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## Material Information- Title:
- Analysis of the sintering force in copper
- Creator:
- Gregg, Roger Allen, 1938- (
*Dissertant*) Rhines, F. N. (*Thesis advisor*) Reed-Hill, R. E. (*Reviewer*) Bailey, T. L. (*Reviewer*) Blake, R. G. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1968
- Copyright Date:
- 1968
- Language:
- English
- Physical Description:
- xiii, 126 leaves. : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Curvature ( jstor )
Density ( jstor ) Geometry ( jstor ) Interfacial tension ( jstor ) Particle density ( jstor ) Particle size classes ( jstor ) Porosity ( jstor ) Sintering ( jstor ) Specimens ( jstor ) Surface areas ( jstor ) Copper ( lcsh ) Dissertations, Academic -- Metallurgical and Materials Engineering -- UF Metallurgical and Materials Engineering thesis Ph. D Sintering ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- The sintering force is defined as the external load which will balance the contractile tendency of a sinter body in the direction of its application. Experimental measurements of the sintering force were made upon uneompacted, sintered copper specimens as a function of apparent density, sintering temperature and particle size. All measurements were obtained from specimens sintering in dry, deoxidized hydrogen. Measurements of void volume and of surface area and average mean curvature of the void-solid interface were made through the procedures of quantitative metallography. The sintering force was found to increase from zero at the onset of sintering, pass through a maximum at approximately 90 per cent of theoretical density and decrease toward zero at bulk density. This general behavior was observed for each particle size. The magnitude of the sintering force at any density increased with a decrease in particle size. There was no effect of temperature on the sintering force in copper over the range 950 to 1050 C. The surface area per unit volume-density relationship was observed to be linear for each particle size. Variation of the average mean surface curvature with density was found to be qualitatively identical to that observed for the sintering force. The results are analyzed on the basis of a balance between the externally applied force and the surface tension forces promoting shrinkage. A quantitative expression of the sintering force in terms of the surface geometry and surface tension of the sinter body is derived from basic concepts of capillarity. A comparison is made of the sintering forces predicted by this relation and experimental value; Sensitivity of the rate of shrinkage to the geometry of the microstructure of the sinter body will be demonstrated through simultaneous application of the concept of the sintering force, an empirical ^treatment of the mechanical behavior of the sinter body and a stress-strain rate relation based on deformation by creep.
- Thesis:
- Thesis - University of Florida.
- Bibliography:
- Bibliography: leaves 122-125.
- Additional Physical Form:
- Also available on World Wide Web
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 001133265 ( AlephBibNum )
20143485 ( OCLC ) AFN0634 ( NOTIS )
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ANALYSIS OF THE SINTERING FORCE IN COPPER By ROGER ALLEN GREGG A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1968 Dedicated to my wife Susan ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to Dr. F. N. Rhines, chairman of his supervisory committee, for suggest- ing the subject of this research and for his invaluable assistance in establishing the conceptual framework of the problem. The author is indebted to Dr. R. T. DeHoff for lending his many talents to the discussion of the problem and especially for his timely contributions to the field of quantitative metallography. The author wishes to thank Dr. R. E. Reed-Hill, Dr. T. L. Bailey and Dr. R. G. Blake for serving on his supervisory committee. The financial support of this research by the Atomic Energy Commission was appreciated, and is hereby acknowledged. TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . . . . . . . . i LIST OF TABLFS . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . viii ABSTRACT . . . . . . . .. . . . . . xii Chapter I. INTRODUCTION . . . . . . . .. . . 1 1.1 General Characteristics of the Sintering Process . . . . . . . . . 1 1.2 Previous Investigations of the Sintering Process . . . . . . . . . 2 1.3 Purpose and Scope of this Research . . . 7 II. EXPERIMENTAL PROCEDURE . . . . . . . .. 13 2.1 Material Specifications . . . . . .. 13 2.2 Particle Size Classification . . . . .. 14 2.3 Specimen Preparation . . . . . . .. 14 2.4 Experimental Determination of the Sintering Force . . . . . . .. 17 2.41 General features of the apparatus . . 17 2.42 Strain measurement . . . . . 21 2.43 Calibration of beam force and deflection versus indicated strain . .. 21 2.44 Test temperature and environment control . . . . . . . .. 23 2.45 Test procedure . . . . . ... 23 TABLE OF CONTENTS (Continued) Chapter II. (Continued) 2.5 Post-test Inspection of the Specimens . . 2.51 Density determination . . . . . 2.52 Metallographic preparation . . . . 2.53 Metallographic examination . . . . III. EXPERIMENTAL RESULTS . . . . . . . . . 3.1 Sintering Force . . . . . . . . 3.2 Quantitative Matallography . . . . . 3.21 Surface area of the void-solid interface 3.22 Area tangent count . . . . . . 3.23 Average mean surface curvature . . . 3.24 Effect of uniaxial constraint on sintering behavior . . . . . . IV. DISCUSSION . . . . . . . . . . . 4.1 Surface Energy and Surface Tension of Solids 4.11 Pressure difference across a curved surface . . . . . . . . 4.12 Measurement of surface tension effects in solids . . . . . . . . 4.2 The Geometry of Sintering . . . . . . 4.21 The evolution of curvature of the void-solid interface . . . . . 4.3 Analysis of the Sintering Force . . . . 4.31 Comparison of predicted and experi- mentally measured sintering forces . . Page 26 27 27 28 30 30 38 45 45 50 50 58 59 61 63 69 73 78 83 TABLE OF CONTENTS (Continued) Chapter Page IV. (Continued) 4.4 Comparison of the Results of This Research with Previous Investigations . . . .. 89 4.5 Application of the Sintering Force to the Measurement of Surface Tension of Solids . 90 4.6 Kinetics of the Sintering Process ..... 92 4.61 Estimation of stresses created by external load . . . . . . 92 4.62 Correlation of the sintering force with shrinkage rates . . . . . 97 4.63 Comments on the mechanism of shrinkage 105 V. CONCLUSIONS . . . . . . . .. . 107 APPENDICES . . . . . . . .. . . . . 109 I. APPLICATION OF THE PRINCIPLES OF TOPOLOGICAL GEOMETRY TO THE DESCRIPTION OF THE SINTERING PROCESS (after F. N. Rhines [l]) . . . . .. 110 II. THE QUANTITATIVE ESTIMATION OF AVERAGE MEAN SURFACE CURVATURE (after R. T. DeHoff [41]) .... 116 REFERENCES. . . . . . . . .. .. . . . 122 BIOGRAPHICAL- SKETCH . . . . . . . . .. .. . 126 LIST OF TABLES Table Page 1. Experimental values of sintering force for the 48 micron particle size as a function of density and temperature . . . . . ... .31 2. Experimental values of sintering force for the 30 micron particle size as a function of density and temperature . . . . . ... .33 3. Experimental values of sintering force for the 12 micron particle size as a function of density and temperature . . . . . ... .35 4. Quantitative metallography data for the 48 micron particle size as a function of density and temperature . . . . . . . .... . 39 5. Quantitative metallography data for the 30 micron particle size as a function of density and temperature . . . . . . . .... . 41 6. Quantitative metallography data for the 12 micron particle size as a function of density and temperature . . . . . . . . .. . 43 LIST OF FIGURES Figure Page 1. Split graphite mold used for presintering the sintering force specimens . . . . . . .. 15 2. Schematic presentation of experimental setup for determination of the sintering force . . . ... 18 3. Apparatus used for measurement of the sintering force . 19 4. Cantilever beam assembly of sintering force apparatus . 20 5. Calibration curves of indicated strain and change in specimen length, AL, versus cantilever beam force . 22 6. Typical specimen behavior during experimental measurement of the sintering force . . . ... 25 7. Experimental values of the sintering force as a function of density for the 4S micron particle size spherical copper powder . . . . . ... 32 8. Experimental values of the sintering force as a function of density for the 30 micron particle size spherical copper powder . . . . . . 34 9. Experimental values of the sintering force as a function of density for the 12 micron particle size spherical copper powder . . . . . ... 36 10. Variation of surface area per unit volume with density for the 48 micron particle size spherical copper powder . . . . . . ... 40 11. Variation of surface area per unit volume with density for the 30 micron particle size spherical copper powder . . . . . . ... 42 12. Variation of surface area per unit volume with density for the 12 micron particle size spherical copper powder . . . . . . ... 44 viii LIST OF FIGURES (Continued) Figure Page 13. Comparison of the linear relationships between density and surface area per unit volume for the 48, 30 and 12 micron particle sizes . . . .. 46 14. Variation of the net area tangent count with density for the 48 micron particle size spherical copper powder . . . . . .... ... 47 15. Variation of the net area tangent count with density for the 30 micron particle size spherical copper powder . . . . ... .. .. . 48 16. Variation of the net area tangent count with density for the 12 micron particle size spherical copper powder . . . . . . . .. 49 17. Variation of the average mean curvature of the void-solid interface with density for the 48 micron particle size spherical copper powder . . 51 18. Variation of the average mean curvature of the void-solid interface with density for the 30 micron particle size spherical copper powder . . 52 19. Variation of the average mean curvature of the void-solid interface with density for the 12 micron particle size spherical copper powder . . 53 20. Correlation of the slope of the surface area- density relationship with initial particle size for spherical copper powder . . . . . . .. 55 21. Cross-sectional areas of sintering force specimens after testing . . . . . . .... . . 56 22. Schematic of surface stresses in a solid . . ... 60 23. Resolution of surface tension into a pressure acting normal to each point of a curved surface ...... 62 24. Two arrangements for balancing the shrinkage of metal foils by inducing creep with an external force 65 25. Schematic of surface tension forces and grain boundary array in wire geometry used for determination of surface tension in metals . . . . . ... 68 LIST OF FIGURES (Continued) Figure Page 26. Variation of genus (connectivity)and number of separate parts of the void-sclid interface with density for the 48 micron special copper powder [43] 72 27. Comparison of the average surface curvature as a function of density for the 48, 30 and 12 micron particle sizes of spherical copper powders . . . 74 28. Variation of average surface curvature with density for a mixture of sizes of an irregular, dendritic, electrolytic copper powder [62] . . . . .... .79 29. Resolution of surface tension forces emanating from the cut surface of a spherical pore in a direction perpendicular to the sectioning plane . . . ... 81 30. Comparison of the experimental values of the sinter- ing force with predicted values for the 48 micron particle size powder . . . . . . .... .84 31. Comparison of the experimental values of the sinter- ing force with predicted values for the 30 micron particle size powder . . . . . . . ... 35 32. Comparison of the experimental values of the sinter- ing force with predicted values for the 12 micron particle size powder . . . . . . . ... 86 33. Room temperature yield stress at 2 per cent elonga- tion as a function of density for a range of particle sizes of the spherical copper powders used in this investigation [67] . . . . .... .94 34. Variation of the average solid area and the effec- tive solid area estimated from experimentally observed mechanical behavior with density for spherical copper powders . . . . . . ... 96 35. Estimated operating stresses created by externally applied force on sintering force specimens . . .. 98 36. The relation between strain rate, e, and stress, C, as affected by the scale of the microstructure for isothermal high temperature deformation of pure metals . . . . . . . . ... . . 100 LIST OF FIGURES (Continued) Figure Page 37. Variation of experimentally determined linear shrinkage rates with density for the 43 [70] and 30 [71] micron particle sizes of spherical copper powder . . .. 101 33. Relationship between linear shrinkage rates and estimated effective stresses for the h8 and 30 micron particle sizes of spherical copper powder ...... 102 39. Correlation of linear shrinkage rate--estimated operating stress relationship with deformation rate law based on stress-directed diffusion . . ... 104 40(a). Examples of bodies bounded by surface of genus zero, one and two . . . . . . . .... ... 112 40(b). Variation of genus with number of contacts per particle . . . . . . . .... .. 112 41(a). Separation of sintering into three stages on the basis of changes in genus of the void-solid interface . . . . . . . . ... .. . 114 41(b). Effect of genus of the void-solid interface on its minimal surface area configuration . . . ... 114 42. Orientation relationships that exist between an arbitrary element of surface (dudv) and a section- ing plane; n is the surface normal; u and v are the directions of principal curvature; I is normal to the sectioning plane . . . . . . .. 120 Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ANALYSIS OF THE SINTERING FORCE IN COPPER By Roger Allen Gregg June, 1968 Chairman: Dr. F. N. Rhines Major Department: Metallurgical and Materials Engineering The sintering force is defined as the external load which will balance the contractile tendency of a sinter body in the direction of its application. Experimental measurements of the sintering force were made upon uncompacted, sintered copper specimens as a function of apparent density, sintering temperature and particle size. All meas- urements were obtained from specimens sintering in dry, deoxidized hydrogen. Measurements of void volume and of surface area and average mean curvature of the void-solid interface were made through the proce- dures of quantitative metallography. The sintering force was found to increase from zero at the onset of sintering, pass through a maximum at approximately 90 per cent of theoretical density and decrease toward zero at bulk density. This general behavior was observed for each particle size. The magni- tude of the sintering force at any density increased with a decrease in particle size. There was no effect of temperature on the sinter- ing force in copper over tha range 950 to 10500C. xii The surface area per unit volume-density relationship was observed to be linear for each particle size. Variation of the aver- age mean surface curvature with density was found to be qualitatively identical to that observed for the sintering force. The results are analyzed on the basis of a balance between the externally applied force and the surface tension forces promoting shrinkage. A quantitative expression of the sintering force in terms of the surface geometry and surface tension of the sinter body is derived from basic concepts of capillarity. A comparison is made of the sintering forces predicted by this relation and experimental values. Sensitivity of the rate of shrinkage to the geometry of the microstructure of the sinter body will be demonstrated through simul- taneous application of the concept of the sintering force, an empir- ical treatment of the mechanical behavior of the sinter body and a stress-strain rate relation based on deformation by creep. xiii CHAPTER I INTRODUCTION The atoms or molecules on the free surface of a solid possess higher energy than those within the interior. Therefore, a solid sys- tem comprised of finely divided powders has higher total energy than a single large particle of the same material and equal miss. By re- arranging material so as to reduce its surface area, the system of powder particles can lower its energy and become thermodynamically more stable. When powder particles are held in contact at a temper- ature close to, but below, their melting point, material rearrangement driven by the excess surface energy can produce permanent adherence between the particles and result in a single, solid framework. This process is known as sintering. Most powdered materials will exhibit sintering under the proper conditions of temperature and environment. This includes crystalline, vitreous and organic materials. 1.1 General Characteristics of the Sintering Process When powdered materials are sintered the following general features are observed. The particles form permanent connections at their points of contact, which increase in size with time. The sur- faces of the pores become smooth and the total volume of porosity within the powder aggregate decreases with time, increasing the appar- ent density of the system. Every gecmerric change driven by surface energy is accompanied by a decrease in the total surface area in, the system. Under the proper conditions, the apparent density of the sinter body may approach the absolute density of the solid material. The evolution of the internal structure of a sinter body is separable into stages. The classification of the stages of sintering as proposed by Rhines [1] will be applied here. In the first stage, particle contacts broaden into weld necks and surface contours are smoothed. In the second stage, the principal geometric change is a decrease in connectivity of the pore network through closure of the connecting links, or channels, of the network. The completion of this process signifies the end of the second stage. The third and final stage is characterized by the elimination of the isolated porosity. 1.2 Previous Investigations of the Sintering Process Considerable effort has been expended in fundamental research of sintering, primarily directed toward determination of the controlling mechanism. There exist several mass transport processes which can act either individually or in concert in their participation in the mater- ial rearrangement observed during sintering. These are: evaporation- condensation, surface diffusion, volume diffusion, viscous flow and bulk deformation by shear processes. Consequently, sintering is a com- plex phenomenon, a fact evident in the diversity of the theories of sintering which have been proposed. Saurwald, in a series of papers, was the first to attempt to formulate a general theory of sintering. He concluded [2] that adhesive Numbers in brackets refer to te references. Numbers in brackets refer to the references. forces between the powder particles are responsible for their con- solidation, which in his opinion, had striking similarities to the recrystallization processes observed in metals. Balshin [3,4,5] attempted a refinement of this approach suggesting that densification would result from recrystallization initiated at particle contacts, while a decrease in density would be produced by recrystallization within the particles. Jones[6] agreed that the forces of atomic cohe- sion determine the sintering process and concluded that since these forces decrease with increasing temperature, the increase in sintering rate with temperature is due to a rapid decrease in the resistance to plastic flow. Baike [7], Wretblad and Wulff [8], and also Rhines [9], were among the first to suggest that surface tension played a major role in promoting sintering. Balke visualized a "zipper action" in closure of the void space which was initiated at the contact points between the particles. Wretblad and Wulff suggested the application of the equations of capillarity in calculating the stresses at the contact points as a function of the shape of the solid surfaces and their energy. They further suggested that the stresses estimated in this manner could exceed the elastic strength of the solid and result in plastic deformation. Rhines concurred that the magnitude of the forces derived from surface tension would be greatest in the regions of highly curved surfaces. Interest in this viewpoint was spurred by theoretical [10,11] and experimental analyses [12] of the viscous flow of metals under the action of surface tension. It is now generally agreed that surface tension is the primary force in achieving the geometric changes observed during sintering. However, agreement has not been reached concerning the manner in which the action of surface tension acccm- plishes these changes. Pines [13] and also Shaler and Wulff [14] suggested that the mechanisms) responsible for smoothing the particle surfaces and round- ing the pores could be distinct from the transport mode producing densification. Shaler and Wulff pointed out that densification requires changes in shape and size of the particle network or "skeleton." In order to achieve this, material must be removed from within the particles along the line joining their centers. It is now generally acknowledged [15,16,17] that transport mechanisms which are restricted to the surface, such as evaporation-condensation and surface diffusion, can produce only surface-rounding, not densification. Rhines [18] made the impor- tant suggestion that a transfer of void "space" from the pores to the external surface, which would produce densification, could occur by a lattice vacancy diffusion process. To avoid the complications of a multiconnected, three- dimensional stack of particles, Kuczynski [19] used simple models of a single sphere on a flat surface and two spheres in contact to eval- uate the controlling mechanism in the first stage of sintering. He invoked the equations of capillarity to calculate the vapor pressure, vacancy concentration and internal pressure in terms of surface geo- metry. For these simple geometries he was able to derive equations relating the rate of growth of the weld neck to the transport mechanism assumed. lis experimental evidence on metals [19] indicated that neck growth was controlled by volume diffusion in coarse particle systems, and by surface diffusion for small particles. In glass [20] he reported a viscous flow mechanism. Mackenzie and Shuttleworth [21] pointed out that the removal of vacancies by volume diffusion to the exterior of the sinter body is too slow to explain the rates of densification observed. In addition, densification by this process would progress inward from the external surface, and the rate of densification would be a function of the size and shape of the system; a situation which does not develop in sintering. Using the technique of wrapping wires on a spool, Geach and Jones [23] observed that the grain boundaries formed at contact points were stable for long periods of time at sintering temperatures. Using the same experimental procedure, Alexander and Balluffi [24] claimed that pores in the wire compact continued to shrink only if they re- mained connected to a grain.boundary. Correa da Silva and Mehl [25] and Pranatis et al. [26] suggested that vacancy removal, required for densirfication by volume diffusion, could occur at grain boundaries through collapse of the vacancies collected there; Herring [11] had previously suggested this mechanism for creep by volume diffusion. Kuczynski [27] adopted the grain boundary effect into his diffusion model and found reasonable agreement between the rates of pore closure predicted by volume diffusion and those observed in photographs of wire-compacts published by Alexander and Balluffi [24,28]. Herring [29] showed that in two-particle systems with similar geometries but differing by a scale factor X, the times required to reach identical stages in sintering should be related through this This has been reported by Rhines et al. [22] to occur only in the pores immediately adjacent to the external surface. factor. If, for example, (powder), is X times larger than (powder)2, then the times required to reach similar stages of sintering are given by At At2 where n = 1, for evaporation-condensation; n = 2, for viscous Flow; n = 3, for volume diffusion; and n = 4, for surface diffusion. Alexander and Balluffi tested this relation for their wire compacts and found reasonable support for volume diffusion. However, DeHoff et al. [30] applied Herring's analysis to the rate of weld neck growth between twisted wires and found the exponent, n, to vary with the size of the wires. Small wires moved together during sintering while large wires did not, thereby resulting in dissimilar geometries and making Herring's analysis inapplicable. Shewmon and Wilson [31] measured the shrinkage and weld neck size during sintering of chains of particles. They reported that less than 10 per cent of the volume of the weld neck resulted from shrink- age, an observation which casts doubt on the ability of Kuczynski's model to reveal the mechanism of densification. They reported that Herring's analysis supported surface diffusion as the mechanism for weld neck growth. Mackenzie and Shuttleworth [21] chose to support the point of view that surface tension, acting through the curvature of the pore surface, creates stresses sufficient to result in macroscopic plastic deformation of the solid framework. They derived a relation for rate of densification based on the interaction of the capillary pressure of the pores in raising the stress level in the solid between the pores past that required for yielding. Clark and White [32] and Clark, Cannon and White [33] developed a model based on viscous flow, within a surface layer, under the forces of surface tension. They applied their model, and also the equation of Mackenzie and Shuttleworth, to experimental densification rates and achieved some degree of fit. Rhines and Cannon [34] found that the application of small compressive loads to a sintering system had the same effect on densi- fication rates as an increase in stress level has on creep rates. In other words, no new mechanism was introduced through the application of small compressive loads to the sinter body. Williams and Westnacott [35] showed that one of the accepted rate equations for transient creep of metals could be used to develop a relation for rate of weld neck growth which is so similar to Kuczynski's equation supporting volume diffusion, that the available experimental data could not dis- tinguish between them. Lenel and co-workers [17,36] have observed that the effect on shrinkage rate produced by varying the level of small compressive stresses is predictable from creep equations based on deformation by dislocation processes. 1.3 Purpose and Scope of this Research In spite of the obvious differences in the conclusions drawn in the theoretical and experimental analyses of sintering, there exist some important consistencies: (a) surface tension is the force through which densification is achieved, and (b) the magnitude of the effect of surface tension is determined by the geometry of the void-solid interface. Recognition of the importance of statement (b) provoked the use of the simple spherical models of Kuczynski, the wire compacts of Alexander and Balluffi and the particle chains of Wilson and Shewmon. The complex theories of Mackenzie and Shuttleworth, and of Clark and co-workers are also based on geometric effects consistent with state- ments (a) and (b). However, direct experimental evidence that surface tension is the force which produces densification has never been sat- isfactorily obtained, nor has the relationship between surface tension and the geometry of the void-solid interface in real sintering systems been defined. The first detailed study of the geometrical evolution of the pore structure during sintering was published by Rhines et al. [22]. They reported that although the total volume and number of pores decreased with increasing density, the average pore size increased. Arthur [37] measured the relative amounts of interconnected and closed porosity and found the porosity to be mainly interconnected; only after reaching a density of approximately 95 per cent of theoretical density does the porosity become completely closed to the external surface. Both of these results were unexpected and clearly revealed the inadequacies of simple models in explaining the behavior of real complex systems. Rhines [1] has shown that application of the principles of descriptive topology to sintered structures provides a simplified method for describing the complex geometric changes observed. In developing this approach (see Appendix I), Rhines pointed out that the minimal surface area possible is a function of the degree of connec- tivity of the void-solid surface; i.e., the topological state. As the connectivity decreases. the minimal area decreases; thus, once minimal configuration has been reached, further decrease in surface area requires a decrease in connectivity of the surface. DeHoff et al. [38] have produced experimental evidence that, in the second stage of sinter- ing, a linear correspondence exists between the density of the sintered system and the surface area it contains in unit volume. This behavior has been found to hold for all shapes, sizes, and compositions of pow- dered materials investigated, demonstrating the existence of a unique geometrical evolution for all sintering systems. In addition, they also reported that the slope of the linear relationship was predeter- mined by the initial connectivity of the particle stack. Recently, Barrett and Yust [39] have demonstrated by serial sectioning technique that the isolated parts of the porosity network which are developed in the second stage of sintering "occur first as widespread volumes of interconnected porosity, definitely nonspherical in shape." They con- cluded that these sections of the void volume become separated from the rest of the porous network, and from the external surface of the sinter body, by the closing of parts of the void; a similar process has previously been described by DeHoff et al. [38] as a closure of channels in the porous network. Techniques for quantitative determination of many of the geo- metric variables which characterize sintering (e.g., volume fraction of porosity, surface area and mean free path in the void or solid phases) have been available for some time [40]. Recently, DeHoff [41] and Cahn [42], in simultaneous and independent efforts, developed a new fundamental relationship of quantitative metallography which permits the experimental determination of the average value of the mean surface curvature over the total void-solid interface. The details of the derivation of this important relation are presented in Appendix II. A technique for experimental determination of the topological properties of sintered structures has been presented by Aigeltinger [43]. In his procedure a series of closely spaced microsections, obtained by the use of a microtome, is used to synthesize the struc- ture. Thus, the degree of connectivity of the void-solid surface and the number of isolated parts of the porosity can be determined as a function of the density of the sinter body. The geometric and topological information now available cer- tainly constitutes a significant basis for understanding the sintering behavior of real systems. However, in order to explain the kinetics of sintering it is necessary to discover the relations which exist between the kinetics and the geometry of the sintered structure. Rhines [44] has suggested that insight into the latter problem might be gained by an analysis of the force of contraction generated by a sintering powder system. Since the mechanical equivalent to the sintering force exists regardless of the mechanism through which the solid responds to the thermodynamic driving forces, no a priori assumptions as to operative mechanism need be made in order to use this approach. Except for the minor effects of gravity [45], sinter bodies shrink isotropically. In order to stop total shrinkage, and in effect balance the force of sintering isotropically, a uniform hydrostatic tension would have to be imposed on the sinter body. Since it is not practical to achieve this condition experimentally, the sintering force has been defined by Rhines [44] as the force necessary to stop the contraction of the sinter body in the direction of an uniaxially applied load. An attempt to measure such forces was made by Young [46]. He used the method which was employed by Udin [12] to measure surface ten- sion effects in fine metal wires. By suspending various weights from a series of identically prepared, sinrered specimens, Young was able to determine the load which just balanced the tendency to contract in the direction of the applied load. He reported that, in irregularly shaped -325 mesh copper powder, the sintering force increased with increasing density over the range of densities he investigated; from 45 to 60 per cent of the theoretical density. However, he was unable to discern any apparent relation between this unexpected functional dependence on density of these external forces, and the internal forces which are expected to be determined by the geometric evolution accom- panying densification. It was the purpose of this research to develop more completely the concept of the sintering force, experimentally measure it and * Intuitively, the force of sintering would be expected to decrease with increasing density since sintering rates decrease sharply (often several orders of magnitude) with increasing density. determine phenomenologically its relation to the physical, geometric and mechanical properties of the sintered structure. If these goals can be achieved, a strong link between geometry and kinetics will have been forged. The sintering behavior of a system of powder particles is largely determined by the characteristics of the powder. Therefore, the choice of the material used in this exploratory research effort was predicated by the necessity for easily controlled experimental variables and a high degree of reproducibility in the qualities of the sintered specimens. For these reasons, copper was chosen and has the following desirable features: (1) reliable experimental values of surface tension, (2) unique compatibility with hydrogen as the sintering environment; (a) all oxides of copper are reduced by hydrogen, (b) the high diffusivity of hydrogen in copper prevents any pressure buildup within closed porosity, (3) a melting point (1083 C) which creates no serious experimental difficulties, and (4) mechanical properties as a function of stress and temperature which are documented as completely as for any metal. CHAPTER II EXPERIMENTAL PROCEDURE 2.1 Material Specifications The work of DeHoff et al. [38] made quite clear the importance of the topological state of the initial particle stack in determining the specific path of geometrical evolution observed during densifica- tion. They pointed out that the most efficient stacking for any given particle shape is never achieved, and it is generally possible to increase the number of contacts per particle by vibratory packing or mechanical compaction. The latter was not a part of the experimental procedure in this research. However, it was necessary to handle the mold after filling it with powder; therefore a powder which would pro- duce a stable and reproducible stack was highly desirable. In this respect particle shape is of primary importance, for it has been known [47,48] for some time that, in general, spherical powders produce the most efficient, and consequently the most stable, loose particle stack- ing. The copper powders employed in this investigation were prepared by the Linde Company by atomization of liquid copper in an inert .gas atmosphere. The powders possessed a high degree of sphericity and were of high purity. Of particular importance to the geometric analy- sis of the sintered structure, the amount of copper oxide within the interior of the particles was negligible and the formation of gas porosity, which results from the reduction of internal copper oxide by the hydrogen sintering atmosphere, was also negligible. 2.2 Particle Size Classification Standard ASTM sieve analysis procedures were used for particle size classification. In this method, the number which is used to designate a sieve size corresponds either to the average number of apertures per square inch in the wire mesh screen or to the average size of the apertures. If the particles will pass through a sieve, a minus (-) sign precedes the sieve size number; a plus (+) sign pre- cedes the number for a sieve through which the particles will not pass. The powder used in this investigation was initially separated into two size classes: -270 + 325 mesh (-52 + 44 microns) and -325 mesh (-44 microns) on a standard Ro-Tap mechanical shaker. Later in the course of the work the -325 mesh powder was used as the source of a -20 micron classification performed on an Allen-Bradley Sonic Sifter. The particle size distribution, as reported by the powder manufacturer, was used to calculate the average particle diameters within each size classi- fication. The average diameters for the -52 + 44. -44 and -20 micron sizes were estimated to be 48, 30 and 12 microns, respectively. 2.3 Specimen Preparation Specimens used for measurement of the sintering force were pre- pared by presintering to the desired shape in the split graphite mold shown in Figure 1. The basic shape of the specimen was similar to a cylindrical tensile test specimen. However, the shoulders ateachendof the reduced section were gently sloped to reduce constraint at these Figure 1. Split graphite mold used for presintering the sintering force specimens. positions while the specimen was shrinking during presintering or cool- ing from the presintering temperature. Connection between the specimen and the load column was provided by presintering 0.9375 inch diameter tungsten rods into each end of the specimen. The tips of the rods embedded in the specimen were notched to provide interlocking between the tungsten and the copper and increase the mechanical strength of the connection. All specimens were presintered at 7000C in dry, deoxidized hydrogen. Presintering times were 15, 20 and 30 minutes for the 48, 30 and 12 micron particle sizes, respectively. The density of the loose particle stack prior to presintering was calculated by determining the volume of the mold (by filling with water) and weighing the volume of the powder which filled the mold. Several determinations of the loose- stack density revealed it to be very reproducible and always between 3 5.1 and 5.3 gms per cm for each particle size. The densities of the presintered specimens were found to lie in the range 5.4 to 5.6 gms 3 per cm As a result of storage and handling in hunid atmospheres the powder particles were covered with thin oxide films. No attempt was made to remove these films before presintering; however, close inspec- tion after presintering revealed all surfaces, both internal and exter- nal to the specimen, were clean and bright. Internal surfaces were inspected in specimens fractured immediately after presintering. 2.4 Experimental Determination of the Sintering Force 2.41 General features of the apparatus The apparatus used for measurement of the sintering force, shown schematically in Figure 2, was designed to provide the following: (1) A continuously variable and manually adjustable force which would permit the application of the instantaneous force required to stop contraction of the sintering specimen at any given density. (2) Maximum sensitivity to change in specimen length; required to measure accurately the sintering force at high densities where shrinkage rates are extremely slow. (3) Easy entry and removal of the specimen from the test rig; required due to the weakness of the presintered specimens and in order to preserve the density which existed at the moment of final force measurement. (4) Minimum deleterious effects from changes in ambient and furnace temperature so that the control possible for these variables would be adequate. Simultaneous application and measurement of the externally applied load as well as direct measurement of length changes in the specimen were accomplished through the use of the cantilever beam assembly in Figure 3. Length changes were revealed by the change in beam deflection: beam force was monitored by measurement of the strain created in the beam by its deflection. The design of the beam is shown in Figure 4. The reduced section of the beam provided a location of concentrated stress in the region of the fillet where measurable strain was maximized for a given deflection. The beam was fabricated from an iron-base material, Iso-elastic alloy, so named for its low temperature coefficient of the elastic modulus. As reported by the alloy supplier, J. Chatillon and Sons, the value of 18 Cantilever beam Strain indicator 0 0 0 s 0o o Resistance --- Figure 2. Schematic presentation of experimental setup for determination of the sintering force. 19 I Figure 3. Apparatus used for measurement of the sintering force. Figure 4. Cantilever beam assembly of sintering force apparatus. this coefficient for this alloy was 20 x 10- per F as compared to a value of -190 x 106 per F for spring steel. In addition, errors due to anelastic and hysteresis effects in this material were also small; .02 and .05 per cent of the total deflection, respectively. 2.42 Strain measurement The output of the four strain gages employed on the beam was measured by a Baldwin-Lima-Hamilton model 120 A strain indicator. Two gages each were placed on the top and bottom sides of the beam near the fillet farthest from the free end of the beam. The arrangement of the gages into the Wheatstone Bridge of the strain indicator served two purposes: (1) Measurable strain was increased by a factor of four over the true strain at this position of the beam. (2) Strains induced in the gages by thermal expansion or contraction of the beam were essentially cancelled. 2.43 Calibration of beam force and deflection versus indicated strain In order to know accurately the beam force as a function of total indicated strain, standard weights were applied to the beam and the resultant strain recorded. The deflection of the beam, as a func- tion of beam force, was determined by direct measurement with an -4 optical system accurate to 0.5 x 10-4 inches. The calibration curves for these quantities are presented in Figure 5. They were found to be reproducible to within 1 per cent of the value of the beam force. It was assumed that all changes in beam deflection could be attributed to changes in specimen length. Based on the sensitivity of the strain Strain 1500 Figure 5. Calibration curves of indicated strain and change in specimen length, AL, versus cantilever beam force. 250 200 150 100 500 1000 AL (x 105 inches); Strain (in/in x 106) indicator to a change in measurable strain of 2 x 10-6, the apparatus was sensitive to a change of 3.5 x 10-6 inches in specimen length which corresponded to a change of 0.3 gms in beam force. 2.44 Test temperature and environment control All sintering force tests were made in an atmosphere of dry, deoxidized hydrogen. The gas pressure within the furnace tube was maintained just above atmospheric pressure by constant gas flow to prevent the entry of oxygen into the system. Test temperatures were provided by a nichrome element resistance furnace which had a temper- ature variation of 2.5 OC over 4 inches of the hot zone. Test tem- peratures were measured with platinum, platinum-10 per cent rhodium thermocouples and controlled by a Leeds and Northrup Precision Set-Point Control System, which maintained the temperature of the hot zone to within 0.10C. 2.45 Test procedure The presintered specimens were integrated into the load column by inserting the L-shaped end of the upper connecting rod into the end of the pull-red. With the furnace at the desired test temperature, the load column was lowered into the hydrogen-filled furnace tube. The bottom connecting rod was inserted into a slot at the bottom of the apparatus, lowered below the restraining ledge and then rotated such that elevation of the L-shaped tip of the bottom connecting rod, either by manual adjustment of the beam position or by shrinkage of the speci- men, would place the specimen under tensile constraint. In practice, it was found that the connecting rods were often presintered into the specimen at slight angles to the specimen axis, a situation which produced bending stresses upon initial loading. Therefore, each specimen was either initially brought into creep by manual adjustment of the beam force, or was allowed to shrink against the constraint of the beam for some time. This procedure helped to straighten the load column and produced approximately uniaxial loading during the remainder of the test. Typical behavior of a specimen during the test is shown in Figure 6. Notice that the use of a cantilever beam to supply the exter- nal force results in an inverse relationship between specimen length and the magnitude of the beam force on the specimen. If the specimen can shrink against the initial load, its decrease in length will increase the force which the beam exerts upon it. If, by manual adjust- ment of the beam position, the force is increased, then the specimen may be made to creep. The elongation of the specimen in creep reduces the beam force on the specimen until the sintering force within the specimen is sufficient to balance the external beam force. This balance of forces is only temporary since radial shrinkage of the specimen is unopposed, and the density of the specimen is constantly increasing. In practice, the time interval over which the forces appeared to be balanced ranged from a few seconds to several hours depending on the shrinkage rate of the specimen, as affected by temperature, particle size and density. The behavior indicated in Figure 6 is that of a specimen whose density would lie in the range where the sintering force was observed to increase with increasing density. Consequently, after Beam ------ force A Specimen length Time Figure 6. Typical specimen behavior during experimental measurement of the sintering force. The dashed line indicates the desired balance of internal and external forces. the initial force balance, the external force against which the speci- men can shrink increases as the specimen density increases. For speci- mens in this range of density, the test was aborted at the moment the specimen first revealed that it had progressed from creep to shrinkage. At this point (indicated by dashed lines in Figure 6) the external force was removed by manually lowering the beam support. The specimen was then immediately raised to the cool end of the furnace tube in order to preserve the density which corresponded to the final force measurement. The bottom of the load column was essentially suspended from the rest of the column, therefore the minimum load on the specimen throughout the test was the beam force plus the weight of the bottom connecting rod. However, since each point along the length of the specimen had to support the weight below it, the average load on the specimen was taken as the sum of the beam force, the weight of the bottom connecting rod and 0.5 times the weight of the specimen. 2.5 Post-test Inspection of the Specimens After testing, each specimen was inspected for excessive bend- ing, necking or obvious failures due to cracks. Any specimen which possessed any of these defects after testing was discarded. As a fur- ther check on uniformity the diameter of the cylindrical specimen was measured at several locations along the gage length. A Jones and Lamson optical comparator with a reported accuracy of 0.0001 inch was used for this purpose. 2.51 Density determination Apparent densities of tested specimens were determined through Archimedes principle using the standard ASTM procedure for porous bodies. Samples for density determination were taken from the middle of the gage length. After obtaining the weight of the sample in air, the pore openings in its surface were sealed by impregnation with liquid wax; the excess wax was wiped from the surface. The weight of the impregnated sample was then obtained in air and in water, the dif- ference in the two being equal to the volume of the sample. The appar- ent density of the sample was calculated by dividing the weight (with- out wax) by the sample volume. Reproducibility of this method was estimated to be 1 per cent of the value of the density. 2.52 Metallographic preparation Samples taken from the specimen for metallographic inspection were mounted either in bakelite or in epoxy mixed with 0.3 micron alumina. Samples were mounted so that polishing e:cposed a section parallel to the cylindrical axis of the specimen. Mounting in bakelite was accomplished by the standard hot-pressing procedure. Mounting in alumina-dispersioned epoxy was carried out in the following way. After blending the epoxy, 0.3 alumina powder was added in a 1 : 1 ratio by volume. The sample was pl-aced in a hollow plastic cylinder sitting on its end on a glass plate. The cylinder was then filled to the desired level with the epoxy-alumina mixture and the entire assembly placed in a vacuum dessicator attached to a mechanical vacuum pump. The pressure on the system was reduced until the air within the porous sample appeared to be removed. Pressure was then reapplied by opening the system to the atmosphere, thus forcing the liquid into the porous net- work of the sample. The alumina particles, due to agglomeration with- in the epoxy, did not make their way into the sample. However, their presence in the epoxy improved the abrasive qualities of the mount and helped to preserve the edges of the sample during polishing. The epoxy within the sample helped in preserving the true nature of the void-solid interface. For this reason, very low density specimens were usually mounted in the epoxy-alumina mixture; mounting in bakelite proved to be more efficient and adequate for intermediate and high density samples. Mounted specimens were rough polished by hand on wet silicon carbide abrasive papers. Fine polishing was performed, in the sequen- tial steps indicated below, on standard rotating-type polishing wheels covered with microcloths saturated with the following materials: (1) 600 grit silicon carbide particles and water, (2) 0.3 micron alumina and water, and (3) 0.25 micron diamond and lapping oil. 2.53 Metallographic examination All specimens were metallographically examined without etching. Bausch and Lomb bench microscopes and metallograph were employed, depending on the magnification required, for all microscopy. The fol- lowing quantitative metallography parameters were experimentally determined: (1) Pp; the fraction of points of a grid, superimposed on the microstructure, which fall within the phase of interest, void or solid. (2) N ; the number of times a test line, superimposed on the microstructure, intercepts the void-solid interface, per unit length of test line. (3) TAnet= TA TA_; TA+ and TA_ are the number of tangents which occur between a test line swept across the micro- structure and the convex and concave segments, respec- tively, of the void-solid interface, per unit area traversed by the sweeping test line. The total length of test line employed in experimental measure- ment of NL was determined by observation of the effect of accumulated length of test line on the accumulated average value of N When the average value of NL fluctuated no more than 5 per cent with further increase in total length of test line, this value was accepted. Experi- mental values of TAnet were accepted when increase in total measured area caused no fluctuations greater than 10 per cent in the accumulated average value. Throughout this work, convex (positive) segments of void-solid interface are those which are convex with respect to the solid phase. CHAPTER III EXPERIMENTAL RESULTS 3.1 Sintering Force Experimental values of the sintering force were determined as a function of density at three temperatures for copper powders with average particle diameters of 48, 30 and 12 microns. These data are compiled in Tables 1, 2 and 3 and presented in Figures 7, 8 and 9. For a given particle size the behavior of the sintering force with increasing density may be described as follows: in the first and second stages of densification the sintering force increases at a decreasing rate; this is followed by a sharp rise to a maximum value from which the sintering force decreases rapidly toward the zero value expected at the theoretical density of copper. This general functional dependence on density was obtained for each of the particle sizes; however, the maximum value appears to shift toward higher density with decreasing particle size. Existence of the maximum value of the sintering force was verified by continuous measurement of the beam force which the specimen imposed upon itself (through shrinkage or creep in the direction of the applied load) as its density was increased primarily through radial shrinkage. In this way the force against which the specimen could shrink was clearly observed to pass through a maximum. Although the Table 1. Experimental values of sintering force for the 48 micron particle size as a function of density and temperature Final Sintering Test Density Force Temperature (gis/cm ) (gms) ( C) 5.75 5.78 5.80 5.90 5.98 6.57 6.90 7.02 7.16 7.26 7.45 7.55 7.60 7.60 7.73 8.18 8.20 25 23 24 28 31 38 40 35 30 48 44 38 52 48 52 75 65* 56 950 950 950 950 950 1000 1000 1000 1050 1000 1000 1050 1050 1050 1050 950 1050 Maximum (peak) sintering force. Maximum (peak) sintering force. 0 950C O 10000C O 10500C o B; 0 0 9_ -- [ ,---^s- O 20L 7.0 8.0 8.5 Density (gms/cm3 ) Figure 7. Experimental values of the sintering force as a function of density for the 48 micron particle size spherical copper powder. The solid lines indicate that the specimens corresponding to the data points passed through a max- imum in sintering force. _ I I I IJ o-O Table 2. Experimental values of sintering force for the 30 micron particle size as a function of density and temperature Sintering Test Force Temperature )(gms) (C) Final Dens ity (gms/cm3' 6.35 6.41 6.48 6.70 6.78 6.94 6.95 6.97 7.19 7.36 7.46 7.48 7.53 7.54 7.86 7.88 7.93 8.00 8.10 8.20 8.23 8.26 8.37 8.40 65 73 79 64 91 90 100 99 95 100 110 100 95 103 120 114 90 103 108 112 138* 129 104 147* 130 145* 125 Maximum (peak) sintering force. 950 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 950 1000 1050 1000 1050 1050 13 0 [] 11 E]E] I \ D/' C 0 6.0 6.5 7.0 7.5 8.0 8.5 Density (gms/cm3 ) Figure 8. Experimental values of the sintering force as a function of density for the 30 micron particle size spherical copper powder. The solid lines indicate that the specimen corresponding to the data point passed through a max- imum in sintering force. 9.0 1.40 120 L 0 9500C O 10000C A 1050C 100 L 80 L El// / Table 3. Experimental values of sintering force for the 12 micron particle size as a function of density and temperature Final Sintering rest Density Force Temperature (gms/cm ) (gms) (C) 6.16 6.50 6.59 7.13 7.26 7.68 7.71 7.82 8.17 8.22 8.52 8.59 8.68 89 120 126 145 150 158 155 154 176 183 198* 182 214* S203 199* 192 1000 1000 1000 1000 1000 1000 950 950 1000 950 1000 1050 1000 Maximum (peak) sintering force. 200- I 0 95000 180- O Iooo00 r o10500C / 160 / 0O 140 a / o i 4 0 120- / / / / 100 / I / I I I I I I I 6.0 6.5 7.0 7.5 8.0 8.5 9.0 Dens ity (gms/cm3) Figure 9. Experimental values of the sintering force as a function of density for the 12 micron particle size spherical copper powder. The solid lines indicate that the specimens corresponding to the data points passed through a maximum in sintering force. evolution of the sintering force could be accurately measured by this procedure, only the final density of the specimen could be determined. Therefore, the lines drawn in Figures 7, 8 and 9 to indicate the behav- ior of the specimens in the region of the maximum sintering force are accurate with respect to force but estimated with respect to density. After the specimen had clearly passed through a maximum in sintering force, it was removed and its density determined. The data points at the ends of the lines drawn correspond to the final force and density. Measurements of the sintering force were made at 950, 1000 and 1050 C. However, a complete documentation of the sintering force at each of these temperatures was not made since initial measurements revealed that the effect over this range of temperatures was negligible compared to the normal scatter of the experimental values. This knowl- edge was put to use in determination of the sintering force at high densities; the higher temperatures provided faster shrinkage rates and therefore greater sensitivity in the measurement of change in specimen length. The lower limit in density at which measurement of the sinter- ing force could be made was determined by the increase in density which the specimen achieved during the time (approximately 5 minutes) allowed for thermal equilibration of the load column after its insertion into the furnace. Consequently, the lowest density for which sintering force data were obtained increased with decreasing particle size. 3.2 Quantitative Metallography The point count was used to calibrate the volume fraction of solid in the polished microstructure against the volume fraction pre- dicted from the apparent density of the specimen. Polished specimens whose point count density was within 2 per cent of their apparent density were accepted for quantitative evaluation; those outside this limit were repolished. Experimental values of NL and TAnet obtained from unetched microsections were used to calculate the surface area per unit volume, SV, and the average value of mean curvature, H, of the void-solid interface according to the relations NL = S (1) and TA -- TAnet H = rr (2) NL SV, H and TAnet data for the 48, 30 and 12 micron particle sizes are compiled in Tables 4, 5 and 6, respectively. They are plotted as a function of density in Figures 10, 11 and 12, and 14 through 19. Since NL is simply related to S it has been omitted from the tables and figures. 1 1 Mean curvature is defined by H = (- + -), where r and r r1 r2 1 2 are the principal normal radii of curvature. Its average value over Sf HdS the surface is defined [41] by H- ; where the integration is i dS performed over the surface, S. Table 4. Quantitative metallography data for the 48 micron particle size as a function of density and temperature Density Temperature SV TAnet H (gms/cm3) (OC) (cm-1) (cm-2x0-2) (cm- 5.90 950 458 340 466 5.98 950 520 225 272 6.57 1000 460 470 642 6.90 1000 362 545 946 7.02 1000 356 572 1010 7.16 1050 330 521 1011 7.55 1050 250 473 1189 7.60 1050 226 395 1098 7.60 1050 235 460 1230 7.73 1050 218 450 1297 8.18 950 137 320 1468 9.0 0 9500C O 10000C 8.0 / 10500C S7.0 6.0 0 0 200 400 600 800 -1-1 SV (cm- ) Figure 10. Variation of surface area per unit volume with density for the 48 micron particle size spherical copper powder. Table 5. Quantitative metallography data for the 30 micron particle size as a function of density and temperature Density Temperature S, TAnet (gms/cm ) (OC) (nc- ) (cm-2x102) (cm-) 6.48 6.70 6.94 6.97 7.36 7.93 8.00 8.10 8.20 8.23 8.26 8.37 8.40 1000 1000 1000 1000 1000 1000 1000 950 1000 1050 1000 1050 1050 940 860 664 830 614 366 322 314 230 136 248 164 126 2520 2232 2793 2280 2710 1382 1604 1206 1511 1192 896 1196 685 1671 1631 2643 1726 2763 2373 3130 2413 4128 5507 2207 4582 3416 8. ' 7.0 S [Q 10000C [ ] A 10500C 6.0 0 200 400 600 800 1000 SV (cm1) Figure 11. Variation of surface area per unit volume with density for the 30 micron particle size spherical copper powder. Table 6. Quantitative metallography data for the 12 micron particle size as a function of density and temperature Density Temperature S TAnet H (gmis/cm3) ( C) (cm-1) (cm-2x 10-) (m- ) 6.16 1000 1778 2560 905 6.59 1000 1624 6570 2542 7.13 1000 1120 6080 3473 7.26 1000 1210 6190 3241 7.68 1000 744 3830 3234 7.82 950 740 4720 4008 .8.17 1000 470 2960 3957 8.22 950 430 2550 3726 8.52 1000 214 2560 7576 8.59 1050 210 1420 4249 8.68 1000 122 930 4790 8.0 7.0 6.0 600 1200 SV (cm-1) Figure 12. Variation of surface area per unit volume with density for the 12 micron particle size spherical copper powder. 1800 8.0 S12 p -A- 7.0 30 p 48 P 6.0 600 1200 1800 S (cm- ) Figure 13. Comparison of the linear relationships between density and surface area per unit volume for the 48, 30 and 12 micron particle sizes. 3.21 Surface area of the void-solid interface It is evident from Figures 10, 11 and 12 that the relationship between SV and density is linear for each particle size. The lines drawn through the data for each particle size are presented for compar- ison in Figure 13. Their extrapolation to zero surface area produced reasonable agreement with the theoretical density of copper; maximum deviation was found for the 30 micron particle size for which extrap- 3 olation yielded a value of 8.8 gms per cm as compared to the theoret- ical density of 8.94 gms per cm . The S data was collected from specimens sintered at each of the three test temperatures, 950, 1000 and 10500C. The variation of temperature over this range had no effect on the surface area-density relationship. 3.22 Area tangent count The algebraic sum, TAnet, of the number of tangents per unit area with convex (TA+) and concave (TA_) segments of the void-solid interface, as determined as a function of density for the three particle sizes, is presented in Figures 14, 15 and 16. It is apparent that TAnet was also insensitive to changes in test temperature. For the loose stack of particles, all surfaces are convex (as previously defined) and TAnet must therefore be positive. As contacts form between particles, concave elements are created and with the con- tinued evolution of the geometry accompanying densification TAnet passes through zero and becomes negative. As evidenced by Figures 14, 15 and 16, TAnet was found to be negative over the entire range of densities for which the sintering force was determined. The functional dependence 6.5 7.0 Density (gms/cm3 ) 7.5 8.0 -l- - O 950C ] 1000C A 1050C X 10000c* Figure 14. Variation of the net area tangent count with density for the 48 micron particle size spherical copper powder. (*Obtained from specimen sintered without external constraint.) 6.0 0 x a u H 9.0 -200 -400 -600 -800 Density (gms/c3 ) v. .u 8.U 9.0 -1000 M H o OeA CN Q . -, -tfX l S -2000 oo - S- 0 9500C a ] o ooo00c -3000 D 10500C -4000 Figure 15. Variation of the net area tangent count with density for the 30 micron particle size spherical copper powder. r 0 7 n - - Density (gms/cm3 ) 7.0 7.5 8.0 I i i 9.0 l 0 O / El/ O'EI WY -4000 L -10 0 950C 0 1000C A 1050C Figure 16. Variation of the net area tangent count with density for the 12 micron particle size spherical copper powder. +2000 -2000 6.0 I 6.5 1 N <-1 0 CN C4 -6000 L -8000 _ _I 1 I a-6 of TAnet on density was similar for each particle size; increasing to a maximum (negative) value in the region of 75 SO per cent of theo- retical density and then decreasing toward the zero value which must exist when densification is complete. 3.23 Average mean surface curvature Average values of the mean surface curvature, H, are presented in Figures 17, 18 and 19 for the 48, 30 and 12 micron particle sizes, respectively. Since H derives its sign from TAnet, the average values of the mean surface curvature were also negative for all sintering force specimens. In addition, since NL and TAnet were used to calculate H, H was also insensitive to changes in test temperature. The general shape of the curve depicting the mean surface curvature-density rela- tionship indicates that H progresses from positive to negative values in the region of 65 70 per cent of theoretical density (approximately 6.0 gms/cm ). The similarities between the functional dependence of i on density and that exhibited by the sintering force (in Figures 7, 8 and 9) are striking and a most significant discovery in terms of the relation- ship between the microstructure of a sinter body and its sintering behav- ior. Further elaboration of this point may be found in Chapter IV. 3.24 Effect of uniaxial constraint on sintering behavior It was reasonably expected that the effect of uniaxial loading on a sinter body, attempting to shrink isotropically, would distort the microstructure of the specimen and in so doing produce artificial behav- ior in the development of the geometric properties. Indeed, it was Density (gms/cm3 ) 6.0 7.0 8.0 9.0 O 9500C ", 1000oC S -500 \ 10500' -1000 CD -1500 Figure 17. Variation of the average mean curvature of the void-solid interface with density for the 48 micron particle size spherical copper powder. Density (gms/cm3 ) 7.0 8.0 1 -% El El- - El I A Figure 18. Variation of the average mean curvature of the void-solid interface with density for the 30 micron particle size spherical copper powder. +1000_ 6.0 -1000 9.0 -2000 -3000 O 950C O 1000 C Z 10500C -4000 -5000 _I _ 1 __ _ Density (gms/cm3 ) 8.0 EL. o~- fE O 9500C 0 10000C L iooo00 A 10500C Figure 19. Variation of the average mean curvature of the void-solid interface with density for the 12 micron particle size spherical copper powder. +2000 6.0 -2000L 9.0 -4000 -6000 . -8000 YI I I I observed, in the very beginning of this research, that extreme over- loading above the force against which the specimen could shrink produced internal necking and severe distortions of the void-solid interface. However, as long as the overload was maintained close to the minimum needed to produce creep of the specimen, it was impossible to detect any effect of the external load on the geometric development of the micro- structure. The veracity of this statement has been experimentally sub- stantiated. For example, NL measurements which were made parallel with, perpendicular to and at a 450 angle to the direction of external load revealed no statistically significant differences from the values meas- ured without respect to specimen orientation. In addition, the slopes of the S -density plots (see Figure 13) were found to correlate with the average particle diameter in the same way as previously reported by DeHoff et al. [38] for unconstrained sinterings made from the same powder material; this correlation is presented in Figure 20. In Figure 21, the cross-sectional areas calculated from the diameters of the specimen after testing are presented as a function of density. The upper boundary in the figure represents the cross- sectional area which would develop during unconstrained isotropicc) shrinkage. The lower boundary is that predicted by assuming that external constraint is applied in a fashion which limits all shrinkage to the radial directions of the cylindrical specimen. Essentially all specimens fall within these boundaries, indicating that the overload- ing of the specimens used either initially in straightening the speci- men in the load column or in determination of the balance point between S Obtained from sintering force specimens O Obtained from uncon- strained sinterings [38] 20 40 60 D (microns) o 80 100 120 Figure 20. Correlation of the slope of the surface area-density relationship with initial particle size for spherical copper powder. .000 .002L .004 .006 ,008L CM4 N U) E 0 0 c-4 M -.010 -.012L _ ___ I I 1 I ? AA 2 AV _ sotropic shrinkage; A 3 V AA AV = -V; radial shrinkage only Average A A \QN NQ N. N 7.0 7.5 8.0 Density (gms/cm3) A A A 8.5 9.0 Cross-sectional areas of sintering force specimens after testing. The areas predicted by isotropic shrinkage lie on the upper boundary; those which should result from radial shrinkage only lie on the lower boundary. .1600 .1500 A 1400O ] 48 A 30 0 12 microns microns microns .1300 .1200 .1100 \ ( Figure 21. I __ _ __ L _ _ 57 elongation and shrinkage, did not unduly distort the specimen macro- scopically. The scatter in the cross-sectional areas was produced by variations in the fraction of the total time at temperature during which the specimens were under external constraint. CHAPTER IV DISCUSSION The results of this investigation have revealed the sintering force to be an experimentally measurable and reproducible property whose functional dependence on density, particle size and temperature are consistent with and relatable to the geometric evolution of the sintered structure. To analyze these results, it is necessary to consider the general relationships between surface geometry and sur- face tension forces and the details of the geometric evolution of the void-solid interface during densification. A method for estimating the magnitude of the surface tension forces in a sinter body will be formulated from these considerations and compared with experimental values of the sintering force. The ultimate objective of this research was to establish, through the concept of the sintering force, a relationship between the microstructure of the sinter body and its sintering kinetics. Sensitivity of the rate of sintering to the geometry of the micro- structure will be demonstrated through the conjoint application of an empirical treatment of the mechanical behavior of the sinter body and a stress-strain rate relation based on deformation by creep. 4.1 Surface Energy and Surface Tension of Solids Specific surface energy, y, is defined as the energy necessary to increase the surface by unit area, and has units of energy per unit area. Surface tension, C, is the force which resists stretching of the surface; it has units of force per unit length and may be considered to act in the surface, perpendicular to any line drawn in the surface. The equality of surface energy and surface tension in liquids has been documented by many experiments in liquid capillarity. Gibbs [49], in his rigorous formulation of the thermodynamics of interfaces, pointed out that this equality does not always exist in solid surfaces. Shuttleworth [50] has developed a general relationship between surface energy and surface tension which reveals the conditions under which these two surface properties have the same magnitude. This relation- ship may be developed, following Shuttleworth's arguments, in the fol- lowing way. Imagine a crystal surface cut by a plane perpendicular to it and penetrating only a short depth below the surface. To maintain the surface on both sides of the cut in equilibrium, it is necessary to apply, in the plane of the surface, equal and opposite forces to each side of the cut. For anisotropic surfaces, the total force per unit length of the cut is defined as a surface stress which may vary with the orientation of the cut with respect to the atomic arrangement of the crystal surface. Consider the deformations dA1 and dA2 of the segment A of the solid surface in Figure 22, by forces working against the surface stresses C0 and 02. If these deformations are performed reversibly and isothermally, then the work done against the surface stresses will be equal to the increase in the total (Helmholtz) free energy, l dA1 + G2dA2 = d(Ay) (3) 1 S dA A dA i-- 2 Figure 22. Schematic of surface stresses in a solid. In the absence of surface anisotropy, as in a liquid, the surface stresses, aC and C2, are equivalent and may be considered as a single value, a surface tension, which characterizes the surface. Under these conditions equation (1) reduces to a = y + A() (4) For a unary system, at constant temperature and volume, in which y is independent of changes in area, y = dF/dA, where F is the Helmholtz free energy. For systems in which Y is a function of area, the total change in free energy will be d(yA). In the deformation of any surface, liquid or solid, where the mobility of the atoms or molecules is sufficiently high to maintain the original surface density of atoms or molecules throughout the deformation, () dA will be zero, and surface tension and surface energy will be equivalent in magnitude. In liquids, this mobility is easily obtained; in solids, it may be reasonably assumed that the conditions required exist dur- ing slow deformation of surfaces at temperatures close to the melting point where atomic mobility is high. Since these conditions were satisfied by the testing procedure in this research, it has been assumed that surface energy and surface tension are equivalent in magnitude. In keeping with this assumption, the symbol for surface energy, y, has been used interchangeably for surface tension and surface energy. In addition, it has been assumed that the random, polycrystalline nature of the sintered structure assures a statistical distribution of all orientations of the crystal surfaces at the void-solid interface which obviates any corrections for anisotropy of surface tension. 4.11 Pressure difference across a curved surface It has been demonstrated that for a surface of zero curvature, the surface tension can be balanced by applying external forces par- allel to the surface at its periphery. The experimental application of this principle will be discussed in the next section. However, for curved surfaces, surface tension forces result in pressures normal to each point of the surface. The magnitude of this normal pressure may be related to the local curvature of the surface in the following way. Consider the square segment of surface shown in Figure 23, with 62 n S 2 e2 I r an Figure 23. Resolution of surface tension into a pressure acting normal to each point of a curved surface. sides of arc length s and normal radii of curvature r1 and r2. From the edge view it is apparent that the components of surface tension parallel to the surface normal are equal on opposite sides of the 9 element. If the element is extremely small, then sin is approx- imately equal to -, and the forces have total components (parallel 2 2 to the surface normal) of Y- and Y-- The difference in pressure, rl r2 AP, between the volumes separated by the surface is the normal pressure at each point of the surface. The normal component of force resulting 2 from this pressure is Aps Therefore, the condition for mechanical equilibrium of the surface is AP = y( + ) (5) rl r2 Thus, the effect of surface tension on an arbitrarily curved segment of surface can be analyzed in terms of the geometry of the surface. L.12 Measurement of surface tension effects in solids Surface forces are small in magnitude and are negligible when compared to the loads required to overcome the elastic strength of solids at normal temperatures. However, as discussed above, their effects become noticeable at high temperatures. The effect of surface tension in solids was first reported by Faraday [51] who observed that thin metal sheets would shrink when heated. However, it remained for Chapman and Porter [52] to attribute this effect to the presence of forces in the metal surface. Schottky [53], Sawai and Nishida [54] and Tammann and Boehme [55] were among the first to use the shrinkage of metal foils to determine values of surface tension. They employed the method, originally suggested by Gibbs [49], of oppos- ing the shrinkage of the foil with known external forces. More recently, Udin et al. [12] measured the surface tension of copper by determining the external load required to stop the shrinkage of fine wires. Since the sintering force, as defined in this research, embodies the same principle of a balance of forces it is appropriate to consider the analyses of the foil and wire experiments. The mechanical conditions for balancing the shrinkage of foils with external loads have been reviewed by Fisher and Dunn [56]. The simplest geometrical arrangement for balancing the surface tension of a foil is presented in Figure 24(a). The top and vertical sides of the foil are constrained while the remaining edge is free. However, the two vertical sides of the foil are allowed to slide up and down without friction. If the thickness, t, of the foil is neglected, the external load which just balances the surface tension forces may be calculated from the relation F = 2yw (6) where y is the specific surface energy (tension) and w is the foil width. However, these experimental conditions cannot be achieved. Figure 24(b) shows the more accessible situation in which the vertical sides are also free. In this condition the horizontal shrinkage of the foil will contribute to the length changes in the vertical direc- tion and requires a different analysis. If it is assumed that a con- dition of plane stress exists, then the stresses in the foil are (b) I F Figure 24. Two arrangements for balancing the shrinkage of metal foils by inducing creep with an external force. ~i~ C_ F 2yw V wt and (7) a = 2 h t where av and oh refer to the vertical and horizontal directions, respectively. Plastic strain in the vertical direction may be esti- mated by S- [v 2(h)] (8) where E is the plastic modulus and Poisson's ratio has been assumed to be 1/2. If it also assumed that the strain rate, E in the vertical direction is proportional to the stresses producing strain in this direction, then the stress state for e = 0 under external load is v 1 v =- T h (9) which, on substitution, reduces to F = yw (10) For the circular cross section wire used by Udin et al. [12] the same principles apply; however, the stress state is now three- dimensional. For zero longitudinal strain rate in the wire, the assump- tion of constant volume results in zero radial strain rate. These con- ditions are consistent with the existence of a hydrostatic stress state. Therefore, the conditions for balance under the external force are a F 2n ry = = (11) 2 r r rrr or F = nry (12) where r is the radius of the wire cross section, and L and U are Sr the longitudinal and radial stresses, respectively. Udin [57] cor- rected equation (12) by deriving a new relationship based on the ener- getics of the process. Included in this analysis was the effect of grain boundaries within the wire. As shown in Figure 25, the grain boundaries were aligned perpendicular to the cylindrical axis of the wire in a configuration referred to as "bamboo" structure. For a reversible process, the change in potential energy of the external load must be balanced by the total energy change in the wire. If the latter is composed only of changes in grain boundary and external sur- face area, then Fdt = ydA + ydA (13) where the prime notation refers to the grain boundaries. If it is assumed that the changes in length and radius of the wire are related by a Poisson's ratio of 1/2, equation (13) becomes F = rrry nn r y' (14) where n is the number of grain boundaries per unit length of the wire. The grain boundary effect -in Udin's data changed the experimental value of the surface tension of copper at 1000C from 1430 to 1670 dynes per cm; an increase of approximately 17 per cent. The use of foil geometry has been critized due to its complicated grain boundary configuration and the assumptions required to analyze its state of 2 2r Grain Boundaries External Force Figure 25. Schematic of surface tension forces and grain boundary array in wire geometry used for determination of surface tension in metals. stress. The few values of the surface tension of metals which have received general acceptance have been determined from the wire geo- metry. It has been demonstrated that the effect of surface tension forces in solids can be experimentally analyzed through knowledge of the geometry of the surface. The shape of the void-solid interface in a sinter body is extremely complex; consequently, it is necessary to develop some understanding of its geometry before attempting to analyze the surface tension forces which exist within it. The analy- sis which will then be developed will not include an effect of grain boundaries. It has been assumed that the effect of grain boundaries on the experimental values of the sintering force is subordinate to the effect of the void-solid interface and within the scatter of the experimental results. This assumption is based on preliminary meas- urement of the ratio of grain boundary to void-solid surface area in the specimens used in this investigation [58], and also consideration of the fact that the energy of grain boundaries in copper is approximately 1/3 of the energy of a free copper surface [59]. 4.2 The Geometry of Sintering The degree to which one may describe the complex structural changes in a three-dimensional sinter body, which occur during its progression frcm a loose particle stack toward complete densification, has been summarized by DeHoff and Rhines [60]. They pointed out that it is now possible to determine experimentally the following geometric parameters of a sintered structure: (1) total void volume, (2) total area of the void-solid interface, (3) average mean curvature of the total void-solid interface [41] (see Appendix II), (4) connectivity of the pore network (expressed as the genus of the void-solid interface, (see Appendix I), and (5) the number of separate parts of the void volume. The first three of these parameters are sensitive to the dimensions of the system; the last two are topological parameters, insensitive to dimension but necessary to characterize the topological shape of the system. Experimental evaluation of these parameters has enabled Rhines and DeHoff to deduce a reasonably complete and comprehensive descrip- tion of the structural evolution of a sintering system. The application of descriptive topology, as developed by Rhines [1], separates the structural evolution into three stages. These sepa- rations are based on the variation in connectivity of the void phase with increasing density. Experimental evaluation of this parameter for the 48 micron copper powder [60] is presented in Figure 26. The forma- tion and growth of interparticle contacts constitutes the first stage of sintering. In this stage the independent processes of surface round- ing (primarily by surface or vapor transport) and shrinkage should pro- ceed without affecting the connectivity of the pore network. However, as evident in Figure 26, the connectivity increases during the first stage. This behavior is the result of the formation of "bridges" in the loose particle stack. Particles in these regions, originally not in contact, are brought together by the shrinkage of the particle framework, thereby increasing the total connectivity of the void phase. The contributions to the decrease in total surface area by surface and vapor transport, and by shrinkage, continue to be independent until the minimal surface area is achieved for the existing values of volume and connectivity of the void phase; this signals the end of the first stage. Further decrease in surface area requires a new set of values of connectivity and total pore volume. Rhines (see Appendix I) has demonstrated that the system can progress to lower values of the min- imal surface area through continual decreases in the connectivity of the void volume. There exist two experimental findings which support the minimal area concept in the second stage. First, the relation between the density of the system and the surface area it contains in unit volume is linear throughout the second stage. Second, the onset of this linear relation coincides with the beginning of a strictly monotonic decrease in the connectivity of the pore network with in- creasing density; as shown in Figure 26. This decrease in connectivity is produced by closure of the channels in the multiply-connected void volume. The criterion for closure of a channel, in terms of local instability of the void-solid interface, is revealed to a degree by the linear relationship between the density and surface area per unit volume; DeHoff and Rhines [60] have concluded that this behavior is indicative of channel closure events which remove elements of the void phase having a fixed ratio of volume to associated surface area. This conclusion is consistent with their previous finding [38] of a constant First Second Third stage stage stage 5xl06 4x1O06 SGenus 6 3x10 E l S2x106 Separate parts 1x106 IxlO6 P C, 6.0 7.0 8.0 8.94 Dens ity (gns/cm3) Figure 26. Variation of genus (connectivity) and number of separate parts of the void-solid inter- face with density for the 48 micron special copper powder [43]. value of the mean lineal intercept of porosity in the second stage of sintering. Channel closure continues to be the dominant structural change throughout the second stage. Eventually the pore network is no longer multiply connected and the genus is then equal to zero. In the third stage, the number of separate parts of the void volume is in- creased through further division of the complex but simply-connected "tree-like" [60] cavities by channel closure. The number of separate parts decreases from the maximum value primarily through a reduction in number resulting from conglobation with other portions of the void volume. 4.21 The evolution of curvature of the void-solid interface To determine the relationship between the surface tension forces and the microstructure of the sinter body, it is necessary to consider the information available on the evolution of the curvature of the void-solid interface. Constant reference by the reader to Fig- ure 26, which depicts the three stages of sintering and to Figure 27, a compilation of the average mean curvature values, H, for the three particle sizes used in this investigation will serve to clarify.the following discussion. In the initial loose stack of particles, the curvature of all void-solid interfaces is positive, by definition. The average value of Mean lineal intercept, is the average length of all lines traversing the phase of interest and is calculable from Fullman's [51] relation, pertain to the phase of interest. relation, \ = 4 where V and S pertain to the phase of interest. V +3000 12 microns S---- 30 microns +2000 -- 48 microns +1000 _ \ Density (gms/cm3) 5.0 6.0 7.0 8.0 8.94 S -1000 -2000 -3000 -4000 -5000 Figure 27. Comparison of the average surface curvature as a function of density for the 48, 30 and 12 micron particle sizes of spherical copper powders. mean surface curvature of the loose stack may be estimated by = ( + -) 2 4 (15) rl r2 r d where d is the average particle diameter for each powder size. These calculated values were used to extend the curves in Figure 27 from the experimentally determined region (approximately 6.0 to 8.3 gms/cm3) to the estimated loose stack density, 5.3 gms/cm3. The formation of permanent contacts, or weld necks, between the particles introduces negative curvature into the system and creates curvature gradients between the nack region and the positively curved surface of the nearly spherical particles. The magnitudes of these gradients control the rates of vapor and surface transport which smooth the surface and reduce the surface area in the system. The amount of negatively curved surface in the system increases with further densification and the average value of mean curvature drops rapidly through zero and becomes 3 negative in the density range of 5.8 to 6.0 gms/cm Average mean curvature becomes an increasingly large negative value throughout the remainder of the first stage. The slope of the average mean curvature- density curves in the first stage, in Figure 27, is related to the change in the magnitude of the curvature gradients; as the gradients dissipate with increasing density, the slope decreases. This effect is evident also in the change in the slope produced by a change in the particle size. Curvature gradients are steeper in a fine particle size and the slope is correspondingly greater. The end of the first stage is revealed in the average mean curvature-density relationship by an almost abrupt decrease in the slope of the curves in Figure 27. This implies that the effective- ness of the remnants of the original curvature gradients has been considerably reduced and is consistent with the attainment of a min- imal surface area at the end of the first stage. The absolute minimum value of surface area possible for a given volume of porosity would exist in a system containing a single sphere. However, Rhines [1] has demonstrated that the shape changes in the first stage are localized and prevent such a structural development. This is due to the fact that the curvature gradients change the direction of material transport by vapor and surface mechanisms within distances approximating half a particle diameter. Under this limitation, the system can minimize its surface area, for a given void volume, by separating the void phase into separate spherical pores. In order to achieve this configura- tion the channels in the pore network must be closed. Therefore, the coincidence of a decrease in connectivity, a linear relationship between density and surface area per unit volume, and a decrease in the rate of change of surface curvature is evidence of a decrease in the abil- ity of curvature gradients to promote shape changes in the pore net- work. If this were not true, tnan further decrease in surface area per unit volume could occur without requiring a decrease in connectiv- ity, and the rate of change of curvature with increasing density would not be so diminished by the end of the first stage. The evolution of average surface curvature is also sensitive to the variation in connectivity of the pore network during the second stage. As shown in Figures 17, 18 and 19, a rise in curvature occurs during the last half of the second stage. The rate of decrease in con- nectivity may be seen to diminish in the same range of density in Figure 26. According to DeHoff and Rhines [60], a channel can decrease in size until it attains the set of critical dimensions which render it unstable and make it collapse. The surface rounding processes of the first stage produce some channels which can close early in the second stage. The remaining channels must decrease in size through shrinkage of the solid framework. This increases the average curvature of the remaining channels and therefore of the system. The separate parts developed by the channel closure process reveal their inability to shrink by decreasing their surface area through coarsening [63] in the third stage. This process is considered to be responsible for the occurrence of a maximum value of average curvature. Thus, the evolu- tion of average mean surface curvature, in agreement with the other geometric parameters which have been measured, strongly indicates that after an initial period (the first stage) of adjustment in the shape of the void-solid interface, the system proceeds along a path of geo- metric evolution consistent with tne concept of a series of minimal surface area configurations. The instantaneous values of the minimal surface area are conditioned on the values of the remaining volume and connectivity of the void phase. Furthermore, and more important to this research effort, is the evidence that the average value of mean surface curvature should be indicative of the average value of the shrinkage forces in the system. The veracity of the latter state- ment is substantiated in the obvious similarities in the variation with density of the average mean curvature (Figures 17, 18 and 19) and of the sintering force (Figures 7, 8 and 9). The scatter in H data and the limited number of values do not clearly reveal the max- imum in average mean curvature implied by the sintering force results. However, a maximum in average curvature has been reported by DeHoff and Slean [62] to occur at a density of 7.9 gms per cm in an electro- lytic copper powder; this is shown in Figure 28. 4.3 Analysis of the Sintering Force On the basis of the geometric arguments presented in the pre- vious section it will be assumed in this analysis that the average surface tension forces in the system may be calculated in terms of its average value of mean surface curvature. Although it is true that the external force applied may not balance the internal forces at every location in the system, it is assumed that it does so on the average. Using these assumptions, the average pressure, P, acting to compress the void phase can be calculated from the basic equation P = yHi (16) If it is further assumed that this pressure is distributed throughout the system without dilution by the solid phase, then the force com- ponent of this pressure which acts perpendicular to an area A of the sinter body is equal to yHA. The average force, F, which counteracts the effect of the external load is given by Density (gms/cm ) 6.0 7.0 -I000L -2000 -3000 Figure 28. Variation of average surface curvature with density for a mixture of sizes of an irregular, dendritic, electrolytic copper powder [62]. +1000 8.0 V ---- ~- -r`l--------T- F = HA (17) where A is the average cross-sectional area perpendicular to the direc- tion of loading. This relation would be directly applicable to a sys- tem composed of porosity separated by very thin interfaces; e.g., a soap froth. An alternative, and more realistic approach, is to consider a simple balance of forces between the external load and the sun of all force components emanating from the void-solid interface in a direc- tion parallel to the applied load. For example, if the desired compo- nent of force per unit length is F. and is single-valued over the length, 1., of the line of intersection between the void-solid inter- face and the plane perpendicular to the applied load, then the total resolved force would be equal to the sun E F.I. taken over the total i L i length of all such intersections. This sun cannot be rigorously calcu- lated from the geometric description of the void-solid interface pres- ently obtainable. However, the general principles of this approach may be demonstrated through the application of a simplified geometry. Con- sider a system containing a homogeneous spatial distribution of spher- ical pores of radius R. On any cross-sectional plane, the circular intersections with these pores will vary in size with the largest having a radius r = R. The force contribution from any single intersection can 2 be calculated in terms of the curvature, R, of the system. As shown in R' Figure 29, the desired force component, F., is constant around the total length of the perimeter of the intersection. Therefore, the total force from this pore intersection, perpendicular to the cross-sectional F. 1i Figure 29. Resolution of surface tension forces emanating from the cut surface of a spherical pore in a direction perpendicular to the sectioning plane. F. plane, is 27 r F. Since sin = r this becomes y()r2 or yHA, where A is the area of the circular intersection. The total force in the system acting across the intersecting plane is equal to the s-um F = yHE A. (18) J J th where A is the area of the j circular intersection. The total area of pore intersections on a cross-sectional plane of area A is equal to AAVA where AAV is the area fraction (equivalent to the volume fraction) of porosity in the system. Therefore, the working form of this rela- tion becomes F = v HAAVA (19) The effect of increase in curvature and decrease in area fraction of porosity (i.e., increase in density) on the sintering force predicted by this relation may be shown as follows. The maximum component of the surface tension occurs when the radius of the circular intersection in Figure 29 is equal to the radius of the sphere; i.e., when the inter- secting plane cuts the diameter of the sphere. Under these conditions e is 900 and the force component is equal to the surface tension, y. Allowing the sphere to shrink is equivalent to increasing the density of the system. However, this decreases the total line length (perimeter of the circle) over which the force, y, exists and therefore predicts a decrease in the sintering force with increasing density. This is contrary to the experimental results which show an increase in the sintering force over the entire second stage of sintering. This reveals the inadequacy of the sphere model in representing the geometrical evo- lution of the multiply-connected pore network in real systems. However, the principle of a balance of forces expressed by the relation derived rigorously for the spherical model is correct. Examination of equation (19) will show that if the increase in curvature, H, is greater than the decrease in the area fraction of porosity, AAV, then an increase in the sintering force, F, is predicted. This is not possible in a system of spheres since their curvature, intersected area and volume relationships are fixed by their shape. However, in real sintering systems, the shape of the pore network is constantly changing and H and AAV can vary somewhat independently. The change in curvature for a system of shrink- ing spheres would be smooth and continuous, but the change in average mean surface curvature for a real sintering system is sensitive to the changes in connectivity of the void network, as is evident by a compar- ison of Figure 26 and Figure 27. Therefore, equation (19) should be applicable to real sintering systems for which the average mean surface curvature, H, and the average area fraction of porosity, AAV on an aver- age cross-sectional area, A, are known. 4.31 Comparison of predicted and experimentally measured sintering forces The sintering forces predicted by equations (17) and (19) are presented as a function of density in Figures 30, 31 and 32, for com- parison with experimentally obtained values. The value of surface ten- sion, y, used for the calculations was 1650 dynes per cm (1.68 gms per cm), the accepted value for copper at 10000C. The temperature coeffi- cient of the surface tension for copper as reported by Udin et al. [12] is approximately 0.50 dynes per cm per C. A variation in temperature of 1000C, corresponding to the range of temperatures (950 to 10500C) investigated in this research, produces a total change in y of only 50 dynes per cm, or approximately 3 per cent. Thus, the effect of tem- perature is well within the normal scatter of the experimental values of the sintering force and was not included in the calculation of the predicted values. The force values predicted by equation (17) are designated as maximum values since they correspond to equation (19) when the area (volume) fraction of porosity is unity; i.e., when the volume of the solid phase is essentially negligible. Although the predictions of equation (17) are, as expected, greater than the experimental values, the qualitative behavior of the two curves as a function of density is strikingly similar. Although qualitative in nature, this observation F = y H AA A F = yHA S- max F exp 6.0 7.0 Density (gms/cm ) Figure 30. Comparison of the experimental values of the sintering force with predicted values for the 48 micron particle size powder. 300[ ^ 200 o a, IooL I i I_ I 8.0 800 F= y AA A max mA 700 ____ F exp 600 500 400 0 300 200 100 1 6.0 7.0 8.0 Density (gms/cm3) Figure 31. Comparison of the experimental values of the sintering force with predicted values for the 30 micron particle size powder. 1000 900 / / / Z 6.0 7.0 Density (gms/cm ) 8.0 Figure 32. Comparison of the experimental values of the sintering force with predicted values for the 12 micron particle size powder. F = v H AA A F Y HA max A Fex exp / 800L 700L 600L 500L 400L 200L 100[ I_ _ definitely relates the sintering force, as defined in this research, to the average value of the mean surface curvature of the sintered structure, and strengthens the assumptions made in regard to this relation. The forces predicted by equation (19) are in better agreement with the experimental values; however, the two curves do not possess the same shape. Although the predicted curve rises rapidly from zero at zero average curvature to the proper level of the sintering force, it begins to decrease early in the second stage of densification. This discrepancy can be qualitatively reconciled in the following way. In the experimental measurement of the sintering force the external force is applied to balance the shrinkage of the system. Therefore, elements of the void phase which contribute the most to the overall shrinkage are primarily responsible for the level of external load required to balance the shrinkage. The values of average surface curvature and area fraction should be more heavily weighted in favor of these elements of the void phase. The recognition of these elements on a two-dimensional microsection cannot be unambiguously performed. It may be assumed that the smaller pores on a two-dimensional section are those which will continue to shrink, or if they are intersections with channels, to collapse, but a quantitative delineation on this basis must be considered arbitrary. It is obvious that division of the void volume into portions based on their relative contribution to shrinkage will reduce the selected area of porosity below the average value, AAVA. However, the portions which do produce volume changes in |