl -10 - = 1.60 (o.65) * 10 7(5953) cm at 371~C (7-9) = 3.12 (+1.22) 10 7(593~) cm The calculated pressure and temperature coefficients in Equations (7-7) to (7-9) are rather inaccurate; the reason for this is discussed in Appendix D. It should be emphasized here that the properties of the interfaces to be calculated from these coefficients in the next sections are at least equally uncertain, i.e., they are reliable to an order of magnitude, but not much more. 2. The Solid/Liquid Interface From the measured pressure coefficient of interfacial tension, it is possible to determine the surface excess of Ni, ip byr Ni(Pb ), by using Equation (2-39). The only other information needed is the molar volume of solid nickel at 371~ and 593~C, which is readily obtainedo From reference (66), the density of Ni at 20~C is 8.902 g/cc. Smithells(96) gives data for the linear coefficient of thermal expansion between 20~C and the temperature of interest. Using these data, the molar volumes of solid Ni at the temperatures of interest are: VNi = 6.70 cc/mol at 3710C'Si = 6.77 cc/mol at 595CC *(7-10) Vri = 6.77 cc/mol at 593~C.

-132The compressibility of Nickel is on the order of 107(psi)- Since the maximum pressure used in these experiments was about 50,000 psi, the maximum change in V.i with pressure is on the order of 0o5% or less. This change is small enough to be ignored. From the data in (7-8), (7-9) and (7-10) and using Equation (2-39),9 the excess concentration of Ni at the solid/liquid interface may be computed. The results are: ri(Pb) = -2539l10 7(5935) mole/cm2 = -144-IO1 7(5930) atomatoms/cm71C (7-11) = -1.19010(-11 2 =12 N(Pb) = -1 19~1L 7(593 ) mols/cm = -7o17101 7(5930) Ni (Pb)2 atoms/cm at 593~C o The excess entropy of the interface, Sb ) can be determined from the temperature coefficient of interfacial tension, using the results in (7-11) above This can be done in two wayso First, one can assume that the solid is pure nickel, and express the differential of chemical potential of nickel in terms of the solid phase. The expression for the temperature coefficient of interfacial tension is theno 7sl/dT = - ) Ni(Pb) i 7-2) where S is the molar entropy of solid nickel. From reference (l01) the molar entropy of solid nickel is: SNi = 12.65 cal/~C-mol = 530~108 ergs/~C-mol at 371~C 8' p ( 7 - 1 5 ) SNi = 14o88 cal/~C-mol = 6o23510 ergs/~C-mol at 5930C Then usingEquation (7-12) and the data from (7-7 ) (7-11), and (7-13) the excess entropies are:

-133S(s) =.102 (5950 (pb) =-130~10- y(593~) ergs/cm2-~C at 371~C S(s) = -0 785 0-2 y(593~) ergs/cm-C at 593~C The second method of obtaining the excess entropy requires that the differential of chemical potential of nickel be expressed in terms of the liquid phase. This leads to Equation (2-33)5 No simplifying assumption is necessary here as to the composition of the phases, but values are required for the partial molar entropy of nickel in liquid lead,, SNi and for -H) d-T both evaluated in the saturated solution. Sufficient experimental data is not available to allow calculation of these quantities, but they may be estimated from a "?sub-regular" solution model. The necessary calculations are shown in Appendix E. Using the results of Appendix En the terms I(pb)Ni and Ni(Pb) -o- dT occurring in Equation (2-33) are computed, giving the following~ ^ ( N - 2oY20-(10 - 593 )'2 o C at 371 C Ci(Eb) 8Ni = 2-2.O3O10 y(595 7) dye cm at 5371C S - 112 10 T (5930)...c at 593 C Ni(Pb) 6 lP d -3.58-10 3 7(593) dyne55 5 93~C NRi( dx -dye s/cm, at 59 ~C Ni(Pb) 6- T P dT = -3a58'zo 5(5930) c00 Using (7-11), (7-15), (7-7), and Equation (2-33), the excess entropy at the solid/liquid interface is~ S) = -1617102 y(593~) ergs/cm2- C at 371~C (p)) = -O072l1l 7 (5950) ergs/cm0c at 59 7-6) s (s) -0.721o10- 2 7(593~) ergs/cm2-~C at 595~C (Pb)=~

-134These results check well with (7-14)o From the values of excess entropy obtained, some inferences may be drawn as to the thickness of the solid/liquid interfacial regiono If one calculates the entropy per unit volume in the solid and liquid phases, thEnthe excess entropy of the interface can never be larger than the difference between these values. This assumes no maxima or minima in the entropy per unit volume as one moves across the interfacial region. From reference (101) the molar entropy of the liquid lead solution and of solid nickel can be determined. Using the molar volumes of these phases at 371~C and 5935~C the entropies per cubic centimeter in each phase are: S = 4~86 107 ergs/cc-~C at 371~C XS = 516 ~ 107 ergs/cc-~C at 593~C Pb^ (7-17) S!i = 7o91 ~ 107 ergs/cc-~C at 3710C Ago = 916 ~ 107 ergs/cc-~C at 593~C The maximum negative differences in entropies at each temperature are: -3505 ~ 107 ergs/cc-~C at 371~C and -4O0 ~ 107 ergs/cc-~C at 593~C. These may be set equal to S( )/T at each temperature, and the minimum possible thicknesses, Tmin areo -7 min = 170 ~ 10 cm at 3710C (7-18) -7 Tmin = 0~o78 ~ 107 cm at 593~C These are computed assuming that 7(5930) = 400 in Equation (7-16)o (See Appendix B for a justification of this assumption ) At both temperatures the minimum thickness corresponds to 3 - 6 atom layerso

-135These figures are not unreasonable for the solid/liquid interface; they are, of course, no more reliable than the numerous assumptions and estimations that have been made up to this point in the discussion. Such a diffuse interface is also to be expected on the basis of the excess concentrations of nickel in Equation (7-11)o Again using 7(593~) = 400, i(Pb) = -5o76 o 1015 atoms/cm2 at 571~C (7-19) Ni(b) = -287 1015 atoms/cm2 at 593~C. Since there are about 21015 atoms of Ni on a close packed plane in the solid, an interface of about 3 - 4 atom layers would be necessary to obtain these valueso It is generally felt that the thickness of a grain boundary between identical phases is on the order of 3 atom layers thicko(5) For interfaces between dissimilar phases, this is probably not the case. Cahn and Hilliard (2 5) for example, predict diffuse interfaces between dissimilar phases from calculations based on a necessarily simplified model. 3. The Solid/Solid Interface It was pointed out in section (5), Chapter II, that one cannot express the temperature and pressure coefficients of grain boundary tension in a form that permits the direct calculation of the excess concentrations and excess entropy at the interfaceo It is possible, however, to make some inferences as to the signs of the excess propertieso Equations (2-44) and (2-46) are given below: d/dP - -F' - ~ (' —- x — 5 ) (7-20) 1 -' X-i lxN

-136dy/dT = -S(S) + r. P s- l. S - 7b s'[o (7-21) i-WNi l- i - 11 - If one substitutes into these equations the molar volumes and entropies of the saturated phases at 3710C and the experimentally derived coefficients, one obtains two equations involving three interfacial quantities rN-i I7b and S(s) 35l2~l1O 0 y(593) = - rN 6o70 - rFb 19o5 (7-22) 90o4-lo-4 y(5935 = -s(S) +; 5o3010 8 + ( Ni^~ M<(7-23) f 9)44010 ergs/cm2-C No solution of these equations for the interfacial quantities is possible, but certain observations can be made. Rewriting Equation (7-22) one obtains Fr = 2.91 r - 4(6610-11 (593) ~ (7-24) One can now show that the surface excess of nickel must be negative, because the surface excess of lead cannot have a s~fficiently large negative value to make the righthand side of (7-24) negative. Let -6 the grain boundary be 10 cm thick, and suppose the solubility of lead in solid nickel is one atom percento Then, the smallest value of rF occurs when the grain boundary is void of leado The molar density of solid nickel is 0149 moles/cc, so Kmin 2 Pb = -.49 ~ 1 9 moles/cm A negative value larger than this would require either a thicker boundary region or a larger solubility of lead in nickel; neither event is likelyo

-137Inserting rin in (7-24) we see that as long as y(593)> 95 ergs/cm2, ri must be negativeo Since the lead atom is much larger than the nickel atom, one expects segregation of lead to the grain boundary, io.e., r 0P The positive segregation of lead implies a negative segregation of nickelo The above result, n < 0 confirms this. Ni If Equations (7-22) and (7-23) are combined to eliminate Ib one obtains: S(s) =-1.60~10 2 7(593) + 2o06~108 i ergs/cm2- ~C (7-25) Since Ni< 0, Equation (7-25) shows that the excess entropy of the Ni grain boundary is negative. This result requires some explanationo The entropy per mole for atoms in the grain boundary is undoubtedly higher than for atoms in grain interiorso But the excess entropy of the boundary is determined by a comparison of the entropies in equal volumes of material in the boundary and in the bulk phaseo Thus, the difference between the molar densities of the boundary and the bulk phase is also important. For a one-component system, a negative excess grain boundary entropy could only occur if the boundary region was considerably less dense than the grain interior. With two kinds of atoms present, quite different in size, a positive segregation of large atoms to the boundary implies a much greater negative segregation of small atomso The excess entropy of theinterace canthen decrease as segregation increases if the ratio of partial molar entropy to partial molar volume is smaller for the large atoms than for the small atomso

-138Some crude calculations can be made to justify the existence of a negative excess interfacial entropy for nickel grain boundaries with lead present. Assume that the solubility of lead in solid nickel is zero; then the excess entropy of the interface is: (s) = T (ni SN - nNi SNi) + T n"b Sp (7-26) where I i= T ( Ni-.) and P b np; n is the thickness of Ni Ni~b $Pb the boundary; n7i and npb the number of moles of nickel and lead, respectively, per unit volume of boundary; nNi the number of moles of nickel per unit volume of solid nickel; SI and S- the partial molar Ni Pb entropies of nickel and lead, respectively, in the grain boundary; and S.i the molar entropy of solid nickel. -Ni The value for S is knowno o ) Assume that Si is larger than S by the entropy of melting of nickel, and assume that St is the same as the molar entropy of liquid leado Thus, Pb 8 s = 5o30 o 10 ergs/mol-~C S2 - 6 30 ~ 10 ergs/mol-~C Ni ~- 108 StP = 9o44 ~ 10 ergs/mol-~C From lattice parameter measurements at room temperature(66) the ratio of atomic volumes of lead and nickel is 20740 The ratio of molar densities of nickel and lead at 371~C gives a value of 2091998) For purposes of calculation, assume Volume of Pb atom 28 Volume of Ni atom If the only factor which influences the number of atoms present in the grain boundary is the displacement of nickel atoms by lead atoms, then,

-13.9n~~ + 2o8 n n = i Ni ^^"Pb Ni If the mole fraction of lead atoms at the boundary is 0o5, then since nNi = 0149 moles/cc, one has: n.i = n = 0305925 moles/cc From these assumptions and Equation (7-26) one finds S(S)/ = - 17 o 107 ergs/cc-~C -7 so if T = 10 cm, then (s) _ lo7 ergs/cm2-c o (7-27) While this calculation indicates how a negative excess entropy can arise, the value of (7-27) is about 4 - 6 times smaller than Equation (7-25) would suggest. In view of the assumptions made to perform the calculation, this is, perhaps, as good agreement as can (s) be expected. Ss) would have a greater negative value if a larger fraction of lead atoms was assumed present in.the boundary or if the boundary was assumed to have a significant fraction of vacancies. However, the most critical values in the calculation (and the least certain) are the- magnitudes assumed for the partial molar entropies and volumes of atoms in the interfaceo 4o Analysis of Results - Tension and Compresssion The experimental data presented in the previous chapter show that applied tensile and compressive stresses change the shape of liquid lead inclusions in nickelo The results indicate further that the orientation of the lead particles with respect to the axis of tension (or compression) influences the magnitude of the changeo

-140If one could prove that the changes in the median angle observed were solely the result of changes in interfacial tensions with applied stress, the problem of obtaining a meaningful dihedral angle from this data would still be formidable. It would be necessary to calculate a mean difference in orientation with respect to the tension axis for each group of angleso It would also be necessary to compute a correction to the distribution function for observed angles; this would require an assumption as to how the dihedral angle changes as a function of orientation with respect to the axis of tension (or compression). It is not at all certain, however, that the changes observed in particle shapes are due solely to changes in interfacial tensions. To see this, let us consider in more detail the criteria of stability of a state of equilibrium which was discussed in section (3) of Chapter II, The sufficient stability condition of an equilibrium state of a system involving interfaces with respect to motion of the interfaces at constant total entropy and mass of the individual components is min - d i +'dAi 1 (7-28) (See section 3)9 Chapter II) That is, the sum of the products of the interfacial tensions and their areas, less the sum of the products of the pressures of the (2,4g5) phases and their volumes must be a minimum (2 43) For fluid phases, the pressure throughout the phase will be constant, hence SP dv. P.V. I I i 1 o

-141For solids the term Pi must stand for a strain energy density which may vary from point to point in the solid. The total amount of this energy in a solid phase is represented by pidvio Let us consider the system of a solid containing an included liquid phase, both phases being subjected to some hydrostatic pressure (see Figure 21(a))o The system is enclosed in a rigid envelope E, and has equal temperatures and chemical potentials in all parts of the system. The volumes of both phases are held constanto Thus the preliminary conditions are satisfied for the application of (7-28) aboveo Now any perturbation of the shape of the included liquid phase will change the amount of liquid/solid interfacial area, but will not produce any significant change in the pressures of the phaseso (Herring 3) points out that the change in particle shape must change.the distribution of stresses set up in the solid to compensate for the surface stress of the interface. He concludes that such changes will be negligibleo) Since neither the pressures nor the volumes of the bulk phases are affected by movement of -the interface, the stability condition (7-28) reduces to min l ydAl (7-29) where 7 is the liquid/solid interfacial tensiono If the solid is crystalline, y will in general vary over the interface, so the condition (7-29) would require a shape which would depend on the manner and magnitude of the variations in y7 If y is assumed constant over the interface, then condition (7-29) requires that the liquid phase assume a shape which has the minimum surface area for the volume contained, Thus, liquid phase has the form of a sphere in its state of stable equilibrium,

-142S rS L E LSL (a) SI E YSL S2 (b) Figure 21. Sketches of Thermodynamic Systems

-143Now consider the more complex system of Figure 21(b), with two identical solid phases separated by a grain boundary and an included liquid phase lying in the grain boundary. The system is under a uniform hydrostatic pressure, and has equal temperature and chemical potentials in all phaseso Let us now consider the changes which occur in the system when the interfaces are allowed to move and the volumes of all the phases are held constant. As before, the pressures in each phase are virtually unaffected, so that the terms Ipdv for each bulk phase are unchanged. Ience, as before, the stability condition (7-28) reduces to a simpler condition, mini S 7 YidAi (7-30) Now the stability condition (7-30) may be applied to the line of intersection of the grain boundary and the solid/liquid interfaces, and produces a condition on the shape of the liquid phase which must be satisfied in a condition of stable equilibrium. If ys1 and Ygb are assumed constant everywhere on their interfaces and are not influenced by the crystallography of the solid phases, then the condition is simply gb = 2Ysl cos 0/2 (7-51) where E is the dihedral angle The point to be emphasized here is that Equation (7-51) is only valid when the stability criterion (7-28) reduces to Equation (7-30). Now let us consider the system of Figure 21(a) when the phases are not under uniform hydrostatic pressure. Let us suppose that a uniaxial tensile stress is applied to the solid so that in the region of the solid

-144surrounding the liquid. particle stresses are set up which are not the same from point to pointo In particular, the state of stress on elements of the solid adjacent to the liquid phase will vary along the solid/liquid interface, and, most important of all, the particular distribution of stresses in the solid will depend on the shape of the liquid particleo The liquid will be under some uniform hydrostatic pressureo The temperature is constant throughout both phases. Let us assume that the chemical potentials of all components everywhere in the system have equilibrium values. The chemical potential of a component in the solid is influenced by the state of stress and this varies within the solido The only process which would produce a homogeneous state of stress, thus eliminating the gradients of chemical potential within the solid, is removal of the included liquid phase. Since this is not possible, the equilibrium conditions under the proposed constraints necessitate unequal chemical potentials in various parts of the solido This situation may be compared to a column of liquid in a gravitational fieldo Gibbs(299 p 146) shows that the chemical potentials of the components of the liquid vary from level to level in the liquid, when the whole system is in equilibriumo It is true, however, that the chemical potentials of all components must be the same everywhere in the liquid phase, and that one must have these same vaLues in the solid immediately adjacent to the liquido Now let the shape of the liquid phase be subjected to a perturbationo The area of the interface will change as before, thus the term'ydA in Equation (7-28) will changeo But as was pointed

out earlier. when the shape of the liquid phase is varied, the distribution of stresses in the solid near the liquid must also vary in order to preserve mechanical equilibriumo In general, the total energy of the solid is a function of the shape of the liquid phase, so the term pdv for the solid phase occurring in Equation (7-28) is not the same for all positions of the interfaceo Since the pressure on the liquid must be the same as the stress normal to the solid at the solid/liquid boundary, it is conceivable that the pressure on the liquid may also vary slightly with the position of the solid/liquid boundary. To summarize, the terms S pdv in Equation (7-28) which have been shown to be independent of the shape of the liquid particle under conditions of hydrostatic stress, are not independent of the liquid particle shape under a system of generalized stresseso Thus, the stability criterion (7-28) does not reduce to the criterion (7-29)0 It is apparent that a consideration of the more complex system of Figure 21(b) results in the same conclusion, ioe.o that the stable shape of the liquid phase depends not on minimization of the total interfacial energy of the system, but on minimization of the sum of bulk and interfacial energies. Since this is the case, Equation (7-31) is not a condition on the shape of the liquid phaseo Thus, the "dihedral angle" under these conditions is not determined solely by the ratio of grain boundary and solid/liquid tensionso This result forces one to conclude that the dihedral angle changes observed in tension and compression experiments are not necessarily due only to changes in interfacial tensions The sort of changes which are observed are consistent with the analysis aboveo

-146For example, the stress concentration about a lead inclusion must depend on the orientation of the inclusion relative to the axis of tension (or compression)o Hence, the observed dependence of the dihedral angle on this orientation is not surprisingo Secondly, the fact that the magnitudes of the observed dihedral angle changes are so much larger for a given stress in these experiments than in hydrostatic compression, suggests that some other factors may be important, To summarize, the foregoing discussion shows that the restricted condition for stability (7-50) is probably not applicable to solid-liquid systems in non-hydrostatic states of stresso Thus, because the energy contribution of the bulk phases depends on the particle shape, dihedral angles are not determined only by the relative magnitudes of solid/ liquid and solid/solid interfacial tensionso Without knowing the amount of extra strain energy arising from.stress concentrations about a particle and how this energy varies with particle shape, it is not possible to isolate any possible effect of applied stress on interfacial tensions. 5~ Summnary It is possible to relate the change in dihedral angles with pressure or temperature to changes in the interfacial tension at the solid/ solid and solid/liquid interfaces. The assumption required is that the surface tensions at each interface change linearly with temperature or pressureo The fact that the data can be correlated on this basis lends credibility to this assumptiono The properties of the solid/liquid interface can be obtained from the pressure and temperature coefficients of solid/liquid interfacial

-147tension. The surface excess concentration of nickel is determined using the assumption that the solid phase is pure nickelo The excess entropy of the interface is computed by either (a) assuming the solid is pure nickel, or (b) estimating the thermodynamic properties of the nickel/ lead liquid. Both methods yield approximately the same results. From the excess entropy and concentration, one can estimate a minimum thickness for the solid/liquid interfaceo This calculation suggests a rather diffuse interface. The thermodynamic relations for the solid/solid boundary cannot be reduced to the formal simplicity that is obtained for the solid/liquid boundaryo This arises because a dividing surface cannot be chosen in such a way as to make one of the surface excess concentrations vanisho Nevertheless, from the derived equations and the temperature and pressure coefficients of grain boundary tension, it is shown that the excess concentration of nickel at the interface is negative, the excess concentration of lead is positive, and the excess entropy is negative. Thermodynamic arguments lead to the conclusion that the dihedral angle in leaded nickel under general conditions of applied stress (not hydrostatic) cannot beinterpreted as a ratio of interfacial tensionso This happens because the strain energy of the bulk phases depends on the position of the interfaceso Since energy contributions from the bulk phases as well as the total interfacial energy change w!hen interfaces move, the dihedral angle changes observed in tension and compression testing represent the combined tendency of the system to minimize its total interfacial energy and the strain energy of its phaseso These conclusions, however, provide no simple basis for explaining the effects observed,

CHAPTER VIII GENERAL SUMMARY The dihedral angle of lead in nickel has been measured as a function of temperature and hydrostatic pressureo The dihedral angle changes from 520 at 371~C to 32~ at 816~Co When pressure is applied at 3710~C the dihedral angle changes from 52~ at zero pressure to 39~ at a pressure of 50,000 psio When pressure is applied at 593~C, the dihedral angle changes from 41~ at zero pressure to 32~ at a pressure of 52,600 psi. Using the assumption that the interfacial tensions which determine the dihedral angle change linearly with temperature and pressure, one may obtain values for the temperature and pressure coefficients of interfacial tension from the observed dihedral angle changeso The results of these calculations, expressed in terms of the nickel grain boundary tension at 593~C 1( (593~)) are-10 d.sl/dP = 1 60 (~0 65) 110 (5935) cm at 371~C dysl/dp = 0o81 (0o63) 0 10 ('593~0) cm at 593~C l/ 407 (~20203) 10" Y(5935) ~C/ a at 0 pressure d gb/dP = 3012 (~la22) o1-107(5930) cm at 371~C dygb/dp = 1635 (~tl32) 10"10 7(5930) cm at 593~C g 90(-0)14 o dynes/cm dgb/dT = 9004 ( 4005) o104 (5930) deC at 0 pressure. From the above coefficients and the thermodynamic relations which express the dependence of these coefficients on the properties of the interface and the properties of the bulk phases, one can calculate -148

-149the excess concentration of nickel at the solid/liquid interface and estimate the excess entropy of the interface, The properties of the grain boundary cannot be directly calculated by the same method The data imply a deficiency of nickel and segregation of lead at the grain boundary. A decrease in the dihedral angle was observed in leaded nickel subjected to uniaxial tension and compression. An explanation for these changes is not proposed for two reasons~ (1) the stress distribution about the liquid lead particles is not known, hence there is no way to determine the stresses at the particle in terms of the applied stresses, and (2) it is unlikely that the thermodynamic conditions which permit the dihedral angle to be expressed as a ratio of interfacial tensions are satisfiedo It has been established, however, that the changes seen are not due to cracks beginning at liquid lead inclusions,

APPENDIX A DERIVATION OF THEORETICAL DISTRIBUTION FUNCTION FOR OBSERVED ANGLES (39) The following derivation is due to Harker and Parker 9) If planes of all orientations are passed through the line of intersection of two surfaces in space meeting in an angle & the distribution of apparent angles j observed on the planes made by the traces of the intersecting surfaces, is dependent only upon o In order to obtain the required distribution function, it is first necessary to relate P and 6 geometrically, in terms of parameters expressing the inclination of the intersecting plane relative to the intersecting surfaceso Let surfaces A and B intersect in space. At a point q on their line of intersection, let the angle between the normals to each surface be it - 0, where' is the dihedral angle. Then define the following vectors, shown in Figure 22~ A: Unit normal to surface A at qo A in the xy planeo B: Unit normal to surface B at q. B in the ky plane. N~ Unit normal to intersecting plane P at q, m: Unit tangent vector to the line of intersection of A and P, perpendicular to N and Ao no Unit tangent vector to the line of intersection of B and P, perpendicular to N and Bo 0: Angle between m and n (the "observed angle" ) Then in terms of the dLhedral angle 0 and the spherical coordinates a, 3 we have: -150

-151x N n ~ m ^/// \ / \ / / Figure 22. Vectors Used in Deriving Relation Between the Observed Angle, the Dihedral Angle and Orientation Parameters.

-152-sin /2 -sin &/2 sin a cos 1 A cos */2; B = -cos /2; N= sin c sin o 0 0i cos a If. is the angle between N and A and d the angle between N and B, the vectors m and n are given by A x N = sin m; N x B sin n n where 1 and 5 may be evaluated from the expressions N~A - cos' and N~B = cos ~o Then since mon = cos 9 and n x m = N sin 9, one can write the vector equation i x m N tan ( -m -n Evaulation of the third component of this equation gives after simplification tan 2 sin cos a (A-l) tan 2 (A-) sin a (cos 23 - cos e ) + 2 cos 9 which is the result given by Harker and Parker. The probability of obtaining values V to t + wA may be obtained by considering all possible orientations of an element of grain edge of length A\. The probability that 0 will have a particular range of orientation parameters is given by the elemental area on the unit sphere, shown in Figure 230 A = sin a 5b 5P But the probability of intersection is dependent on the projected length of L Q on the normal to the intersecting plane. Taking this normal along the z axis, the projected length is L ~ cos a o Thus, the net probability of seeing a particular orientation is

-153Ia ~~/ - ~ma s3 ELEMENT OF AREA ON UNIT SPHERE Ic ACosa - PROJECTED LENGTH ON NORMAL Figure 23. Sketch Showing Relationships Used to Obtain the Probability of Observing a Given Angle. 0.0 Sin2a X I.0 0 7r/2 Figure 24. Probability Plot After Harker and Parker.(39) For a Given Dihedral Angle 9o, the Ratio of the Crosshatched Area to the Total Area of the Diagram is the Fraction of Observed Angles Having a Magnitude Less Than or Equal to Y o. The Curved Line Bordering the Crosshatched Region is Obtained from Equation (A-l) With 9= To and W =-Wo

-154a cos ac sin ac' 5c C L 5 sin2 c 5 o (A-2) The element of edge length,,. may be disregarded when integrating the above expression because the distribution of angles seen will not usually be influenced by the length of grain edge. (See error discussion for an amplification of this point.) From Equations (A-l) and (A-2) it follows that the relative chance of intersecting surfaces, meeting in an angle,9 producing an observed angle less than or equal to /), may be obtained from a plot of sin a vs V, as in.Figure 24o From Figure 24, the probability of observing an angle of magnitude ~o or less is given by the ratio of the crosshatched area to the total area of the plot. Equation (A-l) for specified values of / and e gives the curve? = oO Analytically, the probability represented by the crosshatched area is given by the cumulative distribution function" F(t/;2) =, \ 1/2 Arc cos (2 (in - c- os&) + cos ) dZ + C 2 where Z = sin ct, and when (=3 t K =0, C 0 QK > J= 2 (cos P - cos G ), C 0 (1 - cos )(l + cosj)) 2o~) 2 < K =2 (cos - cos f ), c = (1 + cos eT)(- - cos 2)

-155Equation (A-3) may be integrated numerically using Simpson s rule. The following properties of these curves are helpful for this integration: (1) when sin a = 10, = 2 for all / (2) when sin2 a = 0o0, = 0 for I < G = t/4 for 1 _= 6 =t/2 for y > o Numerical integration of Equation (A-3) was carried out using an IBM 7090 computers producing the family of cumulative distribution curves shown in Figure 25.

100 Z8o 0 TRUE DIHEDRAL ANGLE ii 60 5 0 150o 2 50 3 5~/ 4 5~/ 5 50/ 6 50 750/ 8 50~/ 950/ 1050/ 150 /1250/ 135~/ 1450 550 1650 175 4 60 20 40 60 80 100 120 140 160 I18 OBSERVED ANGLE-DEGREES Figure 25. Theoretical Cumulative Frequency Curves for Various Dihedral Angles.

APPENDIX B ESTIMATION OF THE NICKEL GRAIN BOUNDARY TENSION IN TEE PRESENCE OF LEAD (41) Hayward and Greenough( have determined the surface tension of solid nickel, using the method of Udin, Shaler and Wulffo( ) Since they determined the grain boundary groove angles in the course of their experiments, their measurements also yield values for the grain boundary tension of pure nickel. Their measurements covered the range from 1250~ to 1435~C yielding a value for solid surface tension at the melting point of 1725 ergs/cm2, a value for the change in surface tension with temperature in this range d = - 06 ergs/cm2-~C dT and a grain boundary groove angle of 158~, constant over the temperature range within the accuracy of its measurement. Their results imply that the grain boundary tension increases with decreasing temperature. The temperature coefficient of nickel grain boundary tension in the presence of lead calculated in Chapter VII is positive (Equation 7-7)o This is to be expected when a marked segregation of lead occurs at the grain boundary. The nickel grain boundary tension with lead adsorbed must be lower at all temperatures than the pure nickel grain boundary tension measured by Hayward and Greenougho It seems likely, that the calculated positive temperature coefficient holds up to the monotectic temperature (1340~C)o From the data of Hayward and Greenough the grain boundary tension of pure -157

-158nickel can be computed at this temperature. Using this for the value of the grain boundary tension of leaded nickel, and using the calculated temperature coefficient, one dan find the grain boundary tension at any other temperature (particularly at 593~C0 the reference temperature used in Chapter VII)by means of the relation: 7gb(1340~) = gb(593~) + 9O04104 7gb(593 )(1340-593)- (B-l) This procedure should produce estimates on the high side, because the grain boundary tension of nickel with lead present will be lower than the tension of pure nickel. It is difficult to estimate the decrease to be expected, so this correction will not be madeo (46) The data of Hilliard, Cohen and Averbach' on the Cu-Au system, 2 suggest a decrease on the order of 50-200 ergs/cm might be anticipatedo Taking the surface tension of nickel at 1340~C to be 1794 ergs/cm2, the groove angle to be 158~0 the grain boundary tension at 1340~C iso Ygb = 2s cos 790 = 685 ergs/cm = gb(1340) (B-2) Inserting the value from (B-2) in Equation (B-l) one has: 7gb (3400) = L68 Ygb(593 ) So the grain boundary tension at 593~C is probably no higher than Ygb(593~) = 409 ergs/cm2. (B-) Ygb~~~~~~~~~~~~~(59-3)

APPENDIX C CORRELATION OF DIHEDRAL ANGLE VS TEMPERATURE DATA FROM THE LITERATURE It has been proposed in Chapter VII that the change in dihedral angle with temperature can be satisfactorily explained by assuming that the interfacial tensions determining the dihedral angle are linear functions of temperature. This implies that a plot of (cos /2 cos e~/2)"1 vs (T - T~)1 is a straight line, where 9~ is the dihedral angle at a reference temperature T~o For convenience a graph of the latter functions will be referred to as an "R-ploto" Changes in the dihedral angle as a function of temperature have been reported for only a few binary systems. Most of this data is not sufficiently accurate (or is taken over too narrow a temperature range) to provide a satisfactory test of the proposed correlationo From the data that are available, the correlation seems valid for systems where the liquid composition changes rather uniformly (though not necessarily linearly) with temperature.When drastic changes in liquid composition occur over a narrow temperature range, the tensions seem to vary non-linearly with temperature. The Cu-Pb System Ikeuye and Smith(51) measured the change in dihedral angle with temperature for this system. Their dihedral angles were determined to the nearest 5~~ Figure 26 is the phase diagram for the Cu-Pb system, and Figure 27 shows how.the dihedral angle changes with temperature. From 3500 to 800~C the dihedral angle is constant with temperature. This implies that either dy/dT is zero for each interface over this -159

-1601200 1100 1083 1000 954~ 900 800 0 Q \ w 7 00 I< 600 aw 500 400 1 3260 327~ 3 00 200 0 10 20 30 40 50 60 70 80 90 100 Cu Pb ATOMIC % LEAD Figure 26. The Copper-Lead Phase Diagram. Hansen. 37) 70 I i I I 50 0 - \ i9 \ w 40.J z 30 w 20 IO 10 400 500 600 700 800 900 TEMPERATURE ~C Figure 27. Dihedral Angle vs. Temperatyre Copper-Lead System. Ikeuye and Smith. 51)

-161range or that (dygb/dT) = K (dy l/dT) where K is a constant chosen so that the ratio Yb/Ys does not vary. Above 800~C, the composition of the liquid is changing rapidly with temperature, and the above relationship does not hold. These data indicate that the surface tensions do not vary linearly with temperature from 350 - 950~C. although the linear dependence may be true from 350 - 800~Co The Al-Sn System Figures 28, 29. and 30 show the results for this system, also from the data of Ikeuye and Smith5) The liquid here changes composition rapidly between 550~ and 660~Co The R-plot (Figure 30) shows that the data points at 600~ and 620~C do not fall on the straight line which satisfactorily correlates the rest of the datao The Zn-Sn System The phase diagram for this system is given in Figure 31 (70) and Miller and Williams7 data( is'shown in Figure 32o The R-plot (Figure 33) shows that all this data fits a straight line. There is a close similarity between the phase diagram for this system and the Al-Sn systemo Note that the dihedral angle in the Zn-Sn system falls to zero at about 3355~C with a composition of 50 atomic % tin in the liquid. The most rapid change in liquid solubility occurs above this temperature. This is in contrast to the A1-Sn system where the dihedral angle data cover the region of rapid composition changeso This probably is the reason why all the data in the-Zn-Sn system are linear on an R-plot, while the data from the Al-Sn system are not.

-162700 660 600 - 500 0: 400 300 230 0____ _228.3 232' 232~ 200 100 0 10 20 30 40 50 60 70 80 90 100 Al Sn ATOMIC %/ TIN.(37) Figure 28. Aluminum-Tin Phase Diagram. Hansen. 6050 40 \ Figure 29. Dihedral Angle 0s. Temperature. Aluminum-Tin System. "J 1~~~~0 ~ z \ < 30 o2- \3 - = \ 300 400 500 600 700 TEMPERATURE C Figure 29. Dihedral Angle vs. Temperature. Aluminum-Tin System. Ikeuye and Smith. (51)

-16350 _ / / / / 30 - / / / / 30 / / Cos~/2-Cos0o2 / / / 20 / / / 0 / 0 0.005 0.010 I/AT (~C) - Figure 30. R-plot for Aluminum-Tin System. Reference Point: 55~ Dihedral Angle at 350~C.

-164500. 419.4~ 400 0 200: 300 -: 200,,198~ 2:32~.,,, w I I- 100 0 10 20 30 40 50 60 70 80 90 100 Zn Sn ATOMIC % TIN Figure 31. Zinc-Tin Phase Diagram. Hansen.(37 50 i j 40 z j 30 w 20 0 10 0 \ 1 200 250 300 350 TEMPERATURE ~C Figure 32. Dihedral Angle vs. Temperature. Zinc-Tin System. Miller and Williams. 70)

-16590 80,- / / 70 / x/ 60 I__________~/ / 40 / CosO/2 —Cos022 40 / / 20 /,0- / 0I -_ 10 0 0 0.010 0.020 0.030 0.040 T- T Figure 33. R-Plot for Zinc-Tin System. Reference Point: 50.9~ Dihedral Angle at 2000C.

-166The Sn^-Te System The phase diagram for this system is shown in Figure 345 Riegger5) has measured the dihedral angle made by the Sn-rich liquid and the Te-rich liquid -with the Sn-Te intermetallic compoundo The scatter in these data (Figure 535) is too large for a good test of this correlation. However, from Figure 36 it appears that the data from the Te-rich liquid obey the proposed relation, and the data from the Sn-rich liquid do not. The phase diagram shows that the Sn-rich liquid undergoes a large change in composition between 650~ and 790~C, while the Te-rich liquid changes composition uniformly from 405~ to 790~Co Both curves in the R-plot are drawn using a reference of 0~ dihedral angle at 790~Co This fictitious datum is used because of the four data points for the Te-rich liquid, one appears to be bad. Use of one of the good data points as reference would leave only two good points for the R-plot. The Al-Sb System Riegger5) has measured the dihedral angle of the Al-rich liquid and the Sb-rich liquid in equilibrium with the Al-Sb intermetallic compound (Figures 37 and 38). In this system too, the data from the solution which changes composition rather regularly with temperature (Sb-rich liquid) give a straight line on the R-plot (Figure 39). The Al-rich liquid, which changes composition less regularly, definitely does not obey the proposed linear relationshipo

-167900 800 700 600 0 500/ w 4050 300 300 ~- __ 2320 2 00 - I 00 0 10 20 30 40 50 60 70 80 90 100 Sn Te ATOMIC % TELLURIUM (37) Figure 34. Tin-Tellurium Phase Diagram. Hansen.(37) eo 0 -----—. I. I I.. 0 0 OO\ 70 560- \0 o 5 \ z s \ < 40 - J \ a 3 0 1w 2 0 0 O Sn-RICH LIQUID X Te -RICH LIQUID I O 10 300 400 500 600 700 800 TEMPERATURE ~C Figure 35. Dihedral Angle vs. Temperature. Tin-Tellurium System. Riegger.(85)

-168I/AT (C )-' -0.010 -0.005 0 o Sn-RICH LIQUID -2 X Te-RICH LIQUID -4 ^'~ i —6,^'XxX -1 6.X -8 j /e/ / — 10 X/ j,11/ Ixo -12 x / /-14 / 1 10/ -16 x / -18 -20 Figure 36. R-Plot for Tin-Tellurium System. Reference Point for Both Liquids: 00 Dihedral Angle at 7900C.

-1691200 I 00 - 1065 1000 0 U 900 D / \ n 800/ \ w 700 6570 630.50~ 600 500 0- 10 20 30 40 50 60 70 80 90 100 Al Sb ATOMIC % ANTIMONY F'igure 37. Aluminum-Antimony Phase Diagram. Hansen.(37) oo 70'- 0 Al - RICH LIQUID 7 0 X Sb-RICH LIQUID 50 _ N_ \ \ a 30 w X 20 10 700 800 900 1000 1100 TE M PERATURE 0C Figure 38. Dihedral Angle vs. Temperature. Aluminum-Antimony System. 5LI.I ~Rige (85) Riegger.85

-17050 0 AL- RICH LIQUID X Sb- RICH LIQUID 40 / / / / / COS/2 -CosO/2 // 20 -'/ / / /, /.x / / X/ / 0 0 0.005 0.010 I/AT (OC)-I Figure 39. R-Plot for Aluminum-Antimony System. Reference for Al-rich liquid: 78~ Dihedral Angle at 700~C. Reference for Sb-rich liquid: 56~ Dihedral Angle at 700~C. Starred Points Represent a 0~ Dihedral Angle at 1065~C, the Critical Temperature for the Intermetallic Compound.

-171Comment All of these systems involve the equilibrium of an essentially pure solid with a liquid of varying composition. For this sort of system, equating the differentials of chemical potential for the solid component in both phases gives the relation: -A s ~ l-dx R s z d (C-l) S -S 1 1 dx dT where S is the molar entropy of the solid, S is the partial molal -l 1 entropy of the solid component in the liquid, and x is the mole fraction of the solid component in the liquid. For a solution in which the activity coefficient is independent of composition = R-T (c-2) dX x and Equation (C-1) becomes: S s d In x (C-3) 1 - S =- Rd ln T (C Plots of ln x vs in T are linear for most systems which show a regular change in liquid solubility with increasing temperature. The molar entropy of the solid, S, changes quite uniformly up to the melting point of the solid. A radical change in liquid composition (which changes dn x ) must then be due to a change in the partial (whioh ch a nge s d ln- T molal entropy of the solid component in the solutiono A sudden increase in the solubility of the liquid with increasing temperature indicates a disproportionate increase in S e This may be due to a "randomization" of a solution which was in some sense "orderedo" For example, Strauss, et al(97 99) suggested that dilute solutions of Cu and Ni in lead form an "interstitial-like" solution.

-1T2The sudden increase in liquid solubility below the monotectic temperature in these systems may indicate reversion to a "substitional-type" solutiono A change in liquid solubility will, to some extent, affect the excess concentrations and entropy at the solid/liquid interface, simply because these interfacial quantities are defined with respect to the properties of the bulk phases. But a very marked increase in solubility, may reflect some intrinsic change in the atoms making up the solution; such a change would also be expected to affect interfacial compositionso In many cases, restricted liquid solubility is blamed on a large size difference between solute and solvent atomso A large size difference probably promotes segregation at interfaceso'3 ) If a sudden increase in solubility with temperature is observed in such a systemit may be due to a decrease in the size difference between the atoms present This will also cause a decrease in the segregation expected at the interfaces, and may explain why abnormal changesin the dihedral angle are associated with abnormal solubility changeso In most of the systems for which dihedral angle data are presented,. the size difference between solute and solvent atoms is more than 15% (based on Goldschmidt atomic diameters 19)o Only in the Sn-Te system are the atomic radii of solute and solvent nearly the sameo

APPENDIX D PROCEDURE FOR FITTING WEIGHTED LEAST SQUARES LINE TO DATA One can assume that the variance of the measured dihedral angle is independent of the magnitude of the angle. However, the variance of the function Z =(cos /2 - cos e~/2)which is plotted in Figures 18 - 20 depends on the magnitude of ZO One can calculate the variance of Z, a (z), in terms of the variances of ~ and (o (which are equal) using the relation, 2,. )Z 2 2,^N /Sz \ 2 2 2 (z) + 2 a ) o Thus one has 2( (sin2 /2 + sin2 e~/2) (e) CY(z (cos e/2 - cos eo/2 4which shows that the variance of Z is proportional to Z o Weighted least square lines are fitted to the data using a weighting function W. given by W. = 1/2(zi) o The procedure used is given in reference (68). The variance of the dihedral angle was taken to be 1713 10-4 (radians), which corresponds to a standard deviation of 0o75~o From this fitting procedure, one obtains the slope and intercept of a "best straight line" and the estimated variances of the slope and intercept The slope and intercept are then used to calculate the quantities dy,/d7 and dygb/dX using the Equations (7-6). -173

-174The variances of these quantities are given by the following equations; 2 [, ( )2 Cr ) r + (m) + ( 0) O Sl () 11b2(b) + 22 4 cot2 ~/2 2 d2 L b 2(b) + (sec 2/ (see 2 + b)2 2(m) + ~2(o~) / 2 2 4 cos2 0~/2 cot2 ~/2j where a (b) is the variance of the intercept, a (m) is the variance of the slope, and the other quantities are defined by Equation (7-6). 2 2 The square roots of (y7? ) and a (y7 b) give the standard si gb deviations of these coefficientso The calculated standard deviations are 1/4 to 1/2 the magnitudes of the pressure and temperature coefficients, thus the latter are rather unreliable.The blame for this uncertain.ty lies not so much with the magnitude of the variance taken for the dihedral angle (which is a reasonable value), but rather with the high magnification of error brought about by the calculations. Since the relative standard deviations of the data points in Figures 18 to 20 are roughly portional to the square of the magnitude of the point on the Z-axis, deviations from the fitted straight line for points with high values of Z are relatively unimportanto

APPENDIX E ESTIMATION OF THERMODYNAMIC PROPERTIES OF NICKEL IN LIQUID LEAD J, L. White(19) has fitted the solubility data for the Ni-Pb (38) system to a'sub-regular" solution model, proposed by Hardy. Using -Ni~ dx this model, one can obtain estimates for Si and — ) TP to use in calculating the excess entropy of the solid/liquid interface (Chapter VII). The two parameters obtained from the solubility data for the Ni-Pb system are: AL Hi() = 3050 + 600 cal/mol ~s oo (E-1) AS.(B = -2.50 + 0.6 cal/mol-~C ) The first of these terms is the difference in partial molal enthalpy between Ni in solution and pure liquid Ni at the temperature of interest, at infinite dilutiono The second term is a similar entropy term, except that it is an excess quantity determined relative to an ideal solution. These quantities serve as parameters for determining the partial molal enthalpy and excess partial molal entropy at any solute concentration, according to the relations: AH{i(}) = XiN( b ) " (E(E-2) AiXs A'^00h (1-\X 0 )2 Ni(l) Ni( (1 Ni() where xNi( ) is the mole fraction of Ni in liquid leado The difference in partial molal Gibbs free energy between Ni in solution and in the standard state is given by: -175

-176nc~ai-~ N - TXS a NiO ) G (J * - i - ^i(O)' ^i( ) + RT In xi() Now since GNi(.) = Ni, and G'i is independent of x, then aNi Ni2), )OO RT + o XN( ) Dropping the subscripts on x, multiplying through by dx/dT and rearranging produces the relation~ (E-3) N dTx - TAS l d(l - x)2 d n x (E5) Ni T9P L i() - NiC)J( dT dT In order to evaluate this term for the saturated solution, plots of (l-x)~ vs T and in x vs T must be prepared, and slopes determined. The latter plot, using the available solubility data 2Y79) produces a reasonably good straight line in the temperature range of interesto The former does not; however, in the region from 3710C to 593~C the slope does not change too mucho The slopes to these plots MNi dx and the calculated values of at 571~C and 595~C are given belowO dT- = - 7.96 o 10- 5 (~C)-I at 710C dT = 4 o (c) d 1n x 4.61 0 10 7 (oc)Kl

-177mP) dT = 5.54 cal/mol-~C = 2.32. 10 ergs/mol-~C at 371~C (EBi4)' I)JT d-T = 7.17 cal/mol-~C = 5.00 10 ergs/mol-~C at 593~C The difference between the partial molal entropy of Ni in solution in the standard state is given by: ^Ni S.( ) =- SNi( ) - Rln Ni( ~) = SNi() - Ni Thus, Ni(s) = SNi A SNi() -R in Ni() (E-5'The standard state is pure liquid nickel at the temperature of interest. The entropy in the standard state may be obtained by extrapolation from a given value of the entropy in the liquid above the melting point, using the heat capacity of liquid nickel. From reference (101), the entropy of liquid Ni at 1800~K is 25357 cal/mol-~C, and the heat capacity is 9.2 cal/mol-~Co Thus, Si = 13591 cal/mol-~C at 571~C (E-6) S0i = 16.64 cal/mol-~C at 593~C. So from Equations (E-1), (E-2)^ (E-5), and (E-6) the partial molal entropy of Ni in liquid lead in the saturated solution is: SNi = 22.05 cal/mol-~C at 371~C (E-T) Si = 22.40 cal/mol-~C at 593~C. The values obtained in Equations (E-4) and (E-7) above are used in Equation (7-15).

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-18143~ Co Herring, "The Use of Classical Macroscopic Concepts in Surface Energy Problems," Structure and Properties of Solid Surfaces, Gomer and Smith (Edso) Univ. of Chicago Press, (1955)o 440 J. Bo Hess, "Measurement of Solid-Solid Interfacial Energies," Metal Interfaces, ASM Publication, Cleveland, Ohio, (1952). 45. Jo Eo Hilliard and Jo Wo Cahn, "On the Nature of the Interface Between a Solid Metal and Its Melt,' Acta Met,, 6, 772, (1958)o 46. J. Eo Hilliard, Mo Cohen and B. L. Averback, "Grain Boundary Energies in Copper-Gold Alloys," Acta Met,, 8, 26, (1960)o 47. Co R. Hocott, "Interfacial Tension Between Water and Oil Under Reservoir Conditions," Trans. AIME, 152, 184, (1959)o 48. Eo Wo Hough, Mo Jo Rzasa and B. Bo Wood, Jr., "Interfacial Tensions at Reservoir Pressures and Temperatures; Apparatus and the HO2-CH4 System," Trans AIVE, 192, 57, (1951)o 49. Eo Wo Hough, B, B. Wood, Jro and Mo Jo Rzasa, "Adsorption at H20-He, -CH4, -N2 Interfaces at Pressures to 15,000 psia," Jo Phys. COem_, 56, 996, (1952)o 50. Eo Wo Hough, Go Jo Heuer and Jo W. Walker, "An Improved Pendant Drop, Interfacial Tension Apparatus and Data for Carbon Dioxide and Water," Trans AIME, 216, 469, (1959)o 51o Ko Ko Ikeuye and Co So Smith, "Studies of Interfacial Energies in Some Aluminum and Copper Alloys," Trans AIME, 185, 762, (1949)o 52~ Ro.E, Johnson, "Conflicts between Gibbsian Thermodynamics and Recent Treatments of Interfacial Energies in Solid-Liquid-Vapor Systems," J. Phys. Chem., 63, 1655, (1959), 53. D. T. Jones, "Determination of Surface Tension and Specific Gravity of Crude Oil under Reservoir Conditions," Trans o AIME 118J 81, (19536 ). 54. Go.L Kehl, Principles of Metallographic Laboratory Practice, McGraw-Hill Book Co., New York, (1949)o 55. W. Do Kingery and Mo Humenik, Jr., "Surface Tension at Elevated Temperatureso Io Furnace and Method for Use of the Sessile Drop Method; Surface Tension of Silicon, Iron and Nickel," Jo Phys. Chem., 57_, 359, (1953)o 56. W. D, Kingery, "Densification During Sintering in the Presence of a Liquid Phase. I. Theory," Jo Appl. Physo, 30, 301, (1959). 57. Wo Do.Kingery, "Densification During.Sintering in the Presence of a Liquid Phaseo IIo Experimental," JO Applo Phys., 50, 507, (1959)0

-18258. W. Do Kingery "'Sintering in the Presence of a Liquid Phase," Kinetics of High Temperature Processes, Conference on the Kinetics of High Temperature Processes, MIT, 1958, The Technology Press of MIT and John Wiley and Sons, Inc., New York, (1959). 59. Do Ao Kraai, S. Floreen, C. Ao Stickels, Do V. Ragone and Eo Eo Hucke, "The Effect of Surface Tension of a Liquid Metal Environment on the Fracture Strength of Solid Metals," AFOSR TR-60-116, Contract No, AF 49(638)-422, (July 1960)o 60o Ao Kundt,'Concerning the influence of pressure on the surface tension at the common surface of separation of liquids and gases, and on the relation of this influence to the Cagniard de la Tour State of the Liquid," Annale'n der Physik und Chemie, 12, 538, (1881)o 61o Go N. Lewis and M. Randall, Thermodynamics, McGraw-Hill Book Co., New York, (19235) 62~ Go No Lewis, Mo Randall, Ko S. Pitzer and Lo Brewer, Thermodynamicss 2nd Edo McGraw-Hill Book Coo New York, (1961)o 635 C. JO Lynde, "The Effect of Pressure on. Surface Tension," Phys. Rev., 22, 181, (1906)o 640 Ko Ao McCarthy and B Chalmers, "Energies of Grain Boundaries in Silver Chloride Crystals9, Canad, J. Phys,, 6, 1645, (1958)o 65. Do McLean. Grain Boundaries in Metals, Oxford Univ. Press, London, (19i57T. 660 ___3______ Metals Handbook Volo 19 Properties and Selection of Metals, 8th Edo, ASM Publication, Metals Park, Novelty, Ohio, (1.961 )o 670 Ao S. Michaels and E, A.o Hauser, "Interfacial Tension at Elevated Pressure and Temperature I., Interfacial Properties of Hydrocarbon-.Water Systems9, Jo Phys. Colloid Chem,,,q 408, (1951). 680 Ho So Mickley, To Ko Sherwood and Co E. Reed, Applied Mathematics in Chemical Engineering, McGraw-Hill Book ~o. New York, (1957)69, Ko 0o Miller and Jo F. Elliott, "Phase Relationships in the Systems Fe-Pb-Ni, Fe-Ni-C (sato) and Fe-Pb-Ni-C; 1500~ to 1500~GC, Trans AIME, 218, 900, (1960)o 70, Wo Ao Miller and.W. Mo Williams, "The Microstructure of Zinc-Tin Alloys," (to be published)o 71. W. Wo Mullins, "Theory of Thermal Grooving," Jo Appl. Phys., 28, 333, (1957).

-18372, Ho Mykura, "The Variation of the Surface Tension of Nickel with Crystallographic Orientation," Acta Met, 9, 570, (1961)o 735 Ho Nichols and WO Rostoker, "On the Effective Surface Energy Involved in the Brittle Fracture of 70/30 Brass," Acta Met., 8, 788, (1960). 74o Ho Nichols and Wo Rostoker, "Ductile-Brittle Transition in a-Brass," Acta Met, o, 8 848, (1960). 75 Ho Nichols and Wo Rostoker, "On the Mechanism of Crack Initiation in Embrittlement by Liquid Metals," Acta Met., 9, 504, (1961)o 76~ K. Mo Olsen, Co Fo Larkin and Po Ho Schmitt, Jr,, "Embrittlement of High Purity Nickel," ASM Trans, 53, 349, (1961)o 77o Eo Orowan, "Energy Criteria of Fracture," Welding Jo, 34, 1575, (1955). 78. Ko Pearson (Edo ), The Tables of the Incomplete Beta Function, Cambridge Univo Press, London, (1934+) 79o Eo Pelzel, "Solubility of Copper, Nickel and Cobalt in Molten Lead," Metall, 9, 692, (1955)o 80, No J, Petch, "The Lowering of Fracture-Stress due to Surface Adsorption," London Philo Mag,, 1, 531, (1956)o 81o Wo To Read and W, Shockley, "Dislocation Models of Crystal Grain Boundaries," PhySo Rev,, 78, 275, (1950)o 82, Jo Rice, "The Influence of Surfaces of Discontinuity upon Equilibrium of Heterogeneous Masses, Theory of Capillarity," A Commentary on the Scientific Writings of J, Willard Gibbs, Vol. I, F. Go Donnan.and Ao Haas, (Edso). Yale Univ. Press, New Haven, Conn,, (1936)o 835 0o K. Rice, "The Effect of Pressure on Surface Tension," Jo Chemo Physo, 15, 5355 (1947)" 84. O0 Ko Riegger and L. HB VanVlack, "Dihedral Angle Measurement," Transo AIM, 218, 9553 (1960). 850 O0 K. Riegger, "The Role of the Solid-Liquid Interphase Boundary in Microstructures," PhoDo Thesisp University of Michigan, Ann Arbor, Michigan, (1962), 86, W. Mo Robertson and P. G. SheWmon, "Variation of Surface Tension with Surface Orientation in Copper, " Trans AIME, 224, 804, (1962), 87o Wo Rostoker, J.'No McCaughey and H. Markus, Embrittlement by Liquid Metals, Reinhold. Publo Co,, New York, (1960)o

-184880, Russian Papers on effect of pressure on surface tensiono M. Mo Kusakov, Po Ao Rehbinder, and K, E.Zinchenko, "Surface Phenomena in Petroleum-Filtration Processes," Compt rendo acad. sci. UoR.S.S., 28, 43355 (1940) Mo Mo Kusakov, N. Mo Lubman and -Ao YUo Koshevnik, "An Apparatus for the determination of interfacial tension of oils and of contact angles of wetting (of the racks) under the conditions in beds," Neftyanoe Khoz., 32, (2), 27; (5), 20, (1954 ) Po Eo Bol shakov, "Ability of Solid Substances to be wet by liquids in the presence of compressed gases," Trudy Gosudarst Nauch, -- Issledovatel, Inst. Azoto Prom, 1952, (1), 43; Referato Zhuro FiZo 1955, (1.3720). Mo Ao German, V B, Schneerson and Ao G. Mamikonov, "Effects of Pressure on the Wettability of Oil-Bearing Minerals (Basins)," Investo Akado Nauk. SooSoRo., Otdel, Tekh, Nauk., 1955, 127o Po Eo Bol.shakov and Go T. Levchenko, "Surface Tension at High Pressures at the Liquid-Gas Boundary," Trudy Gosudarsto Nauch. — Issledo-vatelo i Proekto Insto Azot; Promo 1952, (1), 30; Referat. Zhuro FiZo 1955 (15719) M. M. Kusakov and No Do Tairov9, "The Wetting of Quartz by Liquid Hydrocarbons and Water at High Pressures and Temperatures," Doklady Akado Nauk AzerbaYldzhano SoSoRo, 15, 1019, (1959). Do So Tsiklis and YUo No Vasiliev, "Surface Tension at the Interface Between Two Gaseous Phases at High Pressure," Doklady Akad. Nauk S.So.S.R, 136, 394Y (1961)o Yu, V. Efremov and Io F. Golubev, "YSurface Tension at the LiquidGas Interface Under High Pressure," Zho Fiz, Khimo, 56, 1222, (1962)o 89. Ro Ao Sack, "Extension of Griffith's Theory of Rupture to Three Dimensions," Proc. Physo Soc. (London), 58, 729, (1946)o 90O R Shuttleworth, "The Surface Tension of Solids " Proc.Phys. Soco,? A63, 444, (1950)o 91. o So Siegel, Nonparametric Statistics for the Behavioral Sciences, McGraw-Hill Book Coo New York, (L956)o 92. E. Jo Slowinski, Jr,. Eo.Eo Gates and C. Eo Waring, "The Effect of' Pressure on the Surface Tension of Liquids," J. Phys o Chemo, 61, 808, (1957). 935 C.So. Smith, "Grains, Phases, and.Interfaces An Interpretation of Microstructure," Trans AIE_, 1759 15, (1948)o 94. Co S, Smith, "Interphase Interfaces," Inperfectionsin Nearly Perfect Crystals, John Wiley and Sons, tnCo, New York, (1952)0 95. Co S. Smith, "Microstructure," ASM Trans,.45, 533 (1953). 96, Co Jo Smithells, Metals Reference Book, Volo I and II, Butterworth Scientific Publo, London, (.955)o

-18597. S. W. Strauss, J. Lo White and Bo Fo Brown, "The Atomic Size Effect and Alloying Behavior in Liquid Metals,'" Acta Met,, 6, 604, (1958). 980 So W. Strauss, L..Eo Richards and Bo Fo Brown, "The Density of Liquid Lead and of Dilute Solutions of Nickel in Lead," Nuclear Scio Engo, 7, 442, (1960), 99, S. W. Strauss and L. E. Richards, "The Density of Dilute Solutions of Copper in Liquid Lead," Nuclear Scio Eng., _9, 281, (1961)o 100. A, N. Stro'h "A Theory of the Fracture of Metals," Advances in Physics, 6, 418, (1957)o 101. Do Ro Stull and Go C, Sinke, Thermodynamic Properties of The Elements, No. 18, Advances in Chemistry Series, Amero Chemo Soc., Washington, DoCo, (1956)o 102. Co Ao Swartz, "The Variation in the Surface Tension of Gas-Saturated Petroleum with Pressure of Saturation," Physics, 1, 245, (1931)o 103o J. W. Taylor, "An Evaluation of Interface Energies in Metallic Systems," Jo Insto of Metals, 86, 456, (1958)o 104o Ho Udin, Ao J. Shaler and J. Wulff,'"The Surface Tension of Solid Copper," Trans AIME, 185, 186, (1949). 105. Ho Udin, "Measurement of Solid-Gas and Solid-Liquid Interfacial Energies," Metal Interfaces,'ASM Publication, Cleveland, Ohio, 114, (1952). 106. Lo Ho Van Vlack, "Intergranular Energy of Iron and Some Iron Alloys," Transo AIME, 191, 251, (1951)o 107. Ro S. Wagner and Bo Chalmers, "Grain Boundaries in Germanium," J. Appl. Phys., 51, 581, (1960)o 108. Fo Weinberg, "Grain Boundaries in Metals," Progress in Metal Physics, Vol. 8, Pergamon Press, 105, (1959)o. 109. J. L..White, "A Thermodynamic Analysis of Solubility in Liquid Metal Systems,"'UoS. Naval Research Labbo, NRL Rept 5555, Office of Techo Services, UoS. Depto of Commerce, Washington, DoC,,(1960), 11Oo S. S, Wilks, Mathematical Statistics, John Wiley and Sons, New York, (1962) 111. J. Winter and So.Weinig, "The Grain Boundary Adsorption of Solutes," Trans AIME, 215, 74, (1959)0 112 _____ "The Solid/Liquid Interface," Second International Congress of Surface Activity, Volo III, Academic Press, New York, (1957)o.

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