Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UF00089978/00001
## Material Information- Title:
- Metric and topological characterization of the advanced stages of sintering
- Creator:
- Watwe, Arunkumar Shamrao
- Publisher:
- Arunkumar Shamrao Watwe
- Publication Date:
- 1983
- Language:
- English
## Subjects- Subjects / Keywords:
- Curvature ( jstor )
Distance functions ( jstor ) Grain boundaries ( jstor ) Hot pressing ( jstor ) Material concentration ( jstor ) Nickel ( jstor ) Porosity ( jstor ) Property lines ( jstor ) Sintering ( jstor ) Volume ( jstor )
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- 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.
- Resource Identifier:
- 020303788 ( alephbibnum )
12203626 ( oclc )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

METRIC AND TOPOLOGICAL CHARACTERIZATION OF THE ADVANCED STAGES OF SINTERING By ARUNKUMAR SHAMRAO WATWE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1983 Copyright 1983 by Arunkumar Shamrao Watwe Dedicated To My Parents, Mr. Shamrao Vasudeo Watwe and Mrs. Sharada Shamrao Watwe ACKNOWLEDGEMENTS I am grateful for the opportunity to conduct my research under the guidance of Dr. R. T. DeHoff, the chairman of my advisory committee. An ability to approach any scientific matter with objectivity and logic has been blissfully passed on by him to all his students. I thank Drs. R. E. Reed-Hill, J. J. Hren, G. Y. Onoda, Jr.,of the Department of Materials Science and Engineering and Dr. R. L. Scheaffer of the Department of Statistics for serving on my advisory committee. Their helpful advice and encouragement are deeply appreciated. It is a pleasure to thank my colleagues, Mr. Atul B. Gokhale and Mr. Shi Shya Chang, for their collaboration in the experimental aspects of the project. Mr. Rudy Strohschein, Jr., of the Department of Chemistry assisted me beyond and above the call of duty in the fabrication of the sintering apparatus. He saved me a great deal of time and aggravation. All the credit for the preparation of this dissertation in its final form must go to Miss Debbie Perrine for her excellent typing. The financial support of the Center of Excellence of the State of Florida and the Army Research Office is gratefully acknowledged. TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS........................................ ......... .. iv ABSTRACT....................... ........... ...................vii INTRODUCTION..................................................... 1 CHAPTER ONE EVOLUTION OF MICROSTRUCTURE DURING SINTERING............. 5 Introduction ........................................... 5 Metric Properties of the Microstructure.................. 5 Fundamentals of Topology................................. 6 Sintering from a Geometric Viewpoint..................... 12 Importance of the Present Research...................... 35 TWO EXPERIMENTAL PROCEDURE AND RESULTS...................... 39 Introduction..................... Sample Preparation.............. Metallography.................... Topological Measurements........ THREE DISCUSSION.............................................101 Introduction...........................................101 Loose Stack Sintering................................ ..101 Hot Pressing.............................................130 Conventional Sintering...................................134 Comparison of Loose Stack Sintering with Hot Pressing and Conventional Sintering...............................135 FOUR CONCLUSIONS ............................................137 Introduction............................ .............. 137 Conclusions...........................................137 Suggestions for Further Study...........................139 REFERENCES.........................................................140 APPENDIX THE GEOMETRIC MODEL OF THE PORE PHASE....................143 Introduction.........................................143 .................. .................. .................. .................. PAGE Parameters of the Model .................................143 Metric Properties of the Connected Porosity..............149 Surface Corrections................................ 155 BIOGRAPHICAL SKETCH.............. ............... ............. .. 163 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy METRIC AND TOPOLOGICAL CHARACTERIZATION OF THE ADVANCED STAGES OF SINTERING By Arunkumar Shamrao Watwe- August, 1983 Chairman: Dr. R. T. DeHoff Major Department: Materials Science and Engineering Measurements of the metric properties of porosity and the grain boundary network during the advanced stages of loose stack sintering, conventional sintering and hot pressing of spherical nickel powder (average size 5.5 microns) were followed by topological analysis of the loose stack sintered samples. Linearity between area and volume of the pore phase for the loose stack sintered series was approached by the conventionally sintered and hot pressed series whereas the curvature values for these series remained significantly different. An arrest in grain growth during loose stack sintering was concurrent with the removal of most of the isolated porosity. Subsequent resumption of grain growth coincided with the stabilization of connected porosity. It is suggested that isolated, equiaxed pores pin the boundaries more effectively than do the connected pores. Increase in the boundary area accompanies the boundary migration for all orientations of an equiaxed pore whereas this is true only for a limited number of orientations of a connected pore. Consequently, isolated pores are removed via transport of vacancies to the occupied boundaries; subsequent resumption of grain growth slows the reduction of residual connected porosity. Porosity in loose stack sintered samples is modeled as a set of tubular networks and a collection of monodispersed spheres. Comparison of metric properties of loose stack sintered samples with those of conventionally sintered and hot pressed samples led to the speculations that a higher number of isolated pores exist during hot pressing and that the porosity in con- ventionally sintered samples is composed of finer networks and smaller isolated pores. Absence of an arrest in grain growth during hot pressing is believed to be due to boundary migration that is induced by grain boundary sliding. A similar absence of an arrest in grain growth during conventional sin- tering is attributed to the onset of grain growth well before that of isolation events. viii INTRODUCTION Sintering is a coalescence of powder particles into a massive form,1 wherein the densification is accompanied by a variety of profound geo- metrical changes in the pore-solid composite. The mechanical and physical properties of a powder-processed compact are influenced by the geometry of the pore phase.2-9 Thus, the manner in which the reduction of porosity takes place is of great practical and theoretical interest. There are two approaches to the study of sintering.1 The traditional or mechanistic approach involves the study of kinetics and mechanisms of material transport; the geometric viewpoint focuses on the geometry of the pore phase as it evolves during sintering. The latter approach involves estimation of size- and shape-dependent quantities (volume, area, etc.) and topological properties such as the connectivity and the number of separate parts. Mechanistic studies essentially consist of three steps: 1) A laboratory model of a particulate system is selected that is amenable to mathematical treatment of desired sophistication. Assumptions are made regarding the geometrical changes during sintering and the identities of source and sink of matter. 2) Kinetic equations are derived that describe the variation of a measurable parameter (width of an interparticle contact or "neck," density, etc.) for the particular mechanisms) of interest. 3) These equations are compared with the experimentally observed time dependence of the chosen parameter and an attempt is made to identify the operating mechanismss. The details of the geometry determine 1) the initial and boundary conditions for the flow equations, 2) the areas through which the material fluxes are assumed to occur, 3) the separation between sources and sinks, 4) the relationship between the variation of the chosen parameter and densification, and 5) the state of stress (important in plastic and viscous flow). Hence the time dependence derived in the model studies are influenced by the geometric details. Different mechanisms exhibit different variations with temperature; thus, the relative importance of various mechanisms should depend on temperature and the chemical composition of the powder, as observed in several investigations.1316 As pointed out by DeHoff et al.,11 sintering requires that densification, surface rounding, channel closure and removal of pores proceed in cooperation. Since all these involve different geometric events, the time exponent n in the relation x, the monitored parameter = (t)n varies with the particle size. Any mechanistic arguments must ultimately explain the observed geometrical changes taking place during sintering. It is thus evident that study of the changing geometry or microstructure should precede mechanistic investigations. Knowledge of the dependence of this micro- structure on various process parameters such as initial powder character- istics, temperature and external pressure would be very helpful in the control of sintering aimed at desired end properties of the components. Consequently, a major school of thought prevails that favors the geometric approach. The present investigation was undertaken to study the advanced - stages of sintering (wherein the porosity values vary from ten to a few percent of the total volume) from this point of view. A main feature of this approach is the concept of path of evolution of microstructure. A given microstructure is characterized by its geo- metric properties such as volume, area, curvature, connectivity, etc. A microstructural state is defined as a point in a n-dimensional space where each dimension denotes a particular microstructural property. As the microstructure evolves during a process, the resultant locus of such points represents the sequence of microstructural states that is obtained during the process. This sequence is termed the path of evolution of microstructure.17 It is convenient to represent two-dimensional projec- tions of this path (two geometric properties at a time); usually one of them is the relative density or the volume fraction of solid. Previous studies of microstructural evolution during loose stack sintering,6,12,18-21 conventional sintering-cold pressing followed by sintering6,2223 and hot pressing6,21-23 have provided a coherent pic- ture of these processes for all but the last ten percent of the porosity. A detailed study of the late stages (porosity ten percent or lower) of loose stack sintering, conventional sintering and hot pressing would complete the picture of evolution of microstructure during these pro- cesses. The practical interest in the behavior of porosity at these stages stems from the fact that a variety of commercial products made by powder technology are required to have porosities in the range 0.1 or lower.2 The objectives of this research were to determine the paths of evolution of microstructure during the advanced stages of loose stack sintering, conventional sintering and hot pressing. Since the topologi- cal measurements are time-consuming and since an earlier doctoral research25 dealt with topological characterization of loose stack sintering in the porosity range 0.1 and higher, it was planned to estimate the topological parameters for loose stack sintered series only. Metric properties of the pore structure and grain boundary network were estimated for all three series of samples. Previous investigations of this type are discussed in Chapter 1, followed by experimental procedure and results in Chapter 2. These results are discussed in Chapter 3 and the important findings and con- clusions summarized in Chapter 4. CHAPTER 1 EVOLUTION OF MICROSTRUCTURE DURING SINTERING Introduction A microstructure is characterized by its metric and topological properties and therefore the following discussion will be carried out in terms of variation of these quantities as the sintering proceeds. These microstructural properties will be defined and the previous investigations of this type will be discussed in detail; a review of metric studies will be followed by topological analyses. The principles of quantitative stereology employed in the estimation of microstructural properties will be described in the next chapter on experimental procedure and results. Metric Properties of the Microstructure These quantities are estimated in terms of geometric properties of 26 lines, surfaces and regions of space averaged over the whole structure.2 The basic properties are listed in Table 1 and illustrated in Figure 1. Among the properties listed, VV, SV and MV are used to yield two impor- tant global averages of the microstructural properties. These are listed in Table 2. In a sintered structure, there are two regions of space or phases, namely, pore and solid, and two surfaces, pore-solid interface and grain boundaries. Two main linear features of interest are the grain edges and the lines formed as a result of intersection of pore-solid interface and grain boundaries. Superscripts are used to identify the properties that are associated with a particular feature. These notations are listed in Table 3 and illustrated in Figure 2. In addition to the metric properties listed above, the microstruc- ture of a porous body is also characterized by its topological properties. A brief discussion of the fundamentals of topology will precede the sur- vey of microstructural studies of sintering. Fundametals of Topology The subset of topological geometry of present interest is that of closed surfaces,24 that is to say, surfaces that may enclose a region of space. In a sintered body the regions of space are the pore and the solid phases; the pore-solid interface is a closed surface of interest. Such a surface may enclose several regions and have multiple connectivity. A surface is said to be multiply connected if there exist one or more redundant connections that can be severed without separating the surface in two. The genus of such a surface is defined as the number of redundant connections. For complex geometries it becomes difficult to visualize the topological aspects of surfaces. It has been found very convenient :to represent surfaces by equivalent networks of nodes and branches. Such an equivalent network is called the deformation retract of a particular region of space. It is obtained by shrinking the surface without closing any openings or creating new openings,27 until it collapses into the said network that can be represented in the form of a simple line drawing.28 A number of closed surfaces and their equivalent networks are illustrated in Figure 3. The connectivity, P, of a network is equal to the number of nearr Features of Length L L = L V V Sectioning Plane of Area A L _7 A 4 Lines of Intersection of Length L Element of Surface H ( + ) 2 r1 r2 Figure 1. Illustration of basic metric properties. Volume V Table 1 BASIC METRIC PROPERTIES Property Definition LV Length of a linear feature per unit volume. S,, Area of a surface per unit volume. 1 1 1- 2 rl r2 MV =f/ HdS SV Local mean curvature of a surface ata point on the surface, where rl and r2 are the principal radii of curvature. By convention, a radius of curvature is positive if it points into a solid phase. Thus, a convex solid has a positive curvature whereas a convex pore has a negative curvature. Integral mean curvature of per unit volume. a surface Length of a trace of surface per unit area of a plane section. Volume fraction or volume of a particular region of space per unit volume. Regions of Space Feature Lines Surfaces 9 Table 2 DERIVED METRIC PROPERTIES Feature Property Definition MV Surface H = Average mean curvature of SV a surface 4V 4V Mean intercept in a particular Region of Space of space Sv region of space Table 3 METRIC PROPERTIES OF A SINTERED STRUCTURE Property Description Pore-Solid Interface Porosity Solid Phase Grain Edges in the Solid Lines Formed by the Intersection of Grain Boundaries and Pore-Solid Interface Grain Edges Occupied by the Pore Phase LSSS V LSSP V LSSS V(occ) Area of pore-solid interface per unit volume Integral mean curvature of pore-solid interface per unit volume Volume fraction of porosity Volume fraction of a solid Length of grain edges or triple lines per unit volume Length of intersection lines of pore-solid interface and grain boundaries per unit volume Length of occupied grain edges per unit volume Feature Grain Boundaries of Area SSS SS SS SSS LSSS L(occ) LSS (occ) V V SSS v L V V SSS edge SSP lines Figure 2. Illustration of metric properties characterizing grain boundaries, pore-solid interface and their association. branches that can be cut without creating a new isolated part. If b = number of branches, n = number of nodes, Po = number of separate parts, then P1 = b n + P (1) The first Betti number of the network, P1'29 is equal to the genus of the surface it represents. It may be apparent from Figure 3 that there exists some ambiguity as to the number of nodes and branches in a deformation retract. As illustrated in Figure 4, a number of additional nodes and branches can be used to represent the same region of space. Such spurious branches and nodes do not change the value of the connectivity because each spurious node introduces one and only on spurious branch. Quantities such as connectivity and number of separate parts or subnetworks are estimated by examining a series of parallel polished sections that cover a finite volume of sample, as described in Chapter 2. The investigations dealing with the study of sintering from the geometric viewpoint will be discussed presently. Sintering from a Geometric Viewpoint Three Stages of Sintering Rhines30 and Schwarzkopf31 were among the first investigators to point out three more or less geometrically distinct stages that a sin- tering structure traverses. The first stage is characterized by formation of initial inter- particle contacts and their growth until these contact regions or necks begin to impinge each other, as illustrated in Figure 5. Due to differ- ent crystallographic: orientations of adjacent particles, grain boundaries form in the interparticle contact regions. In this stage, the area of pore-solid interface decreases with a moderate amount of shrinkage.32 Throughout this stage, the pore-solid interface has many redundant con- nections. During the second stage, the distinguishing features are not the interparticle contacts or "necks" but the pore channels formed as a result of the impingement of neighboring necks. Virtually all of the porosity is in the form of an interconnected network of channels16'33 that delineate the solid grain edges. The continued reduction in the volume and the area of porosity is accompanied by a decrease in the connectivity of the pore structure.1,35 The decrease in the connec- tivity can be explained by either removal of solid branches or closure of pore channels. According to Rhines,36 the connected pore network coarsens, analogous to a grain edge network in a single phase polycrystal (driven by excess surface energy) as illustrated in Figure 6. In this scenario, a fraction of solid branches (necks) are pinched off and no new pores are isolated. Although a finite number of isolated pores observed during the late second stage35 can be explained only by channel closure events, a closer scrutiny is needed to resolve this issue. The isolated pores may be irregular in shape.1634 The third stage has begun by the time most of the pores are isolated.30'31 The connectivity of a pore network is now a very small number. Coarsening proceeds along with the spheroidization of pores16'18'3537-39 so that the volume of porosity, the number of pores and pore-solid interface area Figure 3. Some closed surfaces and their deformation retracts (dotted lines). A Closed Surface Deformation Retract Spurious Node Spurious Branch Figure 4. Illustration of a one-to-one correspondence between a spurious node and a.spurious branch in a deformation retract. Figure 5. Illustration of neck growth and impingement of growing necks during the first stage of sintering. continue to decrease. If the pores are filled with a gas of low solu- bility or very slow diffusivity, then coarsening leads to an increase in volume of porosity.16,40 If this gas has enough pressure to stabilize the pore-solid interface, the densification rates can be very low.16,41,42 Since exaggerated or secondary grain growth that results from boundaries breaking away from pores43,44 has been observed to be accompanied by slow rates of shrinkage,43'45-47 it has been theorized that the grain boundaries that can act as efficient vacancy sinks are far away from a large number of pores.43'45-47 The end of the third stage is of course the disappearance of all pores, although that is rarely accomplished in practice. The three stages described above provide a common framework for the discussion of microstructural studies that are reviewed presently. This review is expected to demonstrate the potential that the present research has for providing a perspective of sintering that is more profound than the current one. Metric Investigations It has been observed that in loose stack sintered samples SP P(12,18-20) decreased linearly with the decrease in V(1218-20) during the second stage. Surface area may be reduced both by densification and surface SP rounding or by surface rounding alone; the linearity between SP and VV is believed to arise from a balance between surface rounding and densifi- cation. Support for this hypothesis comes from the observation that sur- face rounding dominates in pressed and sintered samples until the balance has been reached,35 as shown schematically in Figure 7. The slope of the SSP versus VV line is inversely proportional to the initial particle size.20 V V (a) S\ 1I (b) Figure 6. Two basic topological events that occur in the network coarsening scenario proposed by Rhines.36 The dotted lines indicate the occupied grain edges. There is evidence to suggest that this path of evolution of microstruc- ture for loose stack sintering is insensitive to temperature.21 SP Data for hot pressed samples indicate that the SV -Vv relationship 22 23 is only approximately linear even in the late second stage.2223 The path of microstructural change was also found to be insensitive to tem- perature.22'23 The effect of pressure on the path was significant; increasing pressure delayed the approach to linearity until a lower P value of VV, as shown in Figure 8. Integral mean curvature per unit volume, MV, has been measured for loose stack sintering, conventional sintering (cold pressing followed by sintering) and hot pressing in the density range characteristic of late second stage. A convex particle has a positive curvature whereas a con- 1821 ,22 vex pore has a negative curvature. There is a minimum in'MV;182122 35 this minimum occurs at lower VV for finer particle size,3 as illustrated in Figure 9. According to the convention used, most of the "SP" surface has positive curvature in the initial stages. Due to decreasing surface area and increasing negative curvatures there occurs a minimum in MV in the second stage. As the sintered density approaches the theoretical density, MV must approach zero and hence the initially high positive value of MV that becomes negative must go through a minimum. For an initial stack of irregularly shaped particles, MV varies with VP at a N V slower rate and has a minimum earlier in the process, compared to an initial stack of spherical powders.25 This is illustrated in Figure 10. In all the cases studied the paths were insensitive to temperature. In the case of hot pressing, the minimum in MV is much more negative and P 23 occurs at a lower value of VV, compared to a loose stack sintered sample; Schematic representation of the variation of surface area with solid volume for loose stack sintering and conventional sintering. The approach to the linear relation from a range of initial conditions is emphasized.35 SV Figure 7. 14.0 12.0 10.0 8.0 -1 cm1 ) 6.0 0.6 0.8 Pore-solid interface area versus conventionally sintered U0222 solid volume fraction for 0 0 KSI S 1-8 KSI - A 10-20 KSI ~ \ 30-40 KSI O 75-90 KSI 1 -\ I I I I I \ sSP (13 (103 4.0 2.0 0 0 Figure 8. 1.0 ).4 MSP (105 cm2) MV (1 cm ) A (-170+200) Spherical Cu O (-200+230) Spherical Cu D (-270+325) Spherical Cu 0.8 0.9 1.0 Variation of integral mean curvature per unit volume with the volume fraction of solid for three representative copper powders sintered in dry hydrogen at 10050C.35 Figure 9. MP (105 cm2) 0 Spherical - A Dendritic I I I I f I I 0.2 0.5 VV 1.0 Spherical ---------- --- I -IIIi- Dendritic I II III Figure 10. Integral mean curvature versus volume fraction of solid for 48 micron spherical and dendritic copper powder.25 these curves become deeper and shift towards lower VV with increasing pressure,22 as shown in Figures 11 and 12. SS The.grain boundary area per unit volume, SV increases until a SS network is formed; subsequent grain growth tends to decrease SS. 18 This was observed for loose stack sintering,8 as shown in Figure 13. It is evident here that the variation of SS with V is independent of the initial particle shape in the late second and early third stages. LSSP increases with decrease in V until the second stage is reached22 V V SSP when it begins to decrease. In the second stage :L- is significantly higher than the case for random intersection of "SP" and "SS" surfaces,48 as illustrated in Figure 14. A new metric property, IA, was discovered in the course of doctoral research carried out by Gehl22 at the University of Florida; IA is the measure of inflection points observed on the traces of a surface per unit area of plane of polish, and is proportional to the integral curvature of asymptotic lines over saddle surfaces (surfaces that have principal radii of curvature of opposite signs at all points on the surface). This was 22 found to decrease smoothly in the second stage22 which means that the saddle surfaces occupy only a small fraction of the pore-solid interface at the end of the second stage. The variation of grain contiguity, grain face contiguity and grain shapes during conventional sintering and hot pressing were studied in some detail by Gehl.22 There were two parameters, CS and CSS, defined for grain contiguity and grain face contiguity, respectively. Four unitless parameters, Fl, F2, F3 and F4, were used to characterize grain and pore shapes. These were defined as follows.22 0 M SP M (105 cm-2) -1 -2 Figure 11. 0.75 0.80 0.85 0.90 0.95 VV Variation of integral mean curvature per unit volume with the volume fraction of solid during hot pressing of RSR 107 nickel (-170+200) at 1500 psi. Data for spherical copper (-170+200) loose stack sintered at 10050C included for comparison.23.25 MS P V (106 cm-2) Figure 12. The effect of pressure on the path of integral mean curvature for hot pressed specimens of U02O22 -20.0 -40.0 60.0 , 300 200 S SP (cm- 1 - I- I Figure 13. Grain boundary area per unit volume versus volume fraction of solid for 48 micron spherical and dendritic copper powder.2b 1.0 [11 t-^ SS CS V C = S (2) 2SV +S V SSS Css = 3L~_ss SSS SSP (3) 3LV +LV 2LSSP SSS 2Lv +3Lv V V F1 S= P SS2 (4) (Sv +2SV ) F2 = 2LSSP/(SSP)2 (5) SSP SS LV +3L F3 SS (6) 2(SVS)2 SSSP SS)2 (7) F4 = LVP/2(SV)2 (7) The fraction of the total area of solid grains shared with other grains is given by CS. Variation of CS with VP for hot pressed samples of U2(22) and loose stack sintered spherical and dendritic copper powder25 is shown in Figure 15. It can be seen from the definition of S S SS C that high C values indicate high SV ; this was believed to arise from polycrystallinity of the particles. It is apparent that as third stage (VVP 0.1) is approached all data tend to fall on a single curve. Pre- compaction seems to increase SS and hence exhibits higher values of C LSP (104 cm-2) V (10 Figure 14. Variation of the length of lines of intersection of grain boundaries and the pore-solid interface (LSSP) with the corresponding value for the random intersection of the abovementioned surfaces (L ) for spherical copper powder loose stack sintered at 10l5C.48 SS SP TrS SSbS 7V -V 4(1-Vv) 1.0 0.4 Figure 15. The variation of grain contiguity with solid volume fraction for loose stack sintered copper and hot pressed U02.22 P SP at the same VP when compared to a loose stack sintered sample; SV was found to vary linearly with CS and the dependence was the same for widely different precompaction pressures up to very late second stage. For hot pressed samples, a maximum was observed in CS, believed to indicate a point where the grain boundary area has increased enough SSP to form a boundary network that subsequently coarsens. Both LSP and LSSS exhibited a maximum when plotted versus CS for conventionally sintered and hot pressed samples. The grain face contiguity parameter, CSS, indicates the fraction of edge length of grain faces that is shared with other grains. It can be shown22 that CS = CSS for the case of random intersection of grain boundaries and pore-solid interface, and CS > CSS when grain boundaries intersect pore-solid interface preferentially. For conven- tionally sintered and hot pressed samples CS was observed to be greater than CSS which indicated preferential association of grain boundaries with the pore-solid interface. The factors FI, F3 and F4 can be used to compare the grain shapes and F2 the pore shapes; Fl, F2, F3 and F4 were observed to be weakly linear with VV, whereas a strong correlation was observed between F2, F3 and F4 and CS for all the samples. It was believed that the above data indicate a strong influence of the extent of grain contiguity (CS) on the grain and pore shapes. Topological Studies It was found12 that the connectivity or genus, G, stays nearly constant during the first stage. More precise measurements made by Aigeltinger and DeHoff18 indicated a definite increase in G during the first stage. This can be viewed as formation of additional interparticle contacts as particles come closer by densification. For irregularly shaped powders, it was observed18 that G decreases during the first stage, due to coalescence of multiple contacts between particles. SP P During the second stage, S decreased linearly with decrease in VV; the slope of this line was found to be proportional to GV, genus per unit volume20 as should be expected from dimensional analysis. Kronsbein et al.49 carried out serial sectioning of sintered copper samples and found P that even for VV = 0.1, very few pores were isolated. This is in agree- ment with similar observations made by Barrett and Yust.34 Aigeltinger and DeHoff18 studied loose stacking sintering of copper powder by measuring metric and topological properties. The genus per unit mass, Gp, number of isolated pores and number of contacts per parti- cle, C, were the measured topological quantities. Variation of Gp and number of isolated pores with V revealed a definite increase in the former during the first stage and identified the end of the second stage (Gp = 0). As shown in Figure 16, Gp and number of isolated pores were inversely proportional to the initial value of mean particle volume. The same plot for dendritic powder showed that the topological path is different up to late third stage and that the third stage (Gp 0) P begins at a higher value of VV as compared to spherical powder, Figure 17. The initial decrease in Gp during the first stage for dendritic Gp or Np (106 gm ) 0 Gp (48 micron) O Gp (115 micron) x 13.7 * Np (48 micron) * Np (115 micron) x 13.7 0.6 VV 1 0 I -- II II Figure 16. Variation of genus per unit mass (Gp) and the number of isolated pores per unit mass (Np) with the volume fraction of solid for 48 micron spherical copper powder loose stack sintered at 10050C. Data for 115 micron spherical copper included for comparisonn.5 > A Gp or Np (107 gm-1) A 0 0.05 a Spherical Dendritic Spherical Dendritic 0 0 Spherical - --- Dendritic - Figure 17. i 1.0 II "---* II a) Genus per gram (Gp) the number of isolated pores per gram (Gp) versus volume fraction of solid for 48 micron dendritic copper powder. Data for 48 micron spherical powder included for comparison. b) Enlarged part of lower right corner of (a).25 0.6 powder is in agreement with higher C = 14 for the initial stack than C = 4 at VV = 0.55. It has been argued that in second stage, on account of fewer pore channels in the sample sintered from dendritic powder, isolation of pores begins at a higher VV value than for the sample made from spherical powder. The maximum in the number of separate parts observed during the third stage was attributed to simultaneous shrinkage and coarsening. Initiation of rapid grain growth coincided with the approach of connectivity towards zero. Importance of the Present Research Microstructural characterizations of the last stages of sintering where VV goes from about 0.1 to nearly zero have been sketchy. The reasons for such a lack of data are evidently 1) For an aggregate of coarse powder particles that is convenient for serial sectioning, very long sintering times are required to obtain samples with such low values of porosity. 2) For a given range of densities, the paths of evolution of microstructure can be determined with a higher degree of confidence if a larger number of distinct microstructural states can be obtained and examined. Thus, it is desirable to have a sufficient number of samples that have the densities in the range VV = 0.1 and lower; this requires that the samples in the series have VP values that are only a percent or so apart from each other. Due to this requirement and that of long sintering times, much preliminary experimental work is necessary to establish the required sintering schedules. 3) The topological measurements are very tedious in any case. The present investigation that dealt with the microstructural characterization of the advanced stages of sintering has a potential for enhancing and quantifying the existing sketchy picture of the late stages of sintering. The theoretical and practical importance of this work can be appreciated from the following discussion. It has been theorized16'18'35'37-39 that the spheroidization of pores proceeds along with coarsening during the advanced stages. It is necessary to couple topological analysis with the metric measurements to study the spheroidization and coarsening of isolated pores. To date, there has been no such direct observation of the behavior of isolated porosity. If a pore of higher than average size is surrounded by a shell of higher than average density with finer pore channels, then early clo- sure of these channels pulls the solid shell away from the large pore so that the continuity of the solid phase is maintained,34 as illustrated in Figure 18. According to Barrett and Yust,34 most of the reports of coarsening are in fact the observed removal of smaller channels before the larger ones. Another disputed contention is that of deceleration of densification due to separation of grain boundaries from isolated pores.43,45-47 A pore that is observed to be isolated on a two dimen- sional section may or may not be so in the third dimension, whether associated with the grain boundaries or not. The topological analysis of grain boundary-porosity association alone can determine the true extent of association of isolated porosity with the boundaries. A detailed geometric study of porosity in the advanced stages will clarify some aspects of microstructural evolution mentioned above. Figure 18. Illustration of coarsening of a relatively large pore channel that results from an early closure of surrounding finer channels so that the solid conti- nuity is maintained.34 Mechanical and physical properties of commercial porous components are influenced by the geometry of the porosity. Thermal conductivity is influenced by VV, pore shapes and the relative fractions of connected and isolated porosity.7'8 Permeability to fluids depends on the connectivity, VV and S P.9 Mechanical strength and thermal shock resistance3 depend on pore shapes whereas ductility is influenced by pore shapes and spacings.2 Thus geometric characterization of porous structures as a function of adjustable process parameters would suggest a number of potential strate- gies to control the final service properties. It is apparent from the review of previous microstructural studies of sintering that the present investigation is expected to offer a much needed general and quantitative picture of the advanced stages of sin- tering. The experimental procedure employed in the present research is described in detail in the next chapter. CHAPTER 2 EXPERIMENTAL PROCEDURE AND RESULTS Introduction Microstructural characterization involved sample preparation, metallography and in the case of loose stack sintering, also serial sectioning. These are described in detail in this chapter, followed by results of this investigation. Sample Preparation This section presents the procedure employed to prepare the sintered samples and the standard for density measurements. Sintered Samples Three series of samples of sintered nickel powder were prepared: 1) loose stack sintered (LS), 2) pressed and sintered (PS) and 3) hot pressed (HP). In order to study the path of evolution of microstructure during the late stages of sintering, it is desirable to obtain samples having densities that are uniformly distributed over the range VV = 0.85 to 1.0. Accordingly, preliminary experiments were designed to determine the processing parameters, such as temperature, pressure and time, that yield the desired series of samples made from the selected metal powder. INCO type 123 nickel powder, illustrated in Figure 19, supplied by the International Nickel Company, Inc., with the chemical and physical prop- erties listed in Tables 4 and 5, was used in the present investigation. Figure 19. INCO 123 nickel powder used in the present investigation (1000 X). Table 4 CHEMICAL COMPOSITION OF INCO Element TYPE 123 NICKEL POWDER Nickel Powder (Wt.%) Carbon (typical) Carbon Oxygen Sulphur Iron Other Elements Nickel 0.03-0.08 0.1 max 0.15 max 0.001 max 0.01 max trace Balance Table 5 PHYSICAL PROPERTIES OF INCO TYPE 123 NICKEL POWDER Particle Shape Roughly spherical with spiky surface Average Particle Siz 5.5 microns Standard Deviation 0.75 microns Surface Area Per Unit Volume 7.65 x 103 cm/cm VS of As-Received Powder 0.25 V It. was found by trial and error that sintering a loose stack of this powder at 1250C produced the required series of samples in convenient sintering times. This sintering temperature was also used for PS and HP series, in order to ensure that the differences among the paths of evolution of microstructure for LS, PS and HP series were not due to different sin- tering temperatures. 1. Loose Stack Sintered Series In order to have the same initial microstructure for all the samples in a series, they were prepared from the same initial loose stack of powder. The first sample of the series was prepared by heating a loose stack of powder (tapped to yield a level top surface) in an alumina boat (6 x 12 x 75 mm) under a flowing dry hydrogen atmosphere for the specified sintering time. A small piece (about 5 mm thick) was cut off and stored for subsequent characterization; the rest of the sintered body was used to yield the remaining samples in the series by the repetition of the procedure described above for an appropriate sequence of accumulated sintering times. It required 11 minutes for the sample transferred from the cold zone to the hot zone to reach the sintering temperature. Although this time was not negligible compared to the time spent at the sintering temperature and although this procedure takes the samples through an increasing number of heating and cooling cycles with longer sintering times, it has been shown that these cycles do not influence the path of microstructural change in metal powders.32 2. Pressed and Sintered Series A CARVER hydraulic hand press was used to prepare cylindrical pellets about 15 mm in diameter and typically 3 mm in height. Cold pressing at 60,000 psi followed by sintering at 1250C yielded the desired series of samples. Due to the small size of these pellets and their patterns of potential inhomogeneity it was not feasible to prepare the series of samples from a single initial compact, as in the loose stack case. Instead, samples in this series were pre- pared individually by sintering the green compacts in an alumina boat under a flowing dry hydrogen atmosphere for preselected sintering times at 12500C 100C. 3. Hot Pressed Series The third series was prepared by hot pressing at 12500C and under a pressure of 2000 psi in a CENTORR high vacuum hot press. A loose stack of powder was placed in a cylindrical boron nitride die 2.54 cm in diameter and tapped; the die with the top punch resting on the powder was placed in the vacuum chamber. After a vacuum of 10-5 Torr was reached the induction coil was switched on. The attainment of sintering temper- ature which nominally required one hour was followed by the application of a pressure of 2000 psi. The pressure was maintained and the tempera- ture controlled to 50C for the specified sintering times; the pressure was then released and the induction coil turned off. After the sample was allowed to cool overnight, air was admitted and the die assembly removed. As in the case of PS series, samples in this series were made individually. Density Measurements The most common procedure for measuring density of a specimen is the liquid displacement method, wherein the volume of a specimen is estimated by measuring the volume of water displaced when it is immersed in water. Since in the cgs system of units the density of water is unity, this volume is numerically equal to the weight of water displaced, which is equal to the decrease in weight of the sample when immersed, according to the Archimedes principle. The major source of error in the case of this method lies in measuring the weight of the sample in water. A thin coating of paraffin wax, typically weighing a few tenths of a percent of the weight of the sample, was used to seal the surface pores during water immersion. The samples were suspended by placing them in a miniature rigid metal pan, thus eliminating the need to tie odd-shaped samples with a wire. Further, the use of this pan made it easy to correct for the volume of water displaced by the immersed part of the pan, whereas a similar correction in the case of a wire is not made easily. An elec- tronic balance accurate to 0.1 mg was used to achieve the required high degree of accuracy. After the sample was weighed in air (W1), it was coated with wax and weighed again (W2). The wax-coated sample was placed in the minia- ture pan in a beaker of distilled water and weighed (W3). The sample was then dropped to the bottom of the beaker by gently tilting the pan. The weight of partially immersed pan was measured (W4). The density of the sample, p, was calculated as follows: Weight of sample p g/c = Volume of sample Weight of sample (Volume of sample + immersed part of pan) - (Volume of immersed part of pan) Wl (W2 + weight of pan in air W3) - (Weight of pan in air W4) Thus W (8) p (g/cc) = + W 3 (8) The densities thus measured were reproducible within 0.2 percent of the mean of ten values with 95 percent confidence. The density of a piece of pure nickel, known to have a density of 8.902 g/cc,5 was measured and found to be within 0.5 percent of the abovementioned value. Metallography The polishing procedure will be described and followed by a brief discussion of principles of quantitative stereology involved in the estimation of metric properties. The estimated microstructural properties will be presented thereafter. Polishing Procedure The wax coating on the samples was dissolved in hexane and the samples were sectioned; a vacuum impregnation method was used to mount the samples, surrounded by a nickel ring, in epoxy. The purpose of the ring will be discussed later in this section. Rough polishing was done on wet silicon carbide papers of increasing fineness from 180 grit through 600 grit. Fine polishing was done by using 6 micron diamond paste, followed by 1 micron diamond paste, 0.3 micron alumina and finally 0.05 micron alumina. quantitative Stereology Metric properties are estimated by making measurements on a two dimensional plane of polish with the help of standard relations of stereology.26 A set of test lines, arranged in a grid pattern, also provide a set of test points and a test area to characterize the plane section; these are usually used to make the measurements listed in Table 6. The relationships between these measurements and the globally averaged properties of the three dimensional microstruc- ture are listed in Table 7. The relations yield estimates of popula- 26 tion or structure properties provided the structure is sampled uniformly.2 Stereological counting procedure and the estimated properties will be discussed presently. Each metallographically prepared surface was calibrated by measuring the volume fraction of porosity by quantitative stereology and comparing the result with the value obtained from density measurements. A definite amount of plastic deformation by the polishing abrasive media leads to a smearing effect that introduces some error in quantifying the information on a polished section. This effect can be viewed as local movements of traces of the pore-solid interface; all the counted events (number, inter- cept, etc.) aretherefore error-prone to some extent. As this investigation Table 6 QUANTITIES MEASURED ON POLISHED SECTION Test Feature Quantity Definition Points Pp Fraction of points of a grid that fall in a phase of interest Lines LL Fraction of length of test lines that lie in a phase of interest PL Number of intercepts that a test line of unit length makes with the trace of a surface on a plane section Area PA Number of points of emergence of linear feature per unit area of plane section NA Number of full features that appear on a section of unit area TA Net number of times a sweeping test line is tangential to the convex and concave traces of surface per unit area of a plane section Table 7 STANDARD RELATIONSHIPS OF STEREOLOGY2 Pp = L = VV PL 2 V P = LV TA 2NA MV TT A = 27TNA = M (9) (10) (11) (12) dealt with relatively small amounts of porosity (10 percent or lower) the error in VV introduced by the polishing technique approached that of the density measurements, namely, about 0.005, as the sintered den- sity approached the bulk density. Thus, the polishing was accepted for further characterization if the metallographically determined VP was V within 15 percent of VP obtained from the water immersion method, except for the samples 97 percent dense and higher for which the limit had to P P be relaxed to 30 percent of VV. Since VV values range from 0.15 to 0.02, the abovementioned limits translate into a few percent of the sample den- sity as measured metallographically. Typically, the samples 97 percent dense and lower exhibited a precision of 0.05 of the VV value obtained from the density measurements. Manual measurements of SP and MSP were made on the accepted polished surfaces using standard stereological techni- ques.26 The measurements of V SSP and MSP were made with at least 30 different fields and at magnifications that allowed at least 15 pores to be viewed in a single field. As a result, the estimates of the properties were within 5 percent with 95 percent confidence, as illustrated in Figures 20 and 21. Plots of SSP and MSP contained metallographically measured values of VV to yield the paths of evolution of microstructure in order to partially compensate for the polishing errors. Measurement of these metric properties was followed by etching the specimens to reveal the grain boundaries. Each sample was immersed in a solution made from equal parts of nitric acid, glacial acetic acid and acetone for about 30 seconds. The grain boundaries were brought out clearly with some evidence of facetting of the initially smooth contours of pore features. Samples in the lower part of the density range exhibited 23 20 15 SP SV (102 cm ) 10 5 0 0.8 0 Loose Stack Sintered 0 Hot Pressed at 2000 psi SCold Pressed at 60,000 psi I I I I IA II I I I 1 1 1 t I l 1 0.9 VV (Metallographic) Figure 20. Variation of surface area of the pore-solid interface with solid volume during loose stack sintering, hot pressing and conventional sintering of INCO 123 nickel powder at 12500C. 1.0 .0 I I I I I I I I I I 0 MP (105 cm-2) -50.0 -90.0 0 O Loose Stack Sintered O Hot Pressed at 2000 psi - A Cold Pressed at 60,000 psi I I I I I I I I I I 1.. I I I I I I I I I. I I I I.. ...1 L L- I. .8 Figure 21. 0.9 (Metallographic) Variation of integral mean curvature with solid volume during loose stack sintering, hot pressing and conven- tional sintering of INCO 123 nickel powder at 12500C. grain structures that were too fine to be studied optically; these samples were not included in the measurement of grain structure prop- erties. A number of etched microstructures are illustrated in Figures 22 through 36. Typically, the scale of the grain structure was such that the information contained in a single plane section was not enough to yield estimates with the desired precision of 10 percent. Conse- SS SSS SSP SSS quently, SV LV LV and LVcc) defined earlier in this report, were measured by repeating the polishing, etching and counting steps a number of times to obtain at least 100 different fields of view. The grain structure properties are illustrated in Figures 37 through 40. The apparent local movements of the traces of the pore-solid interface, mentioned earlier in this section, are likely to introduce some errors in the estimation of grain structure properties whenever the pores are associated with the boundary network. For example, an enlargement-of pore features residing on grain boundaries would underestimate the value of SS the grain boundary area per unit volume, as measured metallo- graphically. However, it was found that these errors are small compared to those in SSP and MSP; the trends of grain structure properties remain t t unaffected whether plotted versus Archimedes density or the stereological density. The quantities in Figures 37 through 40 are thus plotted versus the Archimedes density. The pores observed on a polished and etched surface can be classi- fied as to their association with the grain boundaries, that is, according to whether they appear to be inside a grain, on the grain boundary or on a grain edge. The relative fractions of pore features regarding their association with the boundary network were measured. These are illustrated in Figure 41. Figure 22. Photomicrograph of INCO 123 nickel powder loose stack sintered at 12500C to VV = 0.871, etched (approx. 400 X). qW' * S r * r'-, " Figure 23. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vv = 0.906, etched (approx. 400 X). Figure 24. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to VV = 0.928, etched (approx. 400 X). Figure 25. Photomicrograph of INCO 123 nickel powder loose stack sintered at 12500C to VV = 0.944, etched (approx. 400 X). Figure 26. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to VV = 0.971, etched (approx. 400 X). Figure 27. Photomicrograph of INCO 123.nickel powder loose stack sintered at 1250% to V = 0.979, etched (approx. 400 X). Figure 28. Photomicrograph of INCO 123 nickel powder hot pressed at 12500C to VV = 0.93, etched (approx. 400 X). Figure 29. Photomicrograph of INCO 123 nickel powder hot pressed at 12500C to VV = 0.943, etched (approx. 400 X). Figure 30. Photomicrograph of INCO 123 nickel powder hot pressed at 12500C to VV = 0.958, etched (approx. 400 X). Figure 31. Photomicrograph of INCO 123 nickel powder hot pressed at 12500 to VV = 0.968, etched (approx. 400 X). Figure 32. Photomicrograph of INCO 123 nickel powder hot pressed at 1250C to VV = 0.984, etched (approx. 400 X). Figure 33. Photomicrograph of INCO 123 nickel powder cold pressed at 60,000 psi and sintered at 1250C to VV = 0.942, etched (approx. 400 X). Figure 34. Photomicrograph of INCO 123 nickel powder cold pressed at 60,000 psi and sintered at 12500C to VV = 0.962, etched (approx. 400 X). Figure 35. Photomicrograph of INCO 123 nickel powder cold pressed at 60,000 psi and sintered at 12500C to VV = 0.975, etched (approx. 400 X). Figure 36. Photomicrograph of INCO 123 nickel powder cold pressed at 60,000 psi and sintered at 12500C to VV = 0.983, etched (approx. 400 X). 700 600 500 400- SSS V (cm-1) 300 - 200 -o 100 - 0 90 0.90 1.0 0-95 Vv Figure 37. Dependence of grain boundary area on solid volume during loose stack sintering, hot pressing and conventional sin- tering of INCO 123 nickel powder at 12500C. 3x10" 2x105 Lss (cm-2) 1x105 lxlO 0.90 0-90 Figure 38. Dependence of the length of grain edges per unit volume on the volume fraction of solid during loose stack sin- tering, hot pressing and conventional sintering of INCO 123 nickel powder at 12500C. 0.95 1.0 1.5x106 1x106 Lssp V (cm-2) 5x10' 1x105 C C OLoose Stack Sintered OHot Pressed at 2000 psi T ACold Pressed at 60,000 psi I I I I ).90 I I I 0.95 1.0 Vv Figure 39. Variation of the length of lines of intersection of grain boundary and pore-solid surfaces per unit volume with the volume fraction of solid during loose stack sintering, hot pressing and conventional sintering of INCO 123 nickel powder at 12500C. 2 1 I ' 2x105 S(occ) (cm-2) 1 x105 0.90 Figure 40. Variation of the length of occupied grain edges per unit volume with the volume fraction of solid during loose stack sintering, hot pressing and conventional sintering of INCO 123 nickel powder at 12500C. 0.95 1.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 Fraction of pores 0.3 0.2 0.1 0. 0. Figure 41. 90 0.95 Vv (a) 1.0 0.90 0.95 0.90 0.95 Variation of fractions of pores on the triple edges (filled), on the boundaries and within the grains (open) for a) loose stack sintered, b) hot pressed and c) sintered nickel powder at 1250C. 1.0 (half-filled) pressed and ii Ii I? II~~i~ . . I ill. I i I ii I - i - 0 A The metric measurements of pore structure and grain structure properties were followed by topological characterization of loose stack sintered samples. The experimental procedure employed in the latter is described presently. Topological Measurements As mentioned earlier in Chapter 1 only the loose stack sintered samples having densities that are typical of late stages were analyzed to yield the topological parameters. These samples and their densities are listed in Table 8. The procedure for serial sectioning is described below, followed by the algorithm used and the results of the topological analysis. Serial Sectioning The first step in the technique of serial sectioning is to develop and standardize the procedure for removing a layer of desired thickness. This optimum thickness is such that it is small enough to encounter the smallest structural feature for a number of sections; yet large enough to avoid redundancy of measurements. Patterson51 and Aigeltinger25 tackled this problem very systematically and found that the optimum thickness is of the order of one-fifth of that of scale of the structure. A reliable measure of the scale of the system is XP, the mean intercept of pore phase. Since = 4VV/SV the slope of the straight line in Figure 20 yields the value of 3 : 4.5 microns. Thus the serial sec- tions for LS Series of samples ought to be roughly one micron apart. Table 8 LOOSE STACK SINTERED SAMPLES USED FOR TOPOLOGICAL CHARACTERIZATION Number Sintering Time Volume Fraction of Solid, V 1 128.7 min. 0.906 2 190.0 min. 0.928 3 221.0 min. 0.944 4 352.5 min. 0.971 5 400.0 min. 0.979 This fine scale ruled out the possibility of employing an established procedure for measuring thickness such as using a micrometer. A new, simple procedure was developed to measure the section thicknesses and is presently outlined. A microhardness tester was used to make square-based, pyramid- shaped indentations on the polished surface of a sample. It is known that the apex angle of the diamond indentor is 1360 and hence the ratio of the depth of an indentation to the diagonal of the impression is equal to 0.1428. As illustrated in Figure 42, the decrease in the depth of an indentation is 0.1428 times the decrease in the legnth of the diagonal. The hardness tester has a capability of a wide variety of loads and magnifications, so indentations of a wide variety of sizes can be made and measured with desired accuracy. Thus, the section thick- ness can be easily measured by measuring the decrease in the length of the diagonals of an indentation. A GeotechTM automatic polisher was used to achieve a reproducible combination of polishing speed, load on the sample and polishing time that would yield the desired magnitude of material removal. A trial sample of sintered nickel was polished, indented with 30 indentations and the section thickness was measured several times by repeatedly polishing and measuring the diagonals until the polishing technique and measurement of section thickness were established with a high degree of confidence. An elaborate and rigorous procedure for topological analysis of porous bodies was developed in the course of doctoral research by Aigeltinger.25 The abovementioned investigation dealt with loose stack Ad (section thickness) = d. df S f Ah = hi hf Ah = 0.1428 Ad Figure 42. Illustration of the relation between the decrease in the length of diagonals of a microhardness indentation and the decrease in the depth of the indentation. sintered samples having densities in the range from 50 percent to 90 percent of the bulk value and hence exhibited pore structures of a large variety of scales and complexities. Since the samples used in the present investigation had densities higher than those used in this research, their pore structures were typically relatively simple. This made it possible to streamline and simplify the topological analysis to a great extent. The revised algorithm is presently described in detail. Algorithm for Topological Analysis Two topological parameters, namely, the connectivity and the number of separate pores, weremeasured in this investigation. Since the connec- tivity is a measure of the number of redundant connections, there is an inevitable uncertainty regarding the connections between pores that inter- sect the boundaries of the volume of analysis (which is a very small fraction of the sample volume). It is not possible to determine whether such pore sections intersect each other or meet with themselves outside the volume covered by the series of parallel areas of observation. This has led to the necessity of putting maximum and minimum limits on the estimate of connectivity. As illustrated in Figure 43, an upper limit on connectivity is obtained when all the pores meeting the boundaries of the volume of analysis are regarded as meeting at a common node, and is called Gmax. A lower limit is derived by considering all such pores to be terminating or "capped" at the boundaries, and is called Gmin. The quantity Gmin then consists solely of redundant connections or "loops" observed within the volume of analysis. The number of separate parts is '71 S Figure 43. Illustration of contributions of subnetworks crossing the surface towards the estimate of Gmax, obtained by counting the separate pores that appear and disappear within the volume observed and do not intersect the boundaries. The actual algorithm is as follows. The surface of a loose stack sintered sample, one from the series designated for topological analysis, was conditioned by polishing it on a microcloth with 1 micron diamond paste abrasive for about half an hour. This effectively removed all plastically deformed material, the result of an earlier alumina polish. The sample was cleaned in an ultrasonic cleaner, dried and viewed under a microscope to check for polishing artifacts. If the nickel ring, mentioned earlier in this section, was polished uniformly all around, the sample was examined for undue number of scratches that would hinder the analysis. If the polishing was uniform and had only a small number of scratches, it was deemed ready for further analysis; otherwise it was returned to the polishing step. Since the contours of the hardness indentation are mixed with those of the pore sections when observed for the measurement of diagonals, the thickness measurements become more difficult the more the sample is polished or the smaller the square-shaped impression. The trial sample of sintered nickel mentioned previously was polsihed, ten indentations were made on the nickel ring and the specimen and the sample repolished. This was followed by measuring the diagonals of impressions on both ring and specimen. The repetition of this procedure demonstrated that the extents of polishing (removal of material or layer thickness) of the ring and the sample were not statistically different. Indentations in the ring were therefore used to measure the section thickness. Nine indentations were made on the nickel rings of each sample. Three indentations were made on the specimen so that the same area could be located and photographed after each polishing step. The pattern of indentations is schematically illustrated in Figure 44. The first photomicrograph of the serial sectioning series was taken by positioning the three indentations on the sample in a manner that can be easily reproduced. The magnification was selected so that at least 70 pore features could be observed in a single field of view. A Bausch and Lomb Research Metallograph II was used for all the photo- micrographs. A set of 4"x5" negatives was obtained by repeating the polishing and photographic steps. Each was enlarged to a size of 8"x10" so that even the smallest pores were easily seen. A smaller rectange of 6"x8" was marked on print #1; this identified the area of observation. This manner of delineating the area was adapted to help minimize the misregistry error. A similar rectangle was marked on successive prints such that the pores observed on the consecutive sections were in the same position relative to the boundaries of the rectangle. Xerox copies of these prints were used for further analysis, which involves marking each pore on the area of observation for easy identification. The pores seen on Section #1 were numbered beginning with 1. These are all connected to the "external" networks and thus were not included in the count of separate parts. Pores that first appeared thereafter on successive sections were numbered with a number and a letter N, beginning with 1N. These were regarded as the "internal" networks and were used to measure the number of separate parts. The Microhardness indentations used to measure section thickness Microhardness indentations used to position the Specimen Sample Nickel ring Epoxy mount Figure 44. Schematic diagram of a typical specimen used in serial sectioning. genus or the connectivity and the number of isolated pores were measured by comparing pairs of neighboring sections as follows. There are listed in Table 9 three possible classes of topologi- cal events that can be observed when two consecutive sections are compared, along with the corresponding increments in Gmax, Gmin and Nis0. The significance of each of such observed events will be discussed presently. Two typical consecutive sections are shown schematically in Figure 45, wherein the types of events mentioned above are also illustrated. The simplest of these events is the appearance and disappearance of whole pores or subnetworks. When an external sub- network disappears, the number of possible "loops" or redundant connections that are assumed to exist outside the volume of analysis is reduced by one, as illustrated in Figure 46. When an internal subnetwork appears, it cannot be determined whether the said subnet- work is wholly contained in the volume of analysis or is connected to the external pores. Thus, this event does not change any of the parameters. However, the disappearance of an internal subnetwork signifies a whole separate part and thus the number of separate parts is increased by one. Within a subnetwork, a branch may appear. When that happens, the number of possible loops, terminating in a single external node, is increased by one, as shown in Figure 47. When such a branch is observed to disappear, the abovementioned number is decreased by one, to account for the increase assumed prior to an observation of this event. 84 Table 9 OBSERVABLE TOPOLOGICAL EVENTS Appearance Whole Subnetworks Disappearance Internal External AGmax 0 -1 AGmin ANiso 0 +1 Within a Subnetwork Between Subnetworks Appearance of a branch +1 Disappearance of a branch -1 Connection Different or new 0 Same +1Connection Same +1 [jth section] [(j+l)th section] Figure 45. Two typical consecutive sections studied during serial sectioning that illustrate the topological events listed in Table 9. External Node J\/ a- End of an Original Subnetwork Redundant Connection to be Deleted from the Estimate of Gmax Figure 46. Contribution of the end of an original subnetwork towards the estimate of Gmax. External Node 10 )-- -- \-- N towards Gmax. / I r i / / r I / r t / Branching within a subnetwork Redundant connection increasing the connectivity by one Figure 47. Illustration of the contribution of a branching event towards Gmax. When a connection is observed between different subnetworks, the said subnetworks have to be renumbered to keep track of such connections. All the subnetworks involved in such connections are marked with the lowest of the numbers of these connecting subnetworks. If an internal subnetwork is observed to be connected to an external one, the said internal subnetwork is marked with the number designating the external subnetwork. A connection between previously unconnected subnetworks, those with different numbers, does not change any of the three parameters. A connection between two or more subnetworks with the same number signi- fies a complete loop observed entirely inside the sample volume, and thus increases the count of Gmin by one. Since Gmax includes such internal loops, it is also increased by one. After each comparison of consecutive sections, the counts of Gmax, Gmin and N1so were updated and tabulated as shown in Table 10. The values of interest are the unit volume quantities, Gax, Gmin iso and NV If the features that give rise to these quantities are randomly and uniformly distributed in the sample,25 then it can be expected that there exists a quantity (QV) characteristic of the structure and equal to the unit volume value. Thus AQ (the change in quantity Q) = QV x AV. Dividing both sides by AV leads to S= QV (13) The slope of AQ versus AV plot therefore should be equal to Qv provided AV, the volume covered is large enough for a meaningful sampling. In the previous investigations of this kind25,51 the analyses were continued Table 10 CURRENT ESTIMATES OF Gmax, Gmin and Niso AFTER SECTION #J Height of Sample, Microns -- Counts Gmax Previous total - This section Current total Unit volume value Current total Current volume Gmin -- _ Niso -- until a linearity between AQ and LV was observed. A different criterion was adapted in the present investigation, and is described presently. Since the connectivity and the number of separate parts are two independent quantities, one may begin to exhibit a constant value of Q after several sections whereas the other may be far from levelling off. This makes it difficult to establish a criterion for terminating the serial sectioning for a given sample. If use is made of both the quantities a standard basis for terminating the analysis can be obtained. A typical pore subnetwork with convex, concave and saddle elements of its surface is shown schematically in Figure 48. If a test plane is net swept through a unit volume, a measure of TV, the number of times this plane is tangential to the pore-solid interface, may be obtained in principle. This is related to the connectivity and the number of parts per unit volume26 as follows. Tet = T + T T = 2(NV GV) (14) where TV = number of times a concave element (having both the principal radii of curvature negative by convention) is tangential to test plane, or the number of "ends" of a feature (see Figure 48). ++ TV = number of times a convex element (having both the principal radii of curvature positive by convention) is tangential to to test plane, very small at this stage of sintering. +- T = number of times a saddle element (with principal radii of curvature of different signs by convention) is tangential Figure 48. Convention used in the net tangent count (Tnet) during serial sectioning. 5x107 min Gv -3 1 x10 0. O- Vv V Figure 49. Dependence of connectivity on the volume fraction of solid during loose stack sintering of INCO 123 nickel powder at 12500C. |

Full Text |

130
Thus, a spherical pore has more than twice as much Cy as does a tubular channel. The relative fractions of tubular channels and spheri cal pores can be computed from the number of branches (by in Table 17) and the surface-corrected number of isolated, pores (Figure 59). It can SP be seen from the variation of f fraction of spheres, Figure 64, that the curvature would decrease (in magnitude) relatively slowly with the volume until the maximum in the fraction of spheres; beyond this point the curvature would decrease sharply. This is supported by the variation of My, Figure 21. It is suggested that a kinetic model of this type of geometry should be devised and tested in the future, provided the associ ation of isolated and connected porosity with the grain boundary network can be successfully quantified. The preceding discussion on loose stack sintering will now be used to describe hot pressing and conventional sintering, in that order. Hot Pressing It can be seen from Figure 20 that in the early part of the range SP of observation a hot pressed sample has a higher Sy than a loose stack sintered sample with comparable density. The variation of surface area with Vy approaches the linear relationship for LS series as the density increases. Both hot pressed samples and loose stack sintered samples then continue to exhibit comparable values of surface area. Since the pressure is applied for the whole duration of hot pressing, the abovementioned SP variation of Sy with Vy may have less to do with pressure and thereby particular mechanisms than with the geometry of the pore structure. CHAPTER 2 EXPERIMENTAL PROCEDURE AND RESULTS Introduction Microstructural characterization involved sample preparation, metallography and in the case of loose stack sintering, also serial sectioning. These are described in detail in this chapter, followed by results of this investigation. Sample Preparation This section presents the procedure employed to prepare the sintered samples and the standard for density measurements. Sintered Samples Three series of samples of sintered nickel powder were prepared: 1) loose stack sintered (LS), 2) pressed and sintered (PS) and 3) hot pressed (HP). In order to study the path of evolution of microstructure during the late stages of sintering, it is desirable to obtain samples c having densities that are uniformly distributed over the range Vy = 0.85 to 1.0. Accordingly, preliminary experiments were designed to determine the processing parameters, such as temperature, pressure and time, that yield the desired series of samples made from the selected metal powder. INCO type 123 nickel powder, illustrated in Figure 19, supplied by the International Nickel Company, Inc., with the chemical and physical prop erties listed in Tables 4 and 5, was used in the present investigation. 39 123 L(measured) = / / L Cos6Sin0d0d<> = L/2 <Â¡>=0 0=0 Unit Radius Figure 61. Illustration of the relation between measured and true lengths of a branch. 62 Figure 30. Photomicrograph of INCO 123 nickel powder hot pressed at 1250C to Vy = 0.958, etched (approx. 400 X). 90 until a linearity between aQ and LV was observed. A different criterion was adapted in the present investigation, and is described presently. Since the connectivity and the number of separate parts are two independent quantities, one may begin to exhibit a constant value of ^ after several sections whereas the other may be far from levelling off. This makes it difficult to establish a criterion for terminating the serial sectioning for a given sample. If use is made of both the quantities a standard basis for terminating the analysis can be obtained. A typical pore subnetwork with convex, concave and saddle elements of its surface is shown schematically in Figure 48. If a test plane is net swept through a unit volume, a measure of Ty, the number of times this plane is tangential to the pore-solid interface, may be obtained in principle. This is related to the connectivity and the number of parts pc per unit volume as follows. T?et = TV~ + TV+ TV~ = 2 radii of curvature negative by convention) is tangential to test plane, or the number of "ends" of a feature (see Figure 48). Ty+ = number of times a convex element (having both the principal radii of curvature positive by convention) is tangential to to test plane, very small at this stage of sintering. Ty" = number of times a saddle element (with principal radii of curvature of different signs by convention) is tangential 4 The objectives of this research were to determine the paths of evolution of microstructure during the advanced stages of loose stack sintering, conventional sintering and hot pressing. Since the topologi- 25 cal measurements are time-consuming and since an earlier doctoral research dealt with topological characterization of loose stack sintering in the porosity range 0.1 and higher, it was planned to estimate the topological parameters for loose stack sintered series only. Metric properties of the pore structure and grain boundary network were estimated for all three series of samples. Previous investigations of this type are discussed in Chapter 1, followed by experimental procedure and results in Chapter 2. These results are discussed in Chapter 3 and the important findings and con clusions summarized in Chapter 4. 94 to the test plane per unit volume, or the number of branching and connection events per unit volume (see Figure 48). Cbmputation of is shown;,in Table 11. not If the constancy;, of T is used as the basis for terminating the analysis, both N and G are used and so a more general criterion is obtained. The serial sectioning was terminated for each sample after Tye* was found to level off. The values of the parameters G^ax, G^n and NyS0 are listed in Table 12. Since Gyax depends on the number of pores intersecting the surface that encloses the volume of analysis, it is very sensitive to the surface to volume ratio of the "slice" of the sample used for analysis. Therefore, G^ax does not exhibit a monotonic decrease with an increase in Vy. On the other hand, Gyin is unaffected by the surface to volume ratio mentioned above and hence can be used to draw meaningful inferences about the variation of connectivity. The variation in Gyin and NyS0 are illustrated in Figures 49 and 50, respec tively. It can be seen that even at 98 percent of the bulk density the pore structure has a finite connectivity and thus cannot be regarded as a collection of simple isolated pores. Behavior of Connected and Isolated Porosity A natural reaction to these data would be to wonder what fraction of porosity is connected and how it varies with the total Vy. It is possible to distinguish connected porosity from the isolated pores on any of the series of sections used for topological analysis. Thus, the volume fraction, surface area and integral mean curvature of the connected 125 Table 17 VALUES OF THE MODEL PARAMETERS USED IN CALCULATIONS No. Vv R L microns microns cm -3 1 0.906 2 0.928 3 0.944 4 0.971 2.34 6.63 4.48 x 108 2.1 x 108 2.5 x 108 2.58 8.18 2.42 x 108 6.2 x 107 2.63 x 108 2.79 7.1 8.39 x 107 2.04 x 107 1.02 x 108 2.1 9.56 1.8 x 108 3.0 x 107 2.66 x 108 1.07 25.8 1.29 x 108 3.4 x 107 1.23 x 108 5 0.979 81 Nine indentations were made on the nickel rings of each sample. Three indentations were made on the specimen so that the same area could be located and photographed after each polishing step. The pattern of indentations is schematically illustrated in Figure 44. The first photomicrograph of the serial sectioning series was taken by positioning the three indentations on the sample in a manner that can be easily reproduced. The magnification was selected so that at least 70 pore features could be observed in a single field of view. A Bausch and Lomb Research Metallograph II was used for all the photo micrographs. A set of 4"x5" negatives was obtained by repeating the polishing and photographic steps. Each was enlarged to a size of 8"xl0" so that even the smallest pores were easily seen. A smaller rectange of 6"x8" was marked on print #1; this identified the area of observation. This manner of delineating the area was adapted to help minimize the misregistry error. A similar rectangle was marked on successive prints such that the pores observed on the consecutive sections were in the same position relative to the boundaries of the rectangle. Xerox copies of these prints were used for further analysis, which involves marking each pore on the area of observation for easy identification.' The pores seen on Section #1 were numbered beginning with 1. These are all connected to the "external" networks and thus were not included in the count of separate parts. Pores that first appeared thereafter on successive sections were numbered with a number and a letter N, beginning with IN. These were regarded as the "internal" networks and were used to measure the number of separate parts. The 149 which is the same as equation (26). Thus, the number of branches can be calculated from the number of nodes, and vice versa. Metric Properties of the Connected Porosity Three basic properties, namely, volume fraction, area per unit volume and integral mean curvatre per unit volume are discussed. The conn cconn Mconn i S\ cl net Mt; notations used are listed in Table 20; V are made V dna V up of contributions from branches, one-branch nodes and three-branch nodes; they will be discussed in terms of these individual contributions, listed in Table 21. Three-Branch Node It can be seen from Figure 67 that r, the radius of a cylindrical branch, and R, the radius of a three-branch node, are related by the equation r = RCos(ir/6) or r = '^-R (31) Thus, h, the height of the spherical cap, illustrated in Figure 68, is given by h = R/2 (32) C The volume of this spherical cap, V is given by (33) 2r = 3R Figure 63. Illustration of a typical pore channel in the connected porosity. 147 The abovementioned parameters of the model are discussed in terms of the notations listed in Table 19. Since each one-branch node is associated with a branch and each three-branch node with three branches, by, the total number of branches 1 h and Ny and Ny are related by the following equation + 3Nyb 2 (26) since each branch is counted twice. Alternatively, each one-three branch is associated with one one-branch node and one three-branch node, whereas each three-three branch is associated with two three- branch nodes. Thus ,3b Nr h13 .33 by + 2by (27) since each three-branch node is counted thrice. The number of one-branch nodes, Nyb, is given by Njb = bj3 (28) Equations (26) and (27) give wlb .33 3b NV + 2bV V or 3N?b NJb (29) Thus by is given by K h13 4. h33 bV bv + bV 3|\j3^ N^b My> + 3_Ni_Ny. bv = Nb + 3Nf or 2 (30) 25 Figure 11. Variation of integral mean curvature per unit volume with the volume fraction of solid during hot pressing of RSR 107 nickel (-170+200) at 1500 psi. Data for spherical copper (-170+200) loose stack sintered at 1005C included for comparison.23.25 112 Figure 57. Illustration of three-branch nodes signified by branching and connection events. 42 Table 5 PHYSICAL PROPERTIES OF INCO TYPE 123 NICKEL POWDER Particle Shape Roughly spherical with spiky surface Average Particle Siz 5.5 microns Standard Deviation 0.75 microns Surface Area Per Unit Volume 7.65 x 10^ cm^/cm^ 5 Vy of As-Received Powder 0.25 METRIC AND TOPOLOGICAL CHARACTERIZATION OF THE ADVANCED STAGES OF SINTERING By ARUNKUMAR SHAMRAO WATWE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1983 145 Figure 66. Illustration of a) a one-three branch and b) a three-three branch. 61 Fiaure 29. Photomicrograph of INCO 123 nickel powder hot pressed at 1250C to Vv = 0.943, etched (approx. 400 X). 22 M SP V Figure 9. Variation of integral mean curvature per unit volume with the volume fraction of solid for three representative copper powders sintered in dry hydrogen at 1005C.35 PAGE Parameters of the Model 143 Metric Properties of the Connected Porosity 149 Surface Corrections 155 BIOGRAPHICAL SKETCH 163 VI 34 Figure 17. a) Genus per gram (Gp) the number of isolated pores per gram (Gp) versus volume fraction of solid for 48 micron dendritic copper powder. Data for 48 micron spherical powder included for comparison, b) Enlarged part of lower right corner of (a).2^ INTRODUCTION Sintering is a coalescence of powder particles into a massive formj wherein the densification is accompanied by a variety of profound geo metrical changes in the pore-solid composite. The mechanical and physical properties of a powder-processed compact are influenced by the geometry 2-9 of the pore phase. Thus, the manner in which the reduction of porosity takes place is of great practical and theoretical interest. There are two approaches to the study of sintering.^'frie traditional or mechanistic approach involves the study of kinetics and mechanisms of ^Im material transport;1the geometric viewpoint focuses on the geometry of the pore phase as it evolves during sintering. The latter approach involves estimation of size- and shape-dependent quantities (volume, area, etc.) and topological properties such as the connectivity and the number of separate parts. Mechanistic studies essentially consist of three steps: 1) A laboratory model of a particulate system is selected that is amenable to mathematical treatment of desired sophistication. Assumptions are made regarding the geometrical changes during sintering and the identities of source and sink of matter. 2) Kinetic equations are derived that describe the variation of a measurable parameter (width of an interparticle contact or "neck," density, etc.) for the particular mechanism(s) of interest. 3) These equations are compared with the experimentally observed time dependences of the chosen parameter and an attempt is made to identify the operating mechanism(s). 1 144 Figure 65. Illustration of three-branch and one-branch nodes in a pore network. 146 2r Figure 67. Dihedral angle of the edge at the intersection of a cylindrical branch and a spherical node. 71 Vv Figure 39. Variation of the length of lines of intersection of grain boundary and pore-solid surfaces per unit volume with the volume fraction of solid during loose stack sintering, hot pressing and conventional sintering of INCO 123 nickel powder at 1250C. APPENDIX THE GEOMETRIC MODEL OF THE PORE PHASE Introduction Main features of the proposed model are stated and followed by the derivation of equations relating metric properties and the parameters of the model. The corrections for surface effects are also outlined. Parameters of the Model As said in Chapter 3, the pore phase is modeled as a collection of spherical pores of the same size and a set of networks made up of cylin drical branches and spherical nodes. Only two kinds of nodes and branches are assumed to exist in the connected pores; a node is either a one-branch node or a three-branch node, as illustrated in Figure 65. A branch either terminates in three-branch nodes at both ends, or in a one-branch node at . one and a three-branch node at the other, Figure 66. A branch that termi nates in one-branch nodes at both ends is considered an isolated part and hence is not included in the connected porosity. A one-branch node is assumed to be a semi-spherical cap, the radius of which is the same as that of a cylindrical branch, r, as shown in Figure 66. A three-branch node is considered as a sphere of radius R which*is connected to three cylindrical branches of radius r, Figure 67. An isolated pore is assumed to be a sphere of radius R. All branches have the same length, "L. 143 21 Figure 8. Pore-solid interface area versus solid volume fraction for conventionally sintered 43 It was found by trial and error that sintering a loose stack of this powder at 1250C produced the required series of samples in convenient sintering times. This sintering temperature was also used for PS and HP series, in order to ensure that the differences among the paths of evolution of microstructure for LS, PS and HP series were not due to different sin tering temperatures. 1. Loose Stack Sintered Series In order to have the same initial microstructure for all the samples in a series, they were prepared from the same initial loose stack of powder. The first sample of the series was prepared by heating a loose stack of powder (tapped to yield a level top surface) in an alumina boat (6 x 12 x 75 mm) under a flowing dry hydrogen atmosphere for the specified sintering time. A small piece (about 5 mm thick) was cut off and stored for subsequent characterization; the rest of the sintered body was used to yield the remaining samples in the series by the repetition of the procedure described above for an appropriate sequence of accumulated sintering times. It required 11 minutes for the sample transferred from the cold zone to the hot zone to reach the sintering temperature. Although this time was not negligible compared to the time spent at the sintering temperature and although this procedure takes the samples through an increasing number of heating and cooling cycles with longer sintering times, it has been shown that these cycles do not influence the path of microstructural 32 change in metal powders. ACKNOWLEDGEMENTS I am grateful for the opportunity to conduct my research under the guidance of Dr. R. T. DeHoff, the chairman of my advisory committee. An ability to approach any scientific matter with objectivity and logic has been blissfully passed on by him to all his students. I thank Drs. R. E. Reed-Hill, J. J. Hren, G. Y. Onoda, Jr., of the Department of Materials Science and Engineering and Dr. R. L.Scheaffer of the Department of Statistics for serving on my advisory committee. Their helpful advice and encouragement are deeply appreciated. It is a pleasure to thank my colleagues, Mr. Atul B. Gokhale and Mr. Shi Shya Chang,'for their collaboration in the experimental aspects of the project. Mr. Rudy Strohschein, Jr., of the Department of Chemistry assisted me beyond and above the call of duty in the fabrication of the sintering apparatus. He saved me a great deal of time and aggravation. All the credit for the preparation of this dissertation in its final form must go to Miss Debbie Perrine for her excellent typing. The financial support of the Center of Excellence of the State of Florida and the Army Research Office is gratefully acknowledged. iv 136 1) As the grain boundaries are anchored effectively only by equiaxed pores, a fine grain structure can be obtained if a sufficient number of equiaxed pores is isolated before grain growth begins. Grain growth requires a connected grain edge network and hence a certain minimum grain boundary area, therefore, an initial powder stack with a relatively coarse grain structure (low grain boundary area) would have grain growth beginning at a higher density (after a sufficient number of pores is isolated); as compared to an initial powder stack with a finer grain structure. Thus, for a sintered body with high density and fine grain structure requirements, an initially coarse grain structure is better than a fine one. 2) If sintering is carried out in such atmosphere that the isolated pores trap a gas of low diffusivity, these pores are relatively stable and hence offer effective grain boundary pinning. Care must be taken to delay the isolation events so that only a few pores are isolated; otherwise coarsening of these pores would 42 lead to an increase in volume. 3) At least for the conditions of the present investigation, it can be said that hot pressing leads to both higher density and a finer grain size in a shorter length of time as compared to loose stack sintering or conventional sintering at the same temperature, up to a certain density. Beyond this, a relatively coarser grain structure is obtained during hot pressing or con ventional sintering. The findings of this investigation are summarized and the course of future research suggested in the next chapter. 124 *1 CQ Where Ny = number of isolated pores per unit volume and ,iso = radius of a spherical pore. Thus, 1/3 3V1S0 Ris ( V 4ttN iso ) (22) 1 C A The radius of a spherical -node;:R> was taken to be equal to R The Al 1 L values of L, R, by, Ny and Ny are listed in Table 17. It can be seen that r increases slowly until Vy s 0.97 when it increases significantly. The reduction in connectivity by way of pinching off of a branch decreases L whereas elemination of a three-branch node, accomplished when a 1-3 type branch merges into the parent network, leads to an increase in "L, Figure 62. Since the number of isolation events goes through a maximum, L should eventually increase significantly once most of the isolation events or channel closures have taken place. The apparent maximum in R can be attributed to simultaneous isolation and shrinkage processes. When a pore shrinks it can be thought of as going from one size class to the lower one. Since all isolated pores shrink, although at different rates, there is a "flux" toward the smallest size class in a given size distribution. Isolation events bring new pores into this collection, such that there is an influx in all the size classes. Since the number of isolation events goes through a maximum, the net "flux" in the size distribution is towards the largest size class when isolation events dominate over shrinkage and towards the smallest size class when very few isolation events occur. This leads to a maximum in the estimated average volume of an isolated pore. In order to test the model, the metric properties of the connected porosity were calculated, since those of the isolated fraction were used, 98 w conn vV or V iso V p Vv (metallographic) Figure 51. Dependence of the metallographically determined volume fraction of connected porosity on the volume fraction of porosity during loose stack sintering of nickel at 1250C. Data for the volume fraction of isolated porosity included for comparison. 150 Table 20 CALCULATED METRIC PROPERTIES Property Notation Definition Volume ..conn vV Volume fraction of connected porosity Area -conn Surface area of connected porosity per unit volume Curvature Mconn Integral mean curvature of connected porosity per unit volume Windows Live Hotmail Page 2 of 3 Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: Watwe, Arunkumar TITLE: Metric and topological characterization of the advanced stages of sintering / (record number: 506243' PUBLICATION DATE: 1983 It v5- 'as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees of the University of Florida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be limited to.those specifically allowed by "Fair Use" as prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and preservation of a digital archive copy. Digitization allows the University of Florida to generate image- and text-base'd Versionsas appropriate and to provide and enhance access using search software. This grant of permissions prohibits use of the digitized versions for commercial use or profit. Signature of Copyright Holder /\aJ K "5. Vv/ A ~f A3 Â£= Printed or Typed Name of Copyright Holder/Licensec Personal information blurred 6 Date of Signature Please print, sign and return to: Cathleen Martyniak UF Dissertation Project Preservation Department University of Florida Libraries P.O. Box 117007 Gainesville, FL 32611-7007 http://bll44w.blul44.mail.live.com/mail/ReadMessageLight.aspx?Action=MarkAsNotJun... 6/10/2008 9 Table 2 DERIVED METRIC PROPERTIES Feature Property Definition Surface - MV H =-r Average mean curvature of sv a surface Region of Space x = 4sVv Mean intercept in a particular region of space 7 Figure 1. Illustration of basic metric properties. 3 Consequently, a major school of thought prevails that favors the geometri approach. The present investigation was undertaken to study the advanced stages of sintering (wherein the porosity values vary from ten to a few percent of the total volume) from this point of view. A main feature of this approach is the concept of path of evolution of microstructure. A given microstructure is characterized by its geo metric properties such as volume, area, curvature, connectivity, etc. A microstructural state is defined as a point in a n-dimensional space where each dimension denotes a particular microstructural property. As the microstructure evolves during a process, the resultant locus of such points represents the sequence of microstructural states that is obtained during the process. This sequence is termed the path of evolution of microstructure.^ It is convenient to represent two-dimensional projec tions of this path (two geometric properties at a time); usually one of them is the relative density or the volume fraction of solid. Previous studies of microstructural evolution during loose stack C ip 1 0_0*| sintering, conventional sintering-cold pressing followed by sinteringD,<:, and hot pressing*c ~ have provided a coherent pic ture of these processes for all but the last ten percent of the porosity. A detailed study of the late stages (porosity ten percent or lower) of loose stack sintering, conventional sintering and hot pressing would complete the picture of evolution of microstructure during these pro cesses. The practical interest in the behavior of porosity at these stages stems from the fact that a variety of commercial products made by powder technology are required to have porosities in the range 0.1 24 or lower. 83 genus or the connectivity and the number of isolated pores were measured by comparing pairs of neighboring sections as follows. There are listed in Table 9 three possible classes of topologi cal events that can be observed when two consecutive sections are compared, along with the corresponding increments in Gmax, G111111 and i so N The significance of each of such observed events will be discussed presently. Two typical consecutive sections are shown schematically in Figure 45, wherein the types of events mentioned above are also illustrated. The simplest of these events is the appearance and disappearance of whole pores or subnetworks. When an external sub network disappears, the number of possible "loops" or redundant connections that are assumed to exist outside the volume of analysis is reduced by one, as illustrated in Figure 46. When an internal subnetwork appears, it cannot be determined whether the said subnet work is wholly contained in the volume of analysis or is connected to the external pores. Thus, this event does not change any of the parameters. However, the disappearance of an internal subnetwork signifies a whole separate part and thus the number of separate parts is increased by one. Within a subnetwork, a branch may appear. When that happens, the number of possible loops, terminating in a single external node, is increased by one, as shown in Figure 47. When such a branch is observed to disappear, the abovementioned number is decreased by one, to account for the increase assumed prior to an observation of this event. 26 Figure 12. The effect of pressure on the path of integral mean curvature for hot pressed specimens of UOg.^ Ad (section thickness) = d^ Ah = h^ Ah = 0.1428 Ad Figure 42. Illustration of the relation between the decrease in the length of diagonals of a microhardness indentation and the decrease in the depth of the indentation. no a mechanistic model and hence derive any mechanistic conclusions from the time dependences of geometric properties of the pore microstructure. However, it was possible to construct a variety of geometric models that describe the evolution of microstructure qualitatively. These models incorporate connected and isolated porosity of regular geometry so that the metric properties such as volume, area and integral mean curvature can be calculated. The models are tested by comparing the calculated metric values with the experimentally determined quantities. Since ' these models do not have any parameters that characterize the extent of association with the grain boundary network, properties that depend SSP SSS on this association, such as Ly and Ly(occy could not be calculated and compared with the measured values. These models are described below. Geometric Models All of the geometric models mentioned above describe the porosity as composed of a collection of isolated, spherical pores of the same size and a set of networks of cylindrical pore channels and spherical nodes, as illustrated in Figure 56. The parameters of such a model are listed in Table 13. Here it is assumed that the connected porosity has only three-branch nodes and one-branch nodes; these were measured for a unit volume during the topological characterization in a manner described presently. Each branching or connection event defines a node formed as a result of merging of three pore channels, Figure 57. Thus, the number of three- branch nodes is given by the number of branching and connection events observed per unit volume (Ty). The number of one-branch nodes is the 74 The metric measurements of pore structure and grain structure properties were followed by topological characterization of loose stack sintered samples. The experimental procedure employed in the latter is described presently. Topological Measurements As mentioned earlier in Chapter 1 only the loose stack sintered samples having densities that are typical of late stages were analyzed to yield the topological parameters. These samples and their densities are listed in Table 8. The procedure for serial sectioning is described below, followed by the algorithm used and the results of the topological analysis. Serial Sectioning The first step in the technique of serial sectioning is to develop and standardize the procedure for removing a layer of desired thickness. This optimum thickness is such that it is small enough to encounter the smallest structural feature for a number of sections; yet large enough 51 25 to avoid redundancy of measurements. Patterson and Aigeltinger tackled this problem very systematically and found that the optimum thickness is of the order of one-fifth of that of scale of the structure. -P A reliable measure of the scale of the system is X the mean intercept _p- p cp of pore phase. Since x = 4VV/Sy the slope of the straight line in -P Figure 20 yields the value of X ~ 4.5 microns. Thus the serial sec tions for LS Series of samples ought to be roughly one micron apart. Table 11 COMPUTATION OF T net V Number of Number of Bottom T~~ = Top Ends Section # Top Ends Ends = A + B + Bottom Ends A: Disappearance of internal & external sub networks B: End of a Branch J+l J+2 T+ = Branchings __ and Connections TeL = T T 10 cn 86 Figure 46. Contribution of the end of an original subnetwork towards the estimate of Gmax. 69 Vv Figure 37. Dependence of grain boundary area on solid volume during loose stack sintering, hot pressing and conventional sin tering of INCO 123 nickel powder at 1250C. 128 Table 18 CALCULATED AND ESTIMATED VALUES OF V conn V cconn bV and M conn V No. VS Vv ..conn vV (.conn -1 , cm Mconn Mv , _ 2 cm Calc. Est. Calc. Est. Calc. Est., 1 0.906 0.05 0.09 479 731 -12.4 x 105 -20.3 x 105 2 0.928 0.04 0.053 373 474 -10.3 x 105 -15.2 x 105 3 0.944 0.015 0.017 133 138 -3.6 x 105 -3.1 x 105 4 0.971 0.022 0.017 257 165 -8.9 x 105 -4.0 x 105 5 0.979 0.009 0.02 202 237 -11.2 x 105 -10.7 x 105 105 after about 95 percent density has been reached. Thus, it can be said that the isolation of pores from an interconnected network continues until a point beyond which the residual collection of tree-like pores (those with very low connectivity) does not change appreciably. The isolated pores, on the other hand, continue to shrink and disappear. The preceding discussion of evolution of pore and grain boundary struc tures is expected to present the overall scenario described below. The Overall Scenario It is interesting to note that the density range of effective pinning of the grain boundary network coincides with that of little changes in the connected porosity; the major change is in the isolated porosity, Figure 51. If most of the associated porosity that anchors the grain boundaries is assumed to be a collection of isolated pores, the removal of isolated porosity in this density range can be explained by the proximity of grain boundaries that can act as vacancy sinks. If a balance exists between the number of isolated pores that disappear and the number of pores that are isolated as a result of channel closures, there will be sufficient associated porosity maintained to anchor the grain boundaries. This suggests that the isolated, equiaxed pores anchor the grain boundaries much more effectively than do the connected pores. A hypothesis is offered presently that attempts to rationalize the above contention. Grain boundary migration takes place during grain growth that decreases the grain boundary area. If part of the boundary area is occupied by second phase particles, as illustrated in Figure 54, the 52 Vv (Metal!ographic) Figure 21. Variation of integral mean curvature with solid volume during loose stack sintering, hot pressing and conven tional sintering of INCO 123 nickel powder at 1250C. 72 X/ Figure 40. Variation of the length of occupied grain edges per unit volume with the volume fraction of solid during loose stack sintering, hot pressing and conventional sintering of INCO 123 nickel powder at 1250C. 20 Figure 7. Schematic representation of the variation of surface area with solid volume for loose stack sintering and conventional sintering. The approach to the linear relation from a range of initial conditions is emphasized.35 33 Gp or Np (106 gm"1) Figure 16. Variation of genus per unit mass (Gp) and the number of isolated pores per unit mass (Np) with the volume fraction of solid for 48 micron spherical copper powder loose stack sintered at 1005C. Data for 115 micron spherical copper included for comparision.25 60 Figure 28. Photomicrograph of INCO 123 nickel powder hot pressed at 1250C to Vy = 0.93, etched (approx. 400 X). (a) (b) Figure 69. Illustration of a) "IS11 pore branches and b) "SS" pore branches. 117 Table 14 MEASURED VALUES OF THE NETWORK PARAMETERS No. vv b, cm"3 M3b -3 N\/ $ cm Njb, cm"3 N, cm"3 1 0.906 4.48 x 108 2.1 x 108 2.5 x 108 V 1.07 x 101 2 0.928 2.42 x 108 6.2 x 107 2.63 x 108 3.33 x 101 3 0.944 8.39 x 107 2.04 x 107 1.02 x 108 5.48 x 101 4 0.971 1.8 x 108 3.0 x 107 2.66 x 108 3.95 x 101 5 0.979 1.29 x 108 3.4 x 107 1.23 x 108 1.55 x 101 29 . SSP LV Figure 14. Variation of the length of lines of intersection of grain boundaries and the pore-solid interface (L$SP) with the corresponding value for the random intersection of the abovementioned surfaces (LR) for spherical copper powder loose stack sintered at 10T5C.48 126 Figure 62. Effects of channel closure and surface rounding on L, the average length of a branch. CHAPTER 4 CONCLUSIONS Introduction The discussion of the results of this investigation is summarized in a number of conclusions; an outline of suggested research is also presented. Conclusions 1) The porosity can be modeled as composed of a set of networks of cylindrical channels and a collection of monosized isolated spherical pores during the advanced stages of loose stack sintering, hot pressing and conventional sintering. 2) During loose stack sintering, a highly interconnected network of branches and nodes disintegrates into simpler subnetworks which sub sequently break up to form the isolated pores. The connected or tree-like pores continue shrinking until Vy r 0.95 when the rate of removal of these pores becomes significantly slow for the rest of the range of observation, up to Vy s 0.98. 3) Isolated porosity, on the other hand, goes through a maximum and diminishes to a very low value when connected porosity is observed to have been more or less stabilized. 4) The onset of stabilization of connected porosity is coincident with an arrest of grain growth and rapid reduction in the isolated porosity. 137 51 0.8 0.9 1.0 Vy (Metallographic) Figure 20. Variation of surface area of the pore-solid interface with solid volume during loose stack sintering, hot pressing and conventional sintering of INCO 123 nickel powder at 1250C. 139 12) It is suggested that this higher boundary area brings the onset of grain coarsening at a lower density, well before the pores begin to isolate. The early grain growth in PS leaves only a small opportunity for subsequently isolated pores to associate with the moving boundaries. Thus, an absence of an arrest in grain growth is attributed to the onset of grain growth well before that of isolation events. Suggestions for Further Study It should be apparent from the preceding discussions that topologi cal analysis of the advanced stages of hot pressing and conventional sintering would resolve the speculations about a higher number of iso lated pores in the hot pressed and sintered sample. The associated fractions of isolated and connected porosity during all three processes, when char acterized, would facilitate the mechanistic study of the advanced stages. Thus, the course of further research is outlined as follows. T) An etching procedure should be developed that will facilitate the determination of associated fractions of isolated and con nected porosity. 2) These fractions should be measured on sections in the series studied for topological, characterization. 3) The isolated and connected fractions should be determined on a section with calibrated polish. 4) The advanced stages of hot pressing and conventional sintering should be characterized regarding the topological properties of the pore structure. 80 obtained by counting the separate pores that appear and disappear within the volume observed and do not intersect the boundaries. The actual algorithm is as follows. The surface of a loose stack sintered sample, one from the series \ designated for topological analysis, was conditioned by polishing it on a microcloth with 1 micron diamond paste abrasive for about half an hour. This effectively removed all plastically deformed material, the result of an earlier alumina polish. The sample was cleaned in an ultrasonic cleaner, dried and viewed under a microscope to check for polishing artifacts. If the nickel ring, mentioned earlier in this section, was polished uniformly all around, the sample was examined for undue number of scratches that would hinder the analysis. If the polishing was uniform and had only a small number of scratches, it was deemed ready for further analysis; otherwise it was returned to the polishing step. Since the contours of the hardness indentation are mixed with those of the pore sections when observed for the measurement of diagonals, the thickness measurements become more difficult the more the sample is polished or the smaller the square-shaped impression. The trial sample of sintered nickel mentioned previously was polsihed, ten indentations were made on the nickel ring and the specimen and the sample repolished. This was followed by measuring the diagonals of impressions on both ring and specimen. The repetition of this procedure demonstrated that the extents of polishing (removal of material or layer thickness) of the ring and the sample were not statistically different. Indentations in the ring were therefore used to measure the section thickness. 24 p these curves become deeper and shift towards lower Vy with increasing 22 pressure, as shown in Figures 11 and 12. SS The.grain boundary area per unit volume, Sy > increases until a SS network is formed; subsequent grain growth tends to decrease Sy . 18 This was observed for loose stack sintering, as shown in Figure 13. SS P It is evident here that the variation of Sy with Vy is independent of the initial particle shape in the late second and early third stages. SSP P 22 Ly increases with decrease in Vy until the second stage is reached SSP when it begins to decrease. In the second stage Ly is significantly 48 higher than the case for random intersection of "SP" and "SS" surfaces, as illustrated in Figure 14. A new metric property, 1^, was discovered in the course of doctoral 22 research carried out by Gehl at the University of Florida; 1^ is the measure of inflection points observed on the traces of a surface per unit area of plane of polish, and is proportional to the integral curvature of asymptotic lines over saddle surfaces (surfaces that have principal radii of curvature of opposite signs at all points on the surface). This was 22 found to decrease smoothly in the second stage which means that the saddle surfaces occupy only a small fraction of the pore-solid interface at the end of the second stage. The variation of grain contiguity, grain face contiguity and grain shapes during conventional sintering and hot pressing were studied in 22 S SS some detail by Gehl. There were two parameters, C and C defined for grain contiguity and grain face contiguity, respectively. Four unitless parameters, F-j, Fg, Fg and F^, were used to characterize grain 22 and pore shapes. These were defined as follows. 41 Table 4 CHEMICAL COMPOSITION OF INCO TYPE 123 NICKEL POWDER Element Nickel Powder (Wt.%) Carbon (typical) 0.03-0.08 Carbon 0.1 max Oxygen 0.15 max Sulphur 0.001 max Iron 0.01 max Other Elements trace Nickel Balance 121: Type One-Three Three-Three Table 16 TYPES OF BRANCHES Order of Events Observed During Serial Sectioning New appearance + Branching New Appearance -* Connection Branching -* End of a branch Connection -* Bottom end Branching Bottom end Connection Branching Branching -* Branching Connection -* Connection Branching - Connection 127 in a sense, to estimate L and R. The calculated and estimated (with surface correction) values of Vy0nn, Sy0nn and My0nn are listed in Table 18. The estimation of the above values was made by using two conn P P experimentally determined values, such as Vy /Vy and Vy; the confidence intervals therefore were relatively large. High sample surface to volume ratio led to significant amounts of corrections, both in estimated and measured values. In light of the difficulties and the geometric simpli city of the model the agreement between measured and calculated values seems to indicate that the modeled geometry is qualitatively representa tive of the real microstructure. SP It was said earlier in this section that the linearity between Sy and Vy probably can be attributed to similar area to volume ratios for pore channels and isolated pores. From the Appendix, for a tubular channel with length = T and radius r = (/3/2)R, the area to volume ratio, Ay, is given by a = 2tttL = 2 _4_ = 2J1 V irr2L r n/3R R (23) as illustrated in Figure 63. For an isolated spherical pore, Ay = 3/R which is not far from 2.31/R. However, the curvature to volume ratios, Cy's, are significantly different. For a spherical pore, Cy vis given by Cy (spherical pore) = ^4 = -4> (24) -2TrRJ R and Cy for a tubular channel is given by C V (tubular channel) = ttL 2r irr L 1 2 r 4 = 1.33 3R2 R2 (25) 12 branches that can be cut without creating a new isolated part. If b = number of branches, n = number of nodes, P0 = number of separate parts, then P1 = b n + PQ (1) 29 The first Betti number of the network, P-j, is equal to the genus of the surface it represents. It may be apparent from Figure 3 that there exists some ambiguity as to the number of nodes and branches in a deformation retract. As illustrated in Figure 4, a number of additional nodes and branches can be used to represent the same region of space. Such spurious branches and nodes do not change the value of the connectivity because each spurious node introduces one and only on spurious branch. Quantities such as connectivity and number of separate parts or subnetworks are estimated by examining a series of parallel polished sections that cover a finite volume of sample, as described in Chapter 2. The investigations dealing with the study of sintering from the geometric viewpoint will be discussed presently. Sintering from a Geometric Viewpoint Three Stages of Sintering 30 31 Rhines and Schwarzkopf were among the first investigators to point out three more or less geometrically distinct stages that a sin tering structure traverses. The first stage is characterized by formation of initial inter particle contacts and their growth until these contact regions or necks 76 This fine scale ruled out the possibility of employing an established procedure for measuring thickness such as using a micrometer. A new, simple procedure was developed to measure the section thicknesses and is presently outlined. A microhardness tester was used to make square-based, pyramid shaped indentations on the polished surface of a sample. It is known that the apex angle of the diamond indentor is 136 and hence the ratio of the depth of an indentation to the diagonal of the impression is equal to 0.1428. As illustrated in Figure 42, the decrease in the depth of an indentation is 0.1428 times the decrease in the legnth of the diagonal. The hardness tester has a capability of a wide variety of loads and magnifications, so indentations of a wide variety of sizes can be made and measured with desired accuracy. Thus, the section thick ness can be easily measured by measuring the decrease in the length of the diagonals of an indentation. A Geotech automatic polisher was used to achieve a reproducible combination of polishing speed, load on the sample and polishing time that would yield the desired magnitude of material removal. A trial sample of sintered nickel was polished, indented with 30 indentations and the section thickness was measured several times by repeatedly polishing and measuring the diagonals until the polishing technique and measurement of section thickness were established with a high degree of confidence. An elaborate and rigorous procedure for topological analysis of porous bodies was developed in the course of doctoral research by 25 Aigeltinger. The abovementioned investigation dealt with loose stack 141 19. R. A. Gregg and F. N. Rhines, Met. Trans., 4, 1365 (1973). 20. R. T. DeHoff, R. A. Rummel, H. P. LaBuff and F. N. Rhines, Modern Developments in Powder Metallurgy, p. 310, Plenum Press, New York (1966). 21. W. D. Tuohig, Doctoral Dissertation, University of Florida (1972). 22. S. M. Gehl, Doctoral Dissertation, University of Florida (1977). 23. A. S. Watwe and R. T. DeHoff, unpublished research. 24. J. S. Adams and D. Glover, Metal Progress, August (1977). 25. E. H. Aigeltinger, Doctoral Dissertation, University of Florida (1969). 26. R. T. DeHoff and F. N. Rhines, eds., Quantitative Microscopy, McGraw Hill Book Co., New York (1967). 27. L. K. Barrett and C. S. Yust, ORNL Report, No. 4411 (1969). 28. L. K. Barrett and C. S. Yust, Metallography, 3^, 1 (1970). 29. S. S. Cairns, Introductory Topology, The Ronald Press Company, New York (1961T! 30. F. N. Rhines, Powder Met. Bull., 3^, 28 (1948). 31. P. Schwarzkopf, Powder Met. Bull., 2 74 (1948). 32. F. Thummler and N. Thomma, Met. Review, 12, 69 (1967). 33. R. L. Coble, J. Appl. Physics, 32, 787 (1961). 34. L. K. Barrett and C. S. Yust, Trans. Met. Soc. AIME, 239, 1172 (1967). 35. R. T. DeHoff and F. N. Rhines, Final Report, AEC Contract AT(40-1), 2581 (1969). 36. F. N. Rhines, University of Florida, private communication. 37. G. C. Kuczynski, Acta Met., 4, 58 (1956). 38. P. J. Wray, Acta Met., 24, 125 (1976). 39. W. D. Kingery and B. Francois, Sintering and Related Phenomena, p. 471, Gordon and Breach Publishers, New York (1965). 40. A. J. Markworth, Met. Trans., Â£, 2651 (1973). 41. W. Trzebiatowski, Zhurnal Physik Chem., B24, 75 (1934). 8 Table 1 BASIC METRIC PROPERTIES Feature Property Definition Lines LV Length of a linear feature per unit volume. Surfaces sv Area of a surface per unit volume. H . rl r2 Local mean curvature of a surface ata point on the surface, where r-i and rz are the principal radii of curvature. By convention, a radius of curvature is positive if it points into a solid phase. Thus, a convex solid has a positive curvature whereas a convex pore has a negative curvature. Mv =// HdS SV Integral mean curvature of a surface per unit volume. la Length of a trace of surface per unit area of a plane section. Regions of Space VV Volume fraction or volume of a particular region of space per unit volume. Figure 31. Photomicrograph of INCO 123 nickel powder hot pressed at 1250C to Vy = 0.968, etched (approx. 400 X). Ill Figure 56. Geometric model of connected and isolated porosity during the advanced stages of loose stack sintering of nickel at 1250C. 67 Figure 35. Photomicrograph of INCO 123 nickel powder cold pressed at 60,000 psi and sintered at 1250C to Vy = 0.975, etched (approx. 400 X). 161 If by(corr) is defined as the corrected number of branches per unit volume, T(corr) as the corrected average length of a branch and TT(add) as the weighted average of the length of additional branches, then by(corr)L(corr) = by*L + by(add)*L(add) (63) where by and T refer to the previously measured values of branches per unit volume and their average length. The values of by(corr), TT(corr), Nyb(corr), Ny^ and R1S0 are listed in Table 22; these were used to cal culate the metric properties of the connected porosity. The surface corrections used to modify the measured values of the metric properties of the connected porosity are described presently. It can be seen from equation (56) that the additional number of isolated pores per unit volume, Ny (add), is given by ,iso (64) N Miso, 1 S _V DS Ny (add) 2 Ny 2V . . 1_ 2V Since VyS0, SyS0 and MyS0, the metric properties of isolated poro sity, are directly proportional to the number of isolated pores, the additions to these properties are given by .ISO Vjs0(add) = 4 N SO N\/S(add) or VwS0(add) = vjso {^4 i DS 1 2V (65) Similarly, sjS0(add) = sjS0 v v 1-DS W (66) 119 properties of the connected porosity have also to be corrected; these corrected values are listed in Table 15. The remaining two parameters of the model, T and R, were estimated as follows. Since the model has only one-branch and three-branch nodes, the branches can be either one-three of three-three type. (One-one type branches are isolated pores and thus are not included in the connected porosity.) These branches can be classified further, as shown in Table 16 and illustrated in Figure 60. Since the procedure described in Chapter 3 involves recording of all these events, it was possible to measure the number and apparent lengths of each type of branches. As illustrated in Figure 61 for the case of randomly oriented branches, the true length of a branch is twice -IT the average of lengths measured at different orientations. Thus L and 33 L the true average lengths of 1-3 and 3-3 type branches can be esti mated from the separation between the pertinent events. The average lengths of 1-3 and 3-3 branches that cross the boundaries can be taken as twice the value measured, since for randomly oriented branches the average length of the part of the branch that is contained in the volume of analysis is half of its true length. The overall weighted average of all types of branches yields the required parameter, L. For a collection of isolated, equiaxed pores, VyS0, the volume fraction of isolated porosity, is given by (21) 28 2S SS CJ = 2sf+sSp (2) ,ss 3L SSS SSS., SSP JLv +LV (3) F, = 9I SSP.-, SSS Lv JLv (sgP+2sgS)2 (4) F2 = 2Lf P/(sf )2 (5) F~ = L^Lg3 2(sgS)2 (6) F4 = LgSP/2(sgS)2 (7) The fraction of the total area of solid grains shared with other c S P grains is given by C Variation of C with Vy for hot pressed samples of U02^22^ and loose stack sintered spherical and dendritic copper pc powder is shown in Figure 15. It can be seen from the definition of c c cc C that high C values indicate high Sy ; this was believed to arise from polycrystallinity of the particles. It is apparent that as third stage p (Vy = 0.1) is approached all data tend to fall on a single curve. Pre ss S compaction seems to increase Sy and hence exhibits higher values of C Copyright 1983 by Arunkumar Shamrao Watwe 75 Table 8 LOOSE STACK SINTERED SAMPLES USED FOR TOPOLOGICAL CHARACTERIZATION Number Sintering Time Volume Fraction of Solid 1 128.7 min. 0.906 2 190.0 min. 0.928 3 221.0 min. 0.944 4 352.5 min. 0.971 5 400.0 min. 0.979 91 Figure 48. Convention used in the net tangent count (Twet) . during serial sectioning. v Figure 18. Illustration of coarsening of a relatively large pore channel that results from an early closure of surrounding finer channels so that the solid conti nuity is maintained.34 31 P SP at the same Vy when compared to a loose stack sintered sample; Sy c was found to vary linearly with C and the dependence was the same for widely different precompaction pressures up to very late second C stage. For hot pressed samples, a maximum was observed in C believed to indicate a point where the grain boundary area has increased enough SSP to form a boundary network that subsequently coarsens. Both Ly and ccc c Ly exhibited a maximum when plotted versus C for conventionally sintered and hot pressed samples. SS The grain face contiguity parameter, C indicates the fraction of edge length of grain faces that is shared with other grains. It 22 S SS can be shown that C = C for the case of random intersection of grain boundaries and pore-solid interface, and C > C when grain boundaries intersect pore-solid interface preferentially. For conven ts tionally sintered and hot pressed samples C was observed to be greater SS than C which indicated preferential association of grain boundaries with the pore-solid interface. The factors F-|, Fg and F^ can be used to compare the grain shapes and ?2 the Pore shapes; F-|, Fg, Fg and F^ were observed to be weakly P linear with Vy, whereas a strong correlation was observed between Fg, C Fg and F^ and C for all the samples. It was believed that the above data indicate a strong influence of the extent of grain contiguity (C ) on the grain and pore shapes. 100 Figure 53. Dependence of the integral mean curvature of the connected porosity on the volume fraction of solid during loose stack sintering of nickel at 1250C. 2 The details of the geometry determine 1) the initial and boundary conditions for the flow equations, 2) the areas through which the material fluxes are assumed to occur, 3) the separation between sources and sinks, 4) the relationship between the variation of the chosen parameter and densification, and 5) the state of stress (important in plastic and viscous flow). Hence the time dependences derived in the model studies are influenced by the geometric detailsJ^~^ Different mechanisms exhibit different variations with temperature; thus, the relative importance of various mechanisms should depend on temperature and the chemical composition of the powder, as observed in several investigations. As pointed out by DeHoff et al.,^ sintering requires that densification, surface rounding, channel closure and removal of pores proceed in cooperation. Since all these involve different geometric events, the time exponent n in the relation x, the monitored parameter = (t)n varies with the particle size. Any mechanistic arguments must ultimately explain the observed geometrical changes taking place during sintering. It is thus evident ' that study of the changing geometry or microstructure should precede mechanistic investigations. Knowledge of the dependence of this micro structure on various process parameters such as initial powder character isties, temperature and external pressure would be very helpful in the control of sintering aimed at desired end properties of the components. 32 Topological Studies 12 It was found that the connectivity or genus, G, stays nearly constant during the first stage. More precise measurements 18 made by Aigeltinger and DeHoff indicated a definite increase in G during the first stage. This can be viewed as formation of additional interparticle contacts as particles come closer by densification. For 18 irregularly shaped powders, it was observed that G decreases during the first stage, due to coalescence of multiple contacts between particles cp p During the second stage, Sy decreased linearly with decrease in Vy-, the slope of this line was found to be proportional to Gy, genus per unit 20 volume as should be expected from dimensional analysis. Kronsbein et 49 al. carried out serial sectioning of sintered copper samples and found D that even for Vy = 0.1, very few pores were isolated. This is in agree- 34 ment with similar observations made by Barrett and Yust. 18 Aigeltinger and DeHoff studied loose stacking sintering of copper powder by measuring metric and topological properties. The genus per unit mass, Gp, number of isolated pores and number of contacts per parti cle, C", were the measured topological quantities. Variation of Gp and p number of isolated pores with Vy revealed a definite increase in the former during the first stage and identified the end of the second stage (Gp s 0). As shown in Figure 16, Gp and number of isolated pores were inversely proportional to the initial value of mean particle volume. The same plot for dendritic powder showed that the topological path is different up to late third stage and that the third stage (Gp ~ 0) p begins at a higher value of Vy as compared to spherical powder, Figure 17. The initial decrease in Gp during the first stage for dendritic 152 Figure 68. Height of the spherical cap in an extended, spherical, three-branch node. 45 Density Measurements The most common procedure for measuring density of a specimen is the liquid displacement method, wherein the volume of a specimen is estimated by measuring the volume of water displaced when it is immersed in water. Since in the cgs system of units the density of water is unity, this volume is numerically equal to the weight of water displaced, which is equal to the decrease in weight of the sample when immersed, according to the Archimedes principle. The major source of error in the case of this method lies in measuring the weight of the sample in water. A thin coating of paraffin wax, typically weighing a few tenths of a percent of the weight of the sample, was used to seal the surface pores during water immersion. The samples were suspended by placing them in a miniature rigid metal pan, thus eliminating the need to tie odd-shaped samples with a wire. Further, the use of this pan made it easy to correct for the volume of water displaced by the immersed part of the pan, whereas a similar correction in the case of a wire is not made easily. An elec tronic balance accurate to 0.1 mg was used to achieve the required high degree of accuracy. After the sample was weighed in air (W-|), it was coated with wax and weighed again (Wg). The wax-coated sample was placed in the minia ture pan in a beaker of distilled water and weighed (Wg). The sample was then dropped to the bottom of the beaker by gently tilting the pan. The weight of partially immersed pan was measured (W^). The density of the sample, p, was calculated as follows: 122 New Appearance Branching (b) Figure 60. Various types of a) one-three and b) three-three branches and the topological events signifying each type of branch. 78 sintered samples having densities in the range from 50 percent to 90 percent of the bulk value and hence exhibited pore structures of a large variety of scales and complexities. Since the samples used in the present investigation had densities higher than those used in this research, their pore structures were typically relatively simple. This made it possible to streamline and simplify the topological analysis to a great extent. The revised algorithm is presently described in detail. Algorithm for Topological Analysis Two topological parameters, namely, the connectivity and the number of separate pores, were measured in this investigation. Since the connec tivity is a measure of the number of redundant connections, there is an inevitable uncertainty regarding the connections between pores that inter sect the boundaries of the volume of analysis (which is a very small fraction of the sample volume). It is not possible to determine whether such pore sections intersect each other or meet with themselves outside the volume covered by the series of parallel areas of observation. This has led to the necessity of putting maximum and minimum limits on the estimate of connectivity. As illustrated in Figure 43, an upper limit on connectivity is obtained when all the pores meeting the boundaries of the volume of analysis are regarded as meeting at a common node, and is called Gmax. A lower limit is derived by considering all such pores to be terminating or "capped" at the boundaries, and is called Gmin. The quantity Gmin then consists solely of redundant connections or "loops" observed within the volume of analysis. The number of separate parts is 49 Table 7 STANDARD RELATIONSHIPS OF STEREOLOGY PL = 2 SV PA = 2 LV "tA = 2ttNa = MV (9) 00) (ID 02) 135 Once most of these are eliminated the pores isolated thereafter have only a limited opportunity to intersect the moving boundaries. Since SSP SSS Ly ancj l_v(occ) quantify the degree of association of porosity with the grain boundary network, it is apparent that initially a pressed and sintered sample has a higher amount of associated porosity that decreases rapidly. Before the onset of isolation processes, most of the porosity is in the form of an interconnected network mostly associated with grain edges. If grain coarsening begins well before an appreciable number of pores are isolated, this initially high associated porosity would decrease rapidly since the connected pores cannot anchor the boundaries effectively and are consequently disassociated. The significance of a topological study of the advanced stages of conventional sintering is stressed here as these measurements would characterize the associated fractions of iso lated and connected porosity and thus would test the postulates put for ward earlier in this section. Comparison of Loose Stack Sintering with Hot Pressing and Conventional Sintering If the paths of evolution of microstructure during these processes are examined together, a number of general processing parameter-micro- structure relationships become apparent. Since these relationships have a potential as possible strategies to control the microstructure and hence the service properties of a powder-processed component, they are of both theoretical and practical interest; they are listed below. 92 Figure 49. Dependence of connectivity on the volume fraction of solid during loose stack sintering of INCO 123 nickel powder at 1250C. 84 Table 9 OBSERVABLE TOPOLOGICAL EVENTS Appearance AGmax AGmin AN1S0 Whole Subnetworks Internal 0 0 +1 I Disappearance External -1 0 0 II Within a Appearance of a branch +1 0 0 Subnetwork Disappearance of a branch -1 0 0 III Between Subnetworks Different or Connection Same new 0 +1 0 +1 0 0 115 Figure 58. Illustration of pores of analysis and cross that terminate within the volume the surface, or "IS" branches. 148 Table 19 PARAMETERS OF THE MODEL Feature Notation Definition Branches J3 bV Number of one-three branches per unit volume h33 bV Number of three-three branches per unit vol ume Nodes Number of one-branch nodes per unit volume nf Number of three-branch nodes per unit volume Isolated Pores nJS0 Number of isolated pores observed to be wholly contained, per unit volume 157 Figure 70. Caliper diameters for a convex body. D, the mean caliper diameter, is the average of D's over all possible orientations of measuring planes. External Node Figure 47. Illustration of the contribution of a branching event towards Gmax. 16 I Figure 5. Illustration of neck growth and impingement of growing necks during the first stage of sintering. Table 10 CURRENT ESTIMATES OF Gmax, Gmin and Niso AFTER SECTION #J Height of Sample, Microns Counts .max gmin Previous total -- This section Current total Unit volume value _ Current total Current volume 108 into association with fractions of isolated and connected porosity. Since only simple, isolated pores pin the boundaries effectively, the grain boundary network is stabilized when it finds itself mostly asso ciated with isolated pores. By this time, most of the connected pores are disassociated,as they do not anchor the boundaries. The isolated porosity that is associated with the boundaries is reduced as the boundaries provide vacancy sinks for the necessary material transport. The balance between the number of isolated pores that disappear and the number of pores isolated by channel closures helps maintain a sufficient number of isolated pores that is associated with the boundaries that renders them immobile. Once most of the associated pores have disappeared, the boundaries become free to migrate and are not pinned by the remaining connected porosity. This slows the reduction of connected porosity con siderably. It is apparent from the preceding discussion that the pore phase in the advanced stages of loose stack sintering must be modeled as composed of isolated and connected fractions that vary in a manner IQ oo OC CO described above. Thus, the models that involve only connected 5 33 37 54 55 or isolated porosity * are not appropriate for describing the evolution of microstructure studied in this investigation. mechanis tic model satisfying the abovementioned geometry requirements may be devised, at least in principle, if grain boundary-associated fractions of connected and isolated porosities are measured during the advanced stages of loose stack sintering. These fractions were not measured since the sections studied for topological analysis could not be etched without affecting the pore features. Thus, it was not possible to devise 35 powder is in agreement with higher C = 14 for the initial stack than _ p C = 4 at Vy = 0.55. It has been argued that in second stage, on account of fewer pore channels in the sample sintered from dendritic powder, p isolation of pores begins at a higher Vy value than for the sample made from spherical powder. The maximum in the number of separate parts observed during the third stage was attributed to simultaneous shrinkage and coarsening. Initiation of rapid grain growth coincided with the approach of connectivity towards zero. Importance of the Present Research Microstructural characterizations of the last stages of sintering p where Vy goes from about 0.1 to nearly zero have been sketchy. The reasons for such a lack of data are evidently 1) For an aggregate of coarse powder particles that is convenient for serial sectioning, very long sintering times are required to obtain samples with such low values of porosity. 2) For a given range of densities, the paths of evolution of microstructure can be determined with a higher degree of confidence if a larger number of distinct microstructural states can be obtained and examined. Thus, it is desirable to have a sufficient number of samples that have the densities P in the range Vy = 0.1 and lower; this requires that the samples in the series have Vy values that are only a percent or so apart from each other. Due to this requirement and that of long sintering times, much preliminary experimental work is necessary to establish the required sintering schedules. 3) The topological meeasurements are very tedious in any case. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. R. T. DeHoff, Chairmar Professor of Material'Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. L n Â¡ip V .. R. E. Reed-Hill Professor of Materials Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Professor of Materials Science atar Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Inoda, Jr. Professor of Materials Science and Engineering 68 Figure 36. Photomicrograph of INCO 123 nickel powder cold pressed at 60,000 psi and sintered at 1250C to Vy = 0.983, etched (approx. 400 X). xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008997800001datestamp 2009-02-24setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Metric and topological characterization of the advanced stages of sintering dc:creator Watwe, Arunkumar Shamraodc:publisher Arunkumar Shamrao Watwedc:date 1983dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00089978&v=00001000506243 (alephbibnum)12203626 (oclc)dc:source University of Floridadc:language English 70 X/ Figure 38. Dependence of the length of grain edges per unit volume on the volume fraction of solid during loose stack sin tering, hot pressing and conventional sintering of INCO 123 nickel powder at 1250C. 10 Table 3 METRIC PROPERTIES OF A SINTERED STRUCTURE Feature Property Description Pore-Solid Interface CSP bV Area of pore-solid interface per unit volume MSP Mv Integral mean curvature of pore-solid interface per uni volume Porosity Vp Vv Volume fraction of porosity Solid Phase VS Vv Volume fraction of a solid Grain Edges in the Solid , sss Lv Length of grain edges or triple lines per unit volume Lines Formed by the Intersection of Grain Boundaries and Pore-Solid Interface , SSP Lv Length of intersection lines of pore-solid interface and grain boundaries per unit volume Grain Edges Occupied by the Pore Phase , sss LV(occ) Length of occupied grain edges per unit volume METRIC AND TOPOLOGICAL CHARACTERIZATION OF THE ADVANCED STAGES OF SINTERING By ARUNKUMAR SHAMRAO WATWE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1983 Copyright 1983 by Arunkumar Shamrao Watwe Dedicated To My Parents, Mr. Shamrao Vasudeo Watwe and Mrs. Sharada Shamrao Watwe ACKNOWLEDGEMENTS I am grateful for the opportunity to conduct my research under the guidance of Dr. R. T. DeHoff, the chairman of my advisory committee. An ability to approach any scientific matter with objectivity and logic has been blissfully passed on by him to all his students. I thank Drs. R. E. Reed-Hill, J. J. Hren, G. Y. Onoda, Jr., of the Department of Materials Science and Engineering and Dr. R. L.Scheaffer of the Department of Statistics for serving on my advisory committee. Their helpful advice and encouragement are deeply appreciated. It is a pleasure to thank my colleagues, Mr. Atul B. Gokhale and Mr. Shi Shya Chang,'for their collaboration in the experimental aspects of the project. Mr. Rudy Strohschein, Jr., of the Department of Chemistry assisted me beyond and above the call of duty in the fabrication of the sintering apparatus. He saved me a great deal of time and aggravation. All the credit for the preparation of this dissertation in its final form must go to Miss Debbie Perrine for her excellent typing. The financial support of the Center of Excellence of the State of Florida and the Army Research Office is gratefully acknowledged. iv TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS iv ABSTRACT vii INTRODUCTION 1 CHAPTER ONE EVOLUTION OF MICROSTRUCTURE DURING SINTERING 5 Introduction 5 Metric Properties of the Microstructure 5 Fundamentals of Topology 6 Sintering from a Geometric Viewpoint 12 Importance of the Present Research 35 TWO EXPERIMENTAL PROCEDURE AND RESULTS 39 Introduction 39 Sample Preparation 39 Metallography 46 Topological Measurements 74 THREE DISCUSSION 101 Introduction 101 Loose Stack Sintering 101 Hot Pressing 130 Conventional Sintering 134 Comparison of Loose Stack Sintering with Hot Pressing and Conventional Sintering 135 FOUR CONCLUSIONS 137 Introduction 137 Conclusions 137 Suggestions for Further Study 139 REFERENCES 140 APPENDIX THE GEOMETRIC MODEL OF THE PORE PHASE 143 Introduction 143 v PAGE Parameters of the Model 143 Metric Properties of the Connected Porosity 149 Surface Corrections 155 BIOGRAPHICAL SKETCH 163 VI Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy METRIC AND TOPOLOGICAL CHARACTERIZATION OF THE ADVANCED STAGES OF SINTERING By Arunkumar Shamrao Watwe' August, 1983 Chairman: Dr. R. T. DeHoff Major Department: Materials Science and Engineering Measurements of the metric properties of porosity and the grain boundary network during the advanced stages of loose stack sintering, conventional sintering and hot pressing of spherical nickel powder (average size 5.5 microns) were followed by topological analysis of the loose stack sintered samples. Linearity between area and volume of the pore phase for the loose stack sintered series was approached by the conventionally sintered and hot pressed series whereas the curvature values for these series remained significantly different. An arrest in grain growth during loose stack sintering was concurrent with the removal of most of the isolated porosity. Subsequent resumption of grain growth coincided with the stabilization of connected porosity. It is suggested that isolated, equiaxed pores pin the boundaries more effectively than do the connected pores. Increase in the boundary area accompanies the boundary migration for all orientations of an equiaxed pore whereas this is true only for a limited number of orientations of a connected pore. Consequently, isolated pores are removed via transport of vacancies to the occupied boundaries; subsequent resumption of grain growth slows the reduction of residual connected porosity. Porosity in loose stack sintered samples is modeled as a set of tubular networks and a collection of monodispersed spheres. Comparison of metric properties of loose stack sintered samples with those of conventionally sintered and hot pressed samples led to the speculations that a higher number of isolated pores exist during hot pressing and that the porosity in con ventionally sintered samples is composed of finer networks and smaller isolated pores. Absence of an arrest in grain growth during hot pressing is believed to be due to boundary migration that is induced by grain boundary sliding. A similar absence of an arrest in grain growth during conventional sin tering is attributed to the onset of grain growth well before that of isolation events. viii INTRODUCTION Sintering is a coalescence of powder particles into a massive formj wherein the densification is accompanied by a variety of profound geo metrical changes in the pore-solid composite. The mechanical and physical properties of a powder-processed compact are influenced by the geometry 2-9 of the pore phase. Thus, the manner in which the reduction of porosity takes place is of great practical and theoretical interest. There are two approaches to the study of sintering.^'frie traditional or mechanistic approach involves the study of kinetics and mechanisms of ^Im material transport;1the geometric viewpoint focuses on the geometry of the pore phase as it evolves during sintering. The latter approach involves estimation of size- and shape-dependent quantities (volume, area, etc.) and topological properties such as the connectivity and the number of separate parts. Mechanistic studies essentially consist of three steps: 1) A laboratory model of a particulate system is selected that is amenable to mathematical treatment of desired sophistication. Assumptions are made regarding the geometrical changes during sintering and the identities of source and sink of matter. 2) Kinetic equations are derived that describe the variation of a measurable parameter (width of an interparticle contact or "neck," density, etc.) for the particular mechanism(s) of interest. 3) These equations are compared with the experimentally observed time dependences of the chosen parameter and an attempt is made to identify the operating mechanism(s). 1 2 The details of the geometry determine 1) the initial and boundary conditions for the flow equations, 2) the areas through which the material fluxes are assumed to occur, 3) the separation between sources and sinks, 4) the relationship between the variation of the chosen parameter and densification, and 5) the state of stress (important in plastic and viscous flow). Hence the time dependences derived in the model studies are influenced by the geometric detailsJ^~^ Different mechanisms exhibit different variations with temperature; thus, the relative importance of various mechanisms should depend on temperature and the chemical composition of the powder, as observed in several investigations. As pointed out by DeHoff et al.,^ sintering requires that densification, surface rounding, channel closure and removal of pores proceed in cooperation. Since all these involve different geometric events, the time exponent n in the relation x, the monitored parameter = (t)n varies with the particle size. Any mechanistic arguments must ultimately explain the observed geometrical changes taking place during sintering. It is thus evident ' that study of the changing geometry or microstructure should precede mechanistic investigations. Knowledge of the dependence of this micro structure on various process parameters such as initial powder character isties, temperature and external pressure would be very helpful in the control of sintering aimed at desired end properties of the components. 3 Consequently, a major school of thought prevails that favors the geometri approach. The present investigation was undertaken to study the advanced stages of sintering (wherein the porosity values vary from ten to a few percent of the total volume) from this point of view. A main feature of this approach is the concept of path of evolution of microstructure. A given microstructure is characterized by its geo metric properties such as volume, area, curvature, connectivity, etc. A microstructural state is defined as a point in a n-dimensional space where each dimension denotes a particular microstructural property. As the microstructure evolves during a process, the resultant locus of such points represents the sequence of microstructural states that is obtained during the process. This sequence is termed the path of evolution of microstructure.^ It is convenient to represent two-dimensional projec tions of this path (two geometric properties at a time); usually one of them is the relative density or the volume fraction of solid. Previous studies of microstructural evolution during loose stack C ip 1 0_0*| sintering, conventional sintering-cold pressing followed by sinteringD,<:, and hot pressing*c ~ have provided a coherent pic ture of these processes for all but the last ten percent of the porosity. A detailed study of the late stages (porosity ten percent or lower) of loose stack sintering, conventional sintering and hot pressing would complete the picture of evolution of microstructure during these pro cesses. The practical interest in the behavior of porosity at these stages stems from the fact that a variety of commercial products made by powder technology are required to have porosities in the range 0.1 24 or lower. 4 The objectives of this research were to determine the paths of evolution of microstructure during the advanced stages of loose stack sintering, conventional sintering and hot pressing. Since the topologi- 25 cal measurements are time-consuming and since an earlier doctoral research dealt with topological characterization of loose stack sintering in the porosity range 0.1 and higher, it was planned to estimate the topological parameters for loose stack sintered series only. Metric properties of the pore structure and grain boundary network were estimated for all three series of samples. Previous investigations of this type are discussed in Chapter 1, followed by experimental procedure and results in Chapter 2. These results are discussed in Chapter 3 and the important findings and con clusions summarized in Chapter 4. CHAPTER 1 EVOLUTION OF MICROSTRUCTURE DURING SINTERING Introduction A microstructure is characterized by its metric and topological properties and therefore the following discussion will be carried out in terms of variation of these quantities as the sintering proceeds. These microstructural properties will be defined and the previous investigations of this type will be discussed in detail; a review of metric studies will be followed by topological analyses. The principles of quantitative stereology employed in the estimation of microstructural properties will be described in the next chapter on experimental procedure and results. Metric Properties of the Microstructure These quantities are estimated in terms of geometric properties of 26 lines, surfaces and regions of space averaged over the whole structure. The basic properties are listed in Table 1 and illustrated in Figure 1. Among the properties listed, Vy, Sy and My are used to yield two impor tant global averages of the microstructural properties. These are listed in Table 2. In a sintered structure, there are two regions of space or phases, namely, pore and solid, and two surfaces, pore-solid interface and grain boundaries. Two main linear features of interest are the grain edges and the lines formed as a result of intersection of pore-solid interface and grain boundaries. Superscripts are used to identify the properties that 5 6 are associated with a particular feature. These notations are listed in Table 3 and illustrated in Figure 2. In addition to the metric properties listed above, the microstruc ture of a porous body is also characterized by its topological properties. A brief discussion of the fundamentals of topology will precede the sur vey of microstructural studies of sintering. Fundametals of Topology The subset of topological geometry of present interest is that of 24 closed surfaces, that is to say, surfaces that may enclose a region of space. In a sintered body the regions of space are the pore and the solid phases; the pore-solid interface is a closed surface of interest. Such a surface may enclose several regions and have multiple connectivity. A surface is said to be multiply connected if there exist one or more redundant connections that can be severed without separating the surface in two. The genus of such a surface is defined as the number of redundant connections. For complex geometries it becomes difficult to visualize the topological aspects of surfaces. It has been found very convenient ;to represent surfaces by equivalent networks of nodes and branches. Such an equivalent network is called the deformation retract of a particular region of space. It is obtained by shrinking the surface without closing any 27 openings or creating new openings, until it collapses into the said 28 network that can be represented in the form of a simple line drawing. A number of closed surfaces and their equivalent networks are illustrated in Figure 3. The connectivity, P, of a network is equal to the number of 7 Figure 1. Illustration of basic metric properties. 8 Table 1 BASIC METRIC PROPERTIES Feature Property Definition Lines LV Length of a linear feature per unit volume. Surfaces sv Area of a surface per unit volume. H . rl r2 Local mean curvature of a surface ata point on the surface, where r-i and rz are the principal radii of curvature. By convention, a radius of curvature is positive if it points into a solid phase. Thus, a convex solid has a positive curvature whereas a convex pore has a negative curvature. Mv =// HdS SV Integral mean curvature of a surface per unit volume. la Length of a trace of surface per unit area of a plane section. Regions of Space VV Volume fraction or volume of a particular region of space per unit volume. 9 Table 2 DERIVED METRIC PROPERTIES Feature Property Definition Surface - MV H =-r Average mean curvature of sv a surface Region of Space x = 4sVv Mean intercept in a particular region of space 10 Table 3 METRIC PROPERTIES OF A SINTERED STRUCTURE Feature Property Description Pore-Solid Interface CSP bV Area of pore-solid interface per unit volume MSP Mv Integral mean curvature of pore-solid interface per uni volume Porosity Vp Vv Volume fraction of porosity Solid Phase VS Vv Volume fraction of a solid Grain Edges in the Solid , sss Lv Length of grain edges or triple lines per unit volume Lines Formed by the Intersection of Grain Boundaries and Pore-Solid Interface , SSP Lv Length of intersection lines of pore-solid interface and grain boundaries per unit volume Grain Edges Occupied by the Pore Phase , sss LV(occ) Length of occupied grain edges per unit volume 77 12 branches that can be cut without creating a new isolated part. If b = number of branches, n = number of nodes, P0 = number of separate parts, then P1 = b n + PQ (1) 29 The first Betti number of the network, P-j, is equal to the genus of the surface it represents. It may be apparent from Figure 3 that there exists some ambiguity as to the number of nodes and branches in a deformation retract. As illustrated in Figure 4, a number of additional nodes and branches can be used to represent the same region of space. Such spurious branches and nodes do not change the value of the connectivity because each spurious node introduces one and only on spurious branch. Quantities such as connectivity and number of separate parts or subnetworks are estimated by examining a series of parallel polished sections that cover a finite volume of sample, as described in Chapter 2. The investigations dealing with the study of sintering from the geometric viewpoint will be discussed presently. Sintering from a Geometric Viewpoint Three Stages of Sintering 30 31 Rhines and Schwarzkopf were among the first investigators to point out three more or less geometrically distinct stages that a sin tering structure traverses. The first stage is characterized by formation of initial inter particle contacts and their growth until these contact regions or necks 13 begin to impinge each other, as illustrated in Figure 5. Due to differ ent crystallographic: orientations of adjacent particles, grain boundaries form in the interparticle contact regions. In this stage, the area of 32 pore-solid interface decreases with a moderate amount of shrinkage. Throughout this stage, the pore-solid interface has many redundant con nections J During the second stage, the distinguishing features are not the interparticle contacts or "necks" but the pore channels formed as a result of the impingement of neighboring necks. Virtually all of the 1 6 *3*3 porosity is in the form of an interconnected network of channels * that delineate the solid grain edges. The continued reduction in the volume and the area of porosity is accompanied by a decrease in the 1 35 connectivity of the pore structure. The decrease in the connec tivity can be explained by either removal of solid branches or closure 36 of pore channels. According to Rhines, the connected pore network coarsens, analogous to a grain edge network in a single phase polycrystal (driven by excess surface energy) as illustrated in Figure 6. In this scenario, a fraction of solid branches (necks) are pinched off and no new pores are isolated. Although a finite number of isolated pores observed during the late second stage can be explained only by channel closure events, a closer scrutiny is needed to resolve this issue. The isolated pores may be irregular in shape.16,34 30 31 The third stage has begun by the time most of the pores are isolated. The connectivity of a pore network is now a very small number.1 Coarsening 16 18 35 37-39 proceeds along with the spheroidization of pores * so that the volume of porosity, the number of pores and pore-solid interface area 14 Figure 3 Some closed surfaces and their deformation retracts (dotted lines). 15 Figure 4. Illustration of a one-to-one correspondence between a spurious node and a spurious branch in a deformation retract. 16 I Figure 5. Illustration of neck growth and impingement of growing necks during the first stage of sintering. 17 continue to decrease. If the pores are filled with a gas of low solu bility or very slow diffusivity, then coarsening leads to an increase in volume of porosity.^^0 gas Â¡ias en0Ugh pressure to stabilize 16 41 42 the pore-solid interface, the densification rates can be very low. Since exaggerated or secondary grain growth that results from boundaries 43 44 breaking away from pores has been observed to be accompanied by slow rates of shrinkage,43,45-47 has been theorized that the grain boundaries that can act as efficient vacancy sinks are far away from a large number of 43 45-47 pores. 5 The end of the third stage is of course the disappearance of all pores, although that is rarely accomplished in practice. The three stages described above provide a common framework for the discussion of microstructural studies that are reviewed presently. This review is expected to demonstrate the potential that the present research has for providing a perspective of sintering that is more profound than the current one. Metric Investigations SP It has been observed that in loose stack sintered samples Sy decreased linearly with the decrease in Vy^2*1820) (jur-jng second stage. Surface area may be reduced both by densification and surface SP rounding or by surface rounding alone; the linearity between Sy and Vy is believed to arise from a balance between surface rounding and densifi cation. Support for this hypothesis comes from the observation that sur face rounding dominates in pressed and sintered samples until the balance has been reached, as shown schematically in Figure 7. The slope of the SP 20 Sy versus Vy line is inversely proportional to the initial particle size. 18 Figure 6. Two basic topological events that occur in the network coarsening scenario proposed by Rhines.36 The dotted lines indicate the occupied grain edges. 19 There is evidence to suggest that this path of evolution of microstruc- 21 ture for loose stack sintering is insensitive to temperature. SP Data for hot pressed samples indicate that the Sy -Vy relationship 22 23 is only approximately linear even in the late second stage. The path of microstructural change was also found to be insensitive to tem- 22 23 perature. The effect of pressure on the path was significant; increasing pressure delayed the approach to linearity until a lower p value of Vy, as shown in Figure 8. Integral mean curvature per unit volume, My, has been measured for loose stack sintering, conventional sintering (cold pressing followed by sintering) and hot pressing in the density range characteristic of late second stage. A convex particle has a positive curvature whereas a con- 18 21 22 vex pore has a negative curvature. There is a miniimum in.*.My; * 35 this minimum occurs at lower Vy for finer particle size, as illustrated in Figure 9. According to the convention used, most of the "SP" surface has positive curvature in the initial stages. Due to decreasing surface area and increasing negative curvatures there occurs a minimum in My in the second stage. As the sintered density approaches the theoretical density, My must approach zero and hence the initially high positive value of My that becomes negative must go through a minimum. For an p initial stack of irregularly shaped particles, My varies with Vy at a slower rate and has a minimum earlier in the process, compared to an 25 initial stack of spherical powders. This is illustrated in Figure 10. In all the cases studied the paths were insensitive to temperature. In the case of hot pressing, the minimum in My is much more negative and P 23 occurs at a lower value of Vy, compared to a loose stack sintered sample; 20 Figure 7. Schematic representation of the variation of surface area with solid volume for loose stack sintering and conventional sintering. The approach to the linear relation from a range of initial conditions is emphasized.35 21 Figure 8. Pore-solid interface area versus solid volume fraction for conventionally sintered 22 M SP V Figure 9. Variation of integral mean curvature per unit volume with the volume fraction of solid for three representative copper powders sintered in dry hydrogen at 1005C.35 23 Figure 10. Integral mean curvature versus volume fraction of solid for 48 micron spherical and dendritic copper powder.25 24 p these curves become deeper and shift towards lower Vy with increasing 22 pressure, as shown in Figures 11 and 12. SS The.grain boundary area per unit volume, Sy > increases until a SS network is formed; subsequent grain growth tends to decrease Sy . 18 This was observed for loose stack sintering, as shown in Figure 13. SS P It is evident here that the variation of Sy with Vy is independent of the initial particle shape in the late second and early third stages. SSP P 22 Ly increases with decrease in Vy until the second stage is reached SSP when it begins to decrease. In the second stage Ly is significantly 48 higher than the case for random intersection of "SP" and "SS" surfaces, as illustrated in Figure 14. A new metric property, 1^, was discovered in the course of doctoral 22 research carried out by Gehl at the University of Florida; 1^ is the measure of inflection points observed on the traces of a surface per unit area of plane of polish, and is proportional to the integral curvature of asymptotic lines over saddle surfaces (surfaces that have principal radii of curvature of opposite signs at all points on the surface). This was 22 found to decrease smoothly in the second stage which means that the saddle surfaces occupy only a small fraction of the pore-solid interface at the end of the second stage. The variation of grain contiguity, grain face contiguity and grain shapes during conventional sintering and hot pressing were studied in 22 S SS some detail by Gehl. There were two parameters, C and C defined for grain contiguity and grain face contiguity, respectively. Four unitless parameters, F-j, Fg, Fg and F^, were used to characterize grain 22 and pore shapes. These were defined as follows. 25 Figure 11. Variation of integral mean curvature per unit volume with the volume fraction of solid during hot pressing of RSR 107 nickel (-170+200) at 1500 psi. Data for spherical copper (-170+200) loose stack sintered at 1005C included for comparison.23.25 26 Figure 12. The effect of pressure on the path of integral mean curvature for hot pressed specimens of UOg.^ 27 SyP (cm-1) Figure 13. Grain boundary area per unit volume versus volume fraction of solid_for 48 micron spherical and dendritic copper . powder.25 28 2S SS CJ = 2sf+sSp (2) ,ss 3L SSS SSS., SSP JLv +LV (3) F, = 9I SSP.-, SSS Lv JLv (sgP+2sgS)2 (4) F2 = 2Lf P/(sf )2 (5) F~ = L^Lg3 2(sgS)2 (6) F4 = LgSP/2(sgS)2 (7) The fraction of the total area of solid grains shared with other c S P grains is given by C Variation of C with Vy for hot pressed samples of U02^22^ and loose stack sintered spherical and dendritic copper pc powder is shown in Figure 15. It can be seen from the definition of c c cc C that high C values indicate high Sy ; this was believed to arise from polycrystallinity of the particles. It is apparent that as third stage p (Vy = 0.1) is approached all data tend to fall on a single curve. Pre ss S compaction seems to increase Sy and hence exhibits higher values of C 29 . SSP LV Figure 14. Variation of the length of lines of intersection of grain boundaries and the pore-solid interface (L$SP) with the corresponding value for the random intersection of the abovementioned surfaces (LR) for spherical copper powder loose stack sintered at 10T5C.48 30 Figure 15. The variation of grain contiguity with solid volume fraction for loose stack sintered copper and hot pressed U02.22 31 P SP at the same Vy when compared to a loose stack sintered sample; Sy c was found to vary linearly with C and the dependence was the same for widely different precompaction pressures up to very late second C stage. For hot pressed samples, a maximum was observed in C believed to indicate a point where the grain boundary area has increased enough SSP to form a boundary network that subsequently coarsens. Both Ly and ccc c Ly exhibited a maximum when plotted versus C for conventionally sintered and hot pressed samples. SS The grain face contiguity parameter, C indicates the fraction of edge length of grain faces that is shared with other grains. It 22 S SS can be shown that C = C for the case of random intersection of grain boundaries and pore-solid interface, and C > C when grain boundaries intersect pore-solid interface preferentially. For conven ts tionally sintered and hot pressed samples C was observed to be greater SS than C which indicated preferential association of grain boundaries with the pore-solid interface. The factors F-|, Fg and F^ can be used to compare the grain shapes and ?2 the Pore shapes; F-|, Fg, Fg and F^ were observed to be weakly P linear with Vy, whereas a strong correlation was observed between Fg, C Fg and F^ and C for all the samples. It was believed that the above data indicate a strong influence of the extent of grain contiguity (C ) on the grain and pore shapes. 32 Topological Studies 12 It was found that the connectivity or genus, G, stays nearly constant during the first stage. More precise measurements 18 made by Aigeltinger and DeHoff indicated a definite increase in G during the first stage. This can be viewed as formation of additional interparticle contacts as particles come closer by densification. For 18 irregularly shaped powders, it was observed that G decreases during the first stage, due to coalescence of multiple contacts between particles cp p During the second stage, Sy decreased linearly with decrease in Vy-, the slope of this line was found to be proportional to Gy, genus per unit 20 volume as should be expected from dimensional analysis. Kronsbein et 49 al. carried out serial sectioning of sintered copper samples and found D that even for Vy = 0.1, very few pores were isolated. This is in agree- 34 ment with similar observations made by Barrett and Yust. 18 Aigeltinger and DeHoff studied loose stacking sintering of copper powder by measuring metric and topological properties. The genus per unit mass, Gp, number of isolated pores and number of contacts per parti cle, C", were the measured topological quantities. Variation of Gp and p number of isolated pores with Vy revealed a definite increase in the former during the first stage and identified the end of the second stage (Gp s 0). As shown in Figure 16, Gp and number of isolated pores were inversely proportional to the initial value of mean particle volume. The same plot for dendritic powder showed that the topological path is different up to late third stage and that the third stage (Gp ~ 0) p begins at a higher value of Vy as compared to spherical powder, Figure 17. The initial decrease in Gp during the first stage for dendritic 33 Gp or Np (106 gm"1) Figure 16. Variation of genus per unit mass (Gp) and the number of isolated pores per unit mass (Np) with the volume fraction of solid for 48 micron spherical copper powder loose stack sintered at 1005C. Data for 115 micron spherical copper included for comparision.25 34 Figure 17. a) Genus per gram (Gp) the number of isolated pores per gram (Gp) versus volume fraction of solid for 48 micron dendritic copper powder. Data for 48 micron spherical powder included for comparison, b) Enlarged part of lower right corner of (a).2^ 35 powder is in agreement with higher C = 14 for the initial stack than _ p C = 4 at Vy = 0.55. It has been argued that in second stage, on account of fewer pore channels in the sample sintered from dendritic powder, p isolation of pores begins at a higher Vy value than for the sample made from spherical powder. The maximum in the number of separate parts observed during the third stage was attributed to simultaneous shrinkage and coarsening. Initiation of rapid grain growth coincided with the approach of connectivity towards zero. Importance of the Present Research Microstructural characterizations of the last stages of sintering p where Vy goes from about 0.1 to nearly zero have been sketchy. The reasons for such a lack of data are evidently 1) For an aggregate of coarse powder particles that is convenient for serial sectioning, very long sintering times are required to obtain samples with such low values of porosity. 2) For a given range of densities, the paths of evolution of microstructure can be determined with a higher degree of confidence if a larger number of distinct microstructural states can be obtained and examined. Thus, it is desirable to have a sufficient number of samples that have the densities P in the range Vy = 0.1 and lower; this requires that the samples in the series have Vy values that are only a percent or so apart from each other. Due to this requirement and that of long sintering times, much preliminary experimental work is necessary to establish the required sintering schedules. 3) The topological meeasurements are very tedious in any case. 36 The present investigation that dealt with the microstructural characterization of the advanced stages of sintering has a potential for enhancing and quantifying the existing sketchy picture of the late stages of sintering. The theoretical and practical importance of this work can be appreciated from the following discussion. It has been theorized161835,37-39 that the spheroidization of pores proceeds along with coarsening during the advanced stages. It is necessary to couple topological analysis with the metric measurements to study the spheroidization and coarsening of isolated pores. To date, there has been no such direct observation of the behavior of isolated porosity. If a pore of higher than average size is surrounded by a shell of higher than average density with finer pore channels, then early clo sure of these channels pulls the solid shell away from the large pore so 34 that the continuity of the solid phase is maintained, as illustrated in 34 Figure 18. According to Barrett and Yust, most of the reports of coarsening are in fact the observed removal of smaller channels before the larger ones. Another disputed contention is that of deceleration of densification due to separation of grain boundaries from isolated 43 45-47 pores. A pore that is observed to be isolated on a two dimen sional section may or may not be so in the third dimension, whether associated with the grain boundaries or not. The topological analysis of grain boundary-porosity association alone can determine the true extent of association of isolated porosity with the boundaries. A detailed geometric study of porosity in the advanced stages will clarify some aspects of microstructural evolution mentioned above. Figure 18. Illustration of coarsening of a relatively large pore channel that results from an early closure of surrounding finer channels so that the solid conti nuity is maintained.34 38 Mechanical and physical properties of conmercial porous components are influenced by the geometry of the porosity. Thermal conductivity is P influenced by Vy, pore shapes and the relative fractions of connected and 7 8 4 isolated porosity. Permeability to fluids depends on the connectivity, p cp g 3 Vy and Sy Mechanical strength and thermal shock resistance depend on 2 pore shapes whereas ductility is influenced by pore shapes and spacings. Thus geometric characterization of porous structures as a function of adjustible process parameters would suggest a number of potential strate gies to control the final service properties. It is apparent from the review of previous microstructural studies of sintering that the present investigation is expected to offer a much needed general and quantitative picture of the advanced stages of sin tering. The experimental procedure employed in the present research is described in detail in the next chapter. CHAPTER 2 EXPERIMENTAL PROCEDURE AND RESULTS Introduction Microstructural characterization involved sample preparation, metallography and in the case of loose stack sintering, also serial sectioning. These are described in detail in this chapter, followed by results of this investigation. Sample Preparation This section presents the procedure employed to prepare the sintered samples and the standard for density measurements. Sintered Samples Three series of samples of sintered nickel powder were prepared: 1) loose stack sintered (LS), 2) pressed and sintered (PS) and 3) hot pressed (HP). In order to study the path of evolution of microstructure during the late stages of sintering, it is desirable to obtain samples c having densities that are uniformly distributed over the range Vy = 0.85 to 1.0. Accordingly, preliminary experiments were designed to determine the processing parameters, such as temperature, pressure and time, that yield the desired series of samples made from the selected metal powder. INCO type 123 nickel powder, illustrated in Figure 19, supplied by the International Nickel Company, Inc., with the chemical and physical prop erties listed in Tables 4 and 5, was used in the present investigation. 39 40 Figure 19. INCO 123 nickel powder used in the present investigation (1000 X). 41 Table 4 CHEMICAL COMPOSITION OF INCO TYPE 123 NICKEL POWDER Element Nickel Powder (Wt.%) Carbon (typical) 0.03-0.08 Carbon 0.1 max Oxygen 0.15 max Sulphur 0.001 max Iron 0.01 max Other Elements trace Nickel Balance 42 Table 5 PHYSICAL PROPERTIES OF INCO TYPE 123 NICKEL POWDER Particle Shape Roughly spherical with spiky surface Average Particle Siz 5.5 microns Standard Deviation 0.75 microns Surface Area Per Unit Volume 7.65 x 10^ cm^/cm^ 5 Vy of As-Received Powder 0.25 43 It was found by trial and error that sintering a loose stack of this powder at 1250C produced the required series of samples in convenient sintering times. This sintering temperature was also used for PS and HP series, in order to ensure that the differences among the paths of evolution of microstructure for LS, PS and HP series were not due to different sin tering temperatures. 1. Loose Stack Sintered Series In order to have the same initial microstructure for all the samples in a series, they were prepared from the same initial loose stack of powder. The first sample of the series was prepared by heating a loose stack of powder (tapped to yield a level top surface) in an alumina boat (6 x 12 x 75 mm) under a flowing dry hydrogen atmosphere for the specified sintering time. A small piece (about 5 mm thick) was cut off and stored for subsequent characterization; the rest of the sintered body was used to yield the remaining samples in the series by the repetition of the procedure described above for an appropriate sequence of accumulated sintering times. It required 11 minutes for the sample transferred from the cold zone to the hot zone to reach the sintering temperature. Although this time was not negligible compared to the time spent at the sintering temperature and although this procedure takes the samples through an increasing number of heating and cooling cycles with longer sintering times, it has been shown that these cycles do not influence the path of microstructural 32 change in metal powders. 44 2. Pressed and Sintered Series A CARVER hydraulic hand press was used to prepare cylindrical pellets about 15 mm in diameter and typically 3 mm in height. Cold pressing at 60,000 psi followed by sintering at 1250C yielded the desired series of samples. Due to the small size of these pellets and their patterns of potential inhomogeneity it was not feasible to prepare the series of samples from a single initial compact, as in the loose stack case. Instead, samples in this series were pre pared individually by sintering the green compacts in an alumina boat under a flowing dry hydrogen atmosphere for preselected sintering times at 1250C 10C. 3. Hot Pressed Series The third series was prepared by hot pressing at 1250C and under a pressure of 2000 psi in a CENT0RR high vacuum hot press. A loose stack of powder was placed in a cylindrical boron nitride die 2.54 cm in diameter and tapped; the die with the top punch resting on the powder -5 was placed in the vacuum chamber. After a vacuum of 10 Torr was reached the induction coil was switched on. The attainment of sintering temper ature which nominally required one hour was followed by the application of a pressure of 2000 psi. The pressure was maintained and the tempera ture controlled to 5C for the specified sintering times; the pressure was then released and the induction coil turned off. After the sample was allowed to cool overnight, air was admitted and the die assembly removed. As in the case of PS series, samples in this series were made individually. 45 Density Measurements The most common procedure for measuring density of a specimen is the liquid displacement method, wherein the volume of a specimen is estimated by measuring the volume of water displaced when it is immersed in water. Since in the cgs system of units the density of water is unity, this volume is numerically equal to the weight of water displaced, which is equal to the decrease in weight of the sample when immersed, according to the Archimedes principle. The major source of error in the case of this method lies in measuring the weight of the sample in water. A thin coating of paraffin wax, typically weighing a few tenths of a percent of the weight of the sample, was used to seal the surface pores during water immersion. The samples were suspended by placing them in a miniature rigid metal pan, thus eliminating the need to tie odd-shaped samples with a wire. Further, the use of this pan made it easy to correct for the volume of water displaced by the immersed part of the pan, whereas a similar correction in the case of a wire is not made easily. An elec tronic balance accurate to 0.1 mg was used to achieve the required high degree of accuracy. After the sample was weighed in air (W-|), it was coated with wax and weighed again (Wg). The wax-coated sample was placed in the minia ture pan in a beaker of distilled water and weighed (Wg). The sample was then dropped to the bottom of the beaker by gently tilting the pan. The weight of partially immersed pan was measured (W^). The density of the sample, p, was calculated as follows: 46 /, _\ Weight of sample p^9/ I Volume of sample Weight of sample . (Volume of sample + immersed part of pan) - (Volume of immersed part of pan) M1 (W2 + weight of pan in air Wg) - (Weight of pan in air W4) Thus W1 p (g/cc) = w2 + w4 w3 The densities thus measured were reproducible within 0.2 percent of the mean of ten values with 95 percent confidence. The density of a 50 piece of pure nickel, known to have a density of 8.902 g/cc, was measured and found to be within 0.5 percent of the abovementioned value. Metallography The polishing procedure will be described and followed by a brief discussion of principles of quantitative stereology involved in the estimation of metric properties. The estimated microstructural properties will be presented thereafter. Polishing Procedure The wax coating on the samples was dissolved in hexane and the samples were sectioned; a vacuum impregnation method was used to mount the samples, surrounded by a nickel ring, in epoxy. The purpose 47 of the ring will be discussed later in this section. Rough polishing was done on wet silicon carbide papers of increasing fineness from 180 grit through 600 grit. Fine polishing was done by using 6 micron diamond paste, followed by 1 micron diamond paste, 0.3 micron alumina and finally 0.05 micron alumina. Quantitative Stereo!ogy Metric properties are estimated by making measurements on a two dimensional plane of polish with the help of standard relations pc of stereology. A set of test lines, arranged in a grid pattern, also provide a set of test points and a test area to characterize the plane section; these are usually used to make the measurements listed in Table 6. The relationships between these measurements and the globally averaged properties of the three dimensional microstruc ture are listed in Table 7. The relations yield estimates of popula- pc tion or structure properties provided the structure is sampled uniformly. Stereological counting procedure and the estimated properties will be discussed presently. Each metallographically prepared surface was calibrated by measuring the volume fraction of porosity by quantitative stereology and comparing the result with the value obtained from density measurements. A definite amount of plastic deformation by the polishing abrasive media leads to a smearing effect that introduces some error in quantifying the information on a polished section. This effect can be viewed as local movements of traces of the pore-solid interface; all the counted events (number, inter cept, etc.) are therefore error-prone to some extent. As this investigation 48 Table 6 QUANTITIES MEASURED ON POLISHED SECTION Test Feature Quantity Points Pp Lines Ll Area NA ta Definition Fraction of points of a grid that fall in a phase of interest Fraction of length of test lines that lie in a phase of interest Number of intercepts that a test line of unit length makes with the trace of a surface on a plane section Number of points of emergence of linear feature per unit area of plane section Number of full features that appear on a section of unit area Net number of times a sweeping test line is tangential to the convex and concave traces of surface per unit area of a plane section 49 Table 7 STANDARD RELATIONSHIPS OF STEREOLOGY PL = 2 SV PA = 2 LV "tA = 2ttNa = MV (9) 00) (ID 02) 50 dealt with relatively small amounts of porosity (10 percent or lower) P the error in Vy introduced by the polishing technique approached that of the density measurements, namely, about 0.005, as the sintered den sity approached the bulk density. Thus, the polishing was accepted for p further characterization if the metal!ographically determined Vy was p within 15 percent of Vy obtained from the water immersion method, except for the samples 97 percent dense and higher for which the limit had to be relaxed to 30 percent of Vy. Since Vy values range from 0.15 to 0.02, the abovementioned limits translate into a few percent of the sample den sity as measured metallographically. Typically, the samples 97 percent p dense and lower exhibited a precision of 0.05 of the Vy value obtained SP SP from the density measurements. Manual measurements of Sy and My were made on the accepted polished surfaces using standard stereological techni- ques. The measurements of Vy, Sy and My were made with at least 30 different fields and at magnifications that allowed at least 15 pores to be viewed in a single field. As a result, the estimates of the properties were within 5 percent with 95 percent confidence, as illustrated in Figures SP SP 20 and 21. Plots of Sy and My contained metallographically measured values of Vy to yield the paths of evolution of microstructure in order to partially compensate for the polishing errors. Measurement of these metric properties was followed by etching the specimens to reveal the grain boundaries. Each sample was immersed in a solution made from equal parts of nitric acid, glacial acetic acid and acetone for about 30 seconds. The grain boundaries were brought out clearly with some evidence of facetting of the initially smooth contours of pore features. Samples in the lower part of the density range exhibited 51 0.8 0.9 1.0 Vy (Metallographic) Figure 20. Variation of surface area of the pore-solid interface with solid volume during loose stack sintering, hot pressing and conventional sintering of INCO 123 nickel powder at 1250C. 52 Vv (Metal!ographic) Figure 21. Variation of integral mean curvature with solid volume during loose stack sintering, hot pressing and conven tional sintering of INCO 123 nickel powder at 1250C. 53 grain structures that were too fine to be studied optically; these samples were not included in the measurement of grain structure prop erties. A number of etched microstructures are illustrated in Figures 22 through 36. Typically, the scale of the grain structure was such that the information contained in a single plane section was not enough to yield estimates with the desired precision of 10 percent. Conse quently, SyS, LySS, LySP and Ly^ccj, defined earlier in this report, were measured by repeating the polishing, etching and counting steps a number of times to obtain at least 100 different fields of view. The grain structure properties are illustrated in Figures 37 through 40. The apparent local movements of the traces of the pore-solid interface, mentioned earlier in this section, are likely to introduce some errors in the estimation of grain structure properties whenever the pores are associated with the boundary network. For example, an enlargement.of pore features residing on grain boundaries would underestimate the value of Sy, the grain boundary area per unit volume, as measured metal!o- graphically. However, it was found that these errors are small compared CD CD . to those in Sy and My ; the trends of grain structure properties remain unaffected whether plotted versus Archimedes density or the stereo!ogical density. The quantities in Figures 37 through 40 are thus plotted versus the Archimedes density. The pores observed on a polished and etched surface can be classi fied as to their association with the grain boundaries, that is, according to whether they appear to be inside a grain, on the grain boundary or on a grain edge. The relative fractions of pore features regarding their association with the boundary network were measured. These are illustrated in Figure 41. 54 Figure 22. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vy = 0.871, etched (approx. 400 X). 55 Figure 23. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vy = 0.906, etched (approx. 400 X). 56 Figure 24. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vy = 0.928, etched (approx. 400 X). 57 Figure 25. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vy = 0.944, etched (approx. 400 X). 58 Figure 26. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vy = 0.971, etched (approx. 400 X). 59 Figure 27. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vy = 0.979, etched (approx. 400 X). 60 Figure 28. Photomicrograph of INCO 123 nickel powder hot pressed at 1250C to Vy = 0.93, etched (approx. 400 X). 61 Fiaure 29. Photomicrograph of INCO 123 nickel powder hot pressed at 1250C to Vv = 0.943, etched (approx. 400 X). 62 Figure 30. Photomicrograph of INCO 123 nickel powder hot pressed at 1250C to Vy = 0.958, etched (approx. 400 X). Figure 31. Photomicrograph of INCO 123 nickel powder hot pressed at 1250C to Vy = 0.968, etched (approx. 400 X). 64 Figure 32. Photomicrograph of INCO 123 nickel powder hot pressed at 1250C to Vy = 0.984, etched (approx. 400 X). 65 Figure 33. Photomicrograph of INCO 123 nickel powder cold pressed at 60,000 psi and sintered at 1250C to Vy = 0.942, etched (approx. 400 X). 66 Figure 34. Photomicrograph of INCO 123 nickel powder cold pressed at 60,000 psi and sintered at 1250C to Vy = 0.962, etched (approx. 400 X). v 67 Figure 35. Photomicrograph of INCO 123 nickel powder cold pressed at 60,000 psi and sintered at 1250C to Vy = 0.975, etched (approx. 400 X). 68 Figure 36. Photomicrograph of INCO 123 nickel powder cold pressed at 60,000 psi and sintered at 1250C to Vy = 0.983, etched (approx. 400 X). 69 Vv Figure 37. Dependence of grain boundary area on solid volume during loose stack sintering, hot pressing and conventional sin tering of INCO 123 nickel powder at 1250C. 70 X/ Figure 38. Dependence of the length of grain edges per unit volume on the volume fraction of solid during loose stack sin tering, hot pressing and conventional sintering of INCO 123 nickel powder at 1250C. 71 Vv Figure 39. Variation of the length of lines of intersection of grain boundary and pore-solid surfaces per unit volume with the volume fraction of solid during loose stack sintering, hot pressing and conventional sintering of INCO 123 nickel powder at 1250C. 72 X/ Figure 40. Variation of the length of occupied grain edges per unit volume with the volume fraction of solid during loose stack sintering, hot pressing and conventional sintering of INCO 123 nickel powder at 1250C. Fraction of pores Figure 41. Variation of fractions of pores on the triple edges (filled), on the boundaries (half-filled) and within the grains (open) for a) loose stack sintered, b) hot pressed and c) pressed and sintered nickel powder at 1250C. 74 The metric measurements of pore structure and grain structure properties were followed by topological characterization of loose stack sintered samples. The experimental procedure employed in the latter is described presently. Topological Measurements As mentioned earlier in Chapter 1 only the loose stack sintered samples having densities that are typical of late stages were analyzed to yield the topological parameters. These samples and their densities are listed in Table 8. The procedure for serial sectioning is described below, followed by the algorithm used and the results of the topological analysis. Serial Sectioning The first step in the technique of serial sectioning is to develop and standardize the procedure for removing a layer of desired thickness. This optimum thickness is such that it is small enough to encounter the smallest structural feature for a number of sections; yet large enough 51 25 to avoid redundancy of measurements. Patterson and Aigeltinger tackled this problem very systematically and found that the optimum thickness is of the order of one-fifth of that of scale of the structure. -P A reliable measure of the scale of the system is X the mean intercept _p- p cp of pore phase. Since x = 4VV/Sy the slope of the straight line in -P Figure 20 yields the value of X ~ 4.5 microns. Thus the serial sec tions for LS Series of samples ought to be roughly one micron apart. 75 Table 8 LOOSE STACK SINTERED SAMPLES USED FOR TOPOLOGICAL CHARACTERIZATION Number Sintering Time Volume Fraction of Solid 1 128.7 min. 0.906 2 190.0 min. 0.928 3 221.0 min. 0.944 4 352.5 min. 0.971 5 400.0 min. 0.979 76 This fine scale ruled out the possibility of employing an established procedure for measuring thickness such as using a micrometer. A new, simple procedure was developed to measure the section thicknesses and is presently outlined. A microhardness tester was used to make square-based, pyramid shaped indentations on the polished surface of a sample. It is known that the apex angle of the diamond indentor is 136 and hence the ratio of the depth of an indentation to the diagonal of the impression is equal to 0.1428. As illustrated in Figure 42, the decrease in the depth of an indentation is 0.1428 times the decrease in the legnth of the diagonal. The hardness tester has a capability of a wide variety of loads and magnifications, so indentations of a wide variety of sizes can be made and measured with desired accuracy. Thus, the section thick ness can be easily measured by measuring the decrease in the length of the diagonals of an indentation. A Geotech automatic polisher was used to achieve a reproducible combination of polishing speed, load on the sample and polishing time that would yield the desired magnitude of material removal. A trial sample of sintered nickel was polished, indented with 30 indentations and the section thickness was measured several times by repeatedly polishing and measuring the diagonals until the polishing technique and measurement of section thickness were established with a high degree of confidence. An elaborate and rigorous procedure for topological analysis of porous bodies was developed in the course of doctoral research by 25 Aigeltinger. The abovementioned investigation dealt with loose stack Ad (section thickness) = d^ Ah = h^ Ah = 0.1428 Ad Figure 42. Illustration of the relation between the decrease in the length of diagonals of a microhardness indentation and the decrease in the depth of the indentation. 78 sintered samples having densities in the range from 50 percent to 90 percent of the bulk value and hence exhibited pore structures of a large variety of scales and complexities. Since the samples used in the present investigation had densities higher than those used in this research, their pore structures were typically relatively simple. This made it possible to streamline and simplify the topological analysis to a great extent. The revised algorithm is presently described in detail. Algorithm for Topological Analysis Two topological parameters, namely, the connectivity and the number of separate pores, were measured in this investigation. Since the connec tivity is a measure of the number of redundant connections, there is an inevitable uncertainty regarding the connections between pores that inter sect the boundaries of the volume of analysis (which is a very small fraction of the sample volume). It is not possible to determine whether such pore sections intersect each other or meet with themselves outside the volume covered by the series of parallel areas of observation. This has led to the necessity of putting maximum and minimum limits on the estimate of connectivity. As illustrated in Figure 43, an upper limit on connectivity is obtained when all the pores meeting the boundaries of the volume of analysis are regarded as meeting at a common node, and is called Gmax. A lower limit is derived by considering all such pores to be terminating or "capped" at the boundaries, and is called Gmin. The quantity Gmin then consists solely of redundant connections or "loops" observed within the volume of analysis. The number of separate parts is 79 Figure 43. Illustration of contributions of subnetworks crossing the surface towards the estimate of Gmax. 80 obtained by counting the separate pores that appear and disappear within the volume observed and do not intersect the boundaries. The actual algorithm is as follows. The surface of a loose stack sintered sample, one from the series \ designated for topological analysis, was conditioned by polishing it on a microcloth with 1 micron diamond paste abrasive for about half an hour. This effectively removed all plastically deformed material, the result of an earlier alumina polish. The sample was cleaned in an ultrasonic cleaner, dried and viewed under a microscope to check for polishing artifacts. If the nickel ring, mentioned earlier in this section, was polished uniformly all around, the sample was examined for undue number of scratches that would hinder the analysis. If the polishing was uniform and had only a small number of scratches, it was deemed ready for further analysis; otherwise it was returned to the polishing step. Since the contours of the hardness indentation are mixed with those of the pore sections when observed for the measurement of diagonals, the thickness measurements become more difficult the more the sample is polished or the smaller the square-shaped impression. The trial sample of sintered nickel mentioned previously was polsihed, ten indentations were made on the nickel ring and the specimen and the sample repolished. This was followed by measuring the diagonals of impressions on both ring and specimen. The repetition of this procedure demonstrated that the extents of polishing (removal of material or layer thickness) of the ring and the sample were not statistically different. Indentations in the ring were therefore used to measure the section thickness. 81 Nine indentations were made on the nickel rings of each sample. Three indentations were made on the specimen so that the same area could be located and photographed after each polishing step. The pattern of indentations is schematically illustrated in Figure 44. The first photomicrograph of the serial sectioning series was taken by positioning the three indentations on the sample in a manner that can be easily reproduced. The magnification was selected so that at least 70 pore features could be observed in a single field of view. A Bausch and Lomb Research Metallograph II was used for all the photo micrographs. A set of 4"x5" negatives was obtained by repeating the polishing and photographic steps. Each was enlarged to a size of 8"xl0" so that even the smallest pores were easily seen. A smaller rectange of 6"x8" was marked on print #1; this identified the area of observation. This manner of delineating the area was adapted to help minimize the misregistry error. A similar rectangle was marked on successive prints such that the pores observed on the consecutive sections were in the same position relative to the boundaries of the rectangle. Xerox copies of these prints were used for further analysis, which involves marking each pore on the area of observation for easy identification.' The pores seen on Section #1 were numbered beginning with 1. These are all connected to the "external" networks and thus were not included in the count of separate parts. Pores that first appeared thereafter on successive sections were numbered with a number and a letter N, beginning with IN. These were regarded as the "internal" networks and were used to measure the number of separate parts. The 82 Figure 44. Schematic diagram of a typical specimen used in serial sectioning. 83 genus or the connectivity and the number of isolated pores were measured by comparing pairs of neighboring sections as follows. There are listed in Table 9 three possible classes of topologi cal events that can be observed when two consecutive sections are compared, along with the corresponding increments in Gmax, G111111 and i so N The significance of each of such observed events will be discussed presently. Two typical consecutive sections are shown schematically in Figure 45, wherein the types of events mentioned above are also illustrated. The simplest of these events is the appearance and disappearance of whole pores or subnetworks. When an external sub network disappears, the number of possible "loops" or redundant connections that are assumed to exist outside the volume of analysis is reduced by one, as illustrated in Figure 46. When an internal subnetwork appears, it cannot be determined whether the said subnet work is wholly contained in the volume of analysis or is connected to the external pores. Thus, this event does not change any of the parameters. However, the disappearance of an internal subnetwork signifies a whole separate part and thus the number of separate parts is increased by one. Within a subnetwork, a branch may appear. When that happens, the number of possible loops, terminating in a single external node, is increased by one, as shown in Figure 47. When such a branch is observed to disappear, the abovementioned number is decreased by one, to account for the increase assumed prior to an observation of this event. 84 Table 9 OBSERVABLE TOPOLOGICAL EVENTS Appearance AGmax AGmin AN1S0 Whole Subnetworks Internal 0 0 +1 I Disappearance External -1 0 0 II Within a Appearance of a branch +1 0 0 Subnetwork Disappearance of a branch -1 0 0 III Between Subnetworks Different or Connection Same new 0 +1 0 +1 0 0 85 [(j+l)th section] Figure 45. Two typical consecutive sections studied during serial sectioning that illustrate the topological events listed in Table 9. 86 Figure 46. Contribution of the end of an original subnetwork towards the estimate of Gmax. External Node Figure 47. Illustration of the contribution of a branching event towards Gmax. 88 When a connection is observed between different subnetworks, the said subnetworks have to be renumbered to keep track of such connections. ATI the subnetworks involved in such connections are marked with the lowest of the numbers of these connecting subnetworks. If an internal subnetwork is observed to be connected to an external one, the said internal subnetwork is marked with the number designating the external subnetwork. A connection between previously unconnected subnetworks, those with different numbers, does not change any of the three parameters. A connection between two or more subnetworks with the same number signi fies a complete loop observed entirely inside the sample volume, and thus increases the count of Gmin by one. Since Gmax includes such internal loops, it is also increased by one. After each comparison of consecutive sections, the counts of Gmax, Gmin and N1S0 were updated and tabulated as shown in Table 10. The values of interest are the unit volume quantities, G!J!ax, Gyin i so and Ny If the features that give rise to these quantities are randomly 25 and uniformly distributed in the sample, then it can be expected that there exists a quantity (Qy) characteristic of the structure and equal to the unit volume value. Thus AQ (the change in quantity Q) = Qy x AV. Dividing both sides by aV leads to "AV = The slope of AQ versus AV plot therefore should be equal to Qy provided AV, the volume covered is large enough for a meaningful sampling. In 25 51 the previous investigations of this kind the analyses were continued Table 10 CURRENT ESTIMATES OF Gmax, Gmin and Niso AFTER SECTION #J Height of Sample, Microns Counts .max gmin Previous total -- This section Current total Unit volume value _ Current total Current volume 90 until a linearity between aQ and LV was observed. A different criterion was adapted in the present investigation, and is described presently. Since the connectivity and the number of separate parts are two independent quantities, one may begin to exhibit a constant value of ^ after several sections whereas the other may be far from levelling off. This makes it difficult to establish a criterion for terminating the serial sectioning for a given sample. If use is made of both the quantities a standard basis for terminating the analysis can be obtained. A typical pore subnetwork with convex, concave and saddle elements of its surface is shown schematically in Figure 48. If a test plane is net swept through a unit volume, a measure of Ty, the number of times this plane is tangential to the pore-solid interface, may be obtained in principle. This is related to the connectivity and the number of parts pc per unit volume as follows. T?et = TV~ + TV+ TV~ = 2 radii of curvature negative by convention) is tangential to test plane, or the number of "ends" of a feature (see Figure 48). Ty+ = number of times a convex element (having both the principal radii of curvature positive by convention) is tangential to to test plane, very small at this stage of sintering. Ty" = number of times a saddle element (with principal radii of curvature of different signs by convention) is tangential 91 Figure 48. Convention used in the net tangent count (Twet) . during serial sectioning. v 92 Figure 49. Dependence of connectivity on the volume fraction of solid during loose stack sintering of INCO 123 nickel powder at 1250C. 93 0-90 0-95 10 Figure 50. Variation of the number of isolated pores per unit volume (not corrected for surface effects) with the volume fraction of solid during loose stack sintering of nickel at 1250C. Data for the connectivity is included for comparison. 94 to the test plane per unit volume, or the number of branching and connection events per unit volume (see Figure 48). Cbmputation of is shown;,in Table 11. not If the constancy;, of T is used as the basis for terminating the analysis, both N and G are used and so a more general criterion is obtained. The serial sectioning was terminated for each sample after Tye* was found to level off. The values of the parameters G^ax, G^n and NyS0 are listed in Table 12. Since Gyax depends on the number of pores intersecting the surface that encloses the volume of analysis, it is very sensitive to the surface to volume ratio of the "slice" of the sample used for analysis. Therefore, G^ax does not exhibit a monotonic decrease with an increase in Vy. On the other hand, Gyin is unaffected by the surface to volume ratio mentioned above and hence can be used to draw meaningful inferences about the variation of connectivity. The variation in Gyin and NyS0 are illustrated in Figures 49 and 50, respec tively. It can be seen that even at 98 percent of the bulk density the pore structure has a finite connectivity and thus cannot be regarded as a collection of simple isolated pores. Behavior of Connected and Isolated Porosity A natural reaction to these data would be to wonder what fraction of porosity is connected and how it varies with the total Vy. It is possible to distinguish connected porosity from the isolated pores on any of the series of sections used for topological analysis. Thus, the volume fraction, surface area and integral mean curvature of the connected Table 11 COMPUTATION OF T net V Number of Number of Bottom T~~ = Top Ends Section # Top Ends Ends = A + B + Bottom Ends A: Disappearance of internal & external sub networks B: End of a Branch J+l J+2 T+ = Branchings __ and Connections TeL = T T 10 cn 96 Table 12 TOPOLOGICAL PARAMETERS lin,La ,,S rmax -3 rmin -3 ..iso -3 Number Vy Gy 9 cm Gy 9 cm Ny 9 cm 0.906 1.71 x 108 4.63 x 107 9.55 x 10 0.928 1.39 x 108 1.7 x 107 2.51 x 10 0.944 2.17 x 107 7.23 x 106 4.63 x 10 0.971 3.23 x 107 5.77 x 106 3.33 x 101 0.979 7.6 x 107 3.18 x 106 1.39 x 10( 5 97 porosity, designated respectively as Vy0nn, Sy0nn and My0nn can be measured unambiguously. In the present context, "connected" porosity includes all pores that were not wholly contained in the volume of analysis. The possibility of measuring the properties of the connected porosity was not foreseen before all the series of parallel sections were obtained. Thus, the measurements had to be made on sections with an uncalibrated polish; the ratios Vy0nn/Vy^a^, Sy0nn/Sy0^ and My0nn/My0tal, were estimated. These dimensionless ratios, when multi plied respectively by Vy, Sy and My values determined previously on calibrated polished surfaces, yield the estimated values of Vy0nn, Sy0nn and My0nn. It is thought that the errors introduced by noncali- brated polished surfaces are at least partially compensated for by measuring the abovementioned ratios rather than the absolute values. These are illustrated in Figures 51, 52 and 53. These quantities were estimated from the measurements made on uncalibrated sections; also the amount of isolated porosity in the 98 percent dense sample was too low to yield a sufficient number of measurements for unbiased estimates of metric properties of isolated porosity. Therefore, only the trends, not the actual numbers (at least for the 98 percent dense sample) are significant. It can be seen that after the porosity is reduced to about 0.06, the connected porosity changes very little, whereas the isolated porosity continues to decrease. The results of the present investigation, briefly mentioned in this chapter, are discussed in detail in the next chapter. 98 w conn vV or V iso V p Vv (metallographic) Figure 51. Dependence of the metallographically determined volume fraction of connected porosity on the volume fraction of porosity during loose stack sintering of nickel at 1250C. Data for the volume fraction of isolated porosity included for comparison. 99 t-conn (cm-1) Vy (Metallographic) Figure 52. Dependence of surface area of the connected porosity per unit volume on the volume fraction of solid during loose stack sintering of nickel at 1250C. 100 Figure 53. Dependence of the integral mean curvature of the connected porosity on the volume fraction of solid during loose stack sintering of nickel at 1250C. CHAPTER 3 DISCUSSION Introduction Sintering of a loose stack of powder is the simplest consolidation process and hence forms the basis for investigation of more complex and involved techniques. Loose stack sintering was therefore studied in the greatest detail possible in this investigation. It will be discussed at length in the beginning of this section to establish a framework on which the descriptions of hot pressing and pressing and sintering (also called conventional sintering) will be based. This discussion will be concluded with a number of speculations regarding potential strategies to control the microstructure of a powder-processed component. Loose Stack Sintering The discussion of metric properties will be followed by that of topological properties. It is expected that this study will indicate a likely scenario of various geometric events that lead to the observed paths of evolution of microstructure. Usually such a detailed study also suggests a variety of plausible geometric models of the structure; this study is no exception. The model that is in the best agreement with the data will be described and followed by suggestions for further research necessary to complete an understanding of the process. 101 102 Metric Properties of the Pore Structure SP It is evident from Figure that Sv is linear with V,, during the v v entire range of observation. Prior to this investigation, it was antici pated that this linear relationship is a manifestation of continued removal of pore sections of constant surface to volume ratio. In this scenario, a pore section disappears as soon as it attains a certain surface area to volume ratio, because gradual shrinkage would increase the surface to volume ratio which is inversely related to the size of the pore. Since the porosity consists of connected and isolated fractions during the late second and third stages, this hypothesis must be taken to mean that the channels that pinch off, and isolated pores that disappear, do so as soon as they attain certain area to volume ratio. In this "reverse popcorn" or "instantaeous removal of pore sections" model the continued removal of identical pore sections would also mean a linearity between My and Vy, a speculation not supported by the variation of M.,, Figure T-9. It is suggested here that this apparent contradiction can be explained if it is assumed that the pore phase consists of tubular channels and spher ical isolated pores that have comparable area to volume ratios but signifi cantly different curvature to volume ratios. If the relative fractions of these channels and isolated pores vary during the sintering process, the area would still be linear to the volume but the curvature may not be. This is discussed further in the section on the geometric model of the porosity. 103 Metric Properties of the Grain Boundary Structure SS In addition to the four parameters mentioned earlier, namely, Sy , SSS SSS SSP Lv Ly^occj and Ly ; the association of pores with the grain boundaries was also characterized. An etched section of the sample was examined to measure the number fraction of pores observed to reside within the grains, on the boundaries and on the grain edges. The variations of these frac tions are illustrated in Figure 41. These numbers indicate the fractions of porosity associated with and not associated with the grain boundary network. The twin boundaries do not participate in grain coarsening and hence were not included in the characterization of the associated porosity. The grain boundary area per unit volume, Sy illustrated in Figure 37, can be seen to change only a little in the density range from Vy = 0.93 to Vy = 0.97. Thus, the grain growth or decrease in grain boundary area appears to have been appreciably inhibited in this density range. It is suggested that the pore structure is changing in such a manner that the associated porosity is able to pin the boundaries during this phase of the process. This aspect of the grain boundary structure will be discussed and explored further in the course of the description of the other grain boundary properties. Variations of LySS, LySP and -vfoCC) 1",lustrated in Figures 38, 39 SS and 40, repsectively, also exhibit the arrest observed for Sy in the same density range. The fraction of porosity associated with the grain boun dary network decrease in this range of Vy, Figure 41. Thus, the more or less stable grain boundary network seems to facilitate the reduction of associated porosity. The following discussion of 104 topological properties of the pore structure will be used to offer an explanation for the observed interplay between grain boundary network and the pore phase. Topological Properties Samples with Vy in the range from 0.90 to 0.98 were character ized to obtain the connectivity and the number of isolated pores per unit volume, illustrated respectively in Figures 49 and 50. The connectivity (Gyin) monotonically decreases to a very small value, but remains finite even at 98 percent density. The numbers of iso lated pores first increases, Figure 50, as the pore network present at the end of second stage disintegrates. A large part of the porosity present at this point consists of rather highly interconnected but isolated separate parts as opposed to isolated, equiaxed parts. Sub sequent channel closure rapidly reduces these connected pores to smaller separate parts with increasing density since, in the absence of any significant amount of redundant connections, each channel closure event produces a new separate part. As the sintering proceeds, more and more parts are isolated; at the same time, some of the simple isolated pores shrink and disappear. The maximum in the number of isolated pores seems to indicate that beyond a certain point, very few pores are isolated and hence from there on the number of isolated pores continues to decrease. It can be seen from Figure 51 that the volume fraction of isolated poro sity also goes through a maximum and then decreases to a very small value, whereas the volume fraction of connected porosity changes only a little 105 after about 95 percent density has been reached. Thus, it can be said that the isolation of pores from an interconnected network continues until a point beyond which the residual collection of tree-like pores (those with very low connectivity) does not change appreciably. The isolated pores, on the other hand, continue to shrink and disappear. The preceding discussion of evolution of pore and grain boundary struc tures is expected to present the overall scenario described below. The Overall Scenario It is interesting to note that the density range of effective pinning of the grain boundary network coincides with that of little changes in the connected porosity; the major change is in the isolated porosity, Figure 51. If most of the associated porosity that anchors the grain boundaries is assumed to be a collection of isolated pores, the removal of isolated porosity in this density range can be explained by the proximity of grain boundaries that can act as vacancy sinks. If a balance exists between the number of isolated pores that disappear and the number of pores that are isolated as a result of channel closures, there will be sufficient associated porosity maintained to anchor the grain boundaries. This suggests that the isolated, equiaxed pores anchor the grain boundaries much more effectively than do the connected pores. A hypothesis is offered presently that attempts to rationalize the above contention. Grain boundary migration takes place during grain growth that decreases the grain boundary area. If part of the boundary area is occupied by second phase particles, as illustrated in Figure 54, the 106 Occupying the Grain Boundary Figure 54. Illustration of an increase in the grain boundary area that follows the motion of the boundary occupied by a second phase particle. 107 motion of these boundaries away from the occupying particles requires that additional boundary area be created. Since this increases the surface energy of the boundary network, there is a hindrance to the 52 boundary motion by associated, particles. However, if the pore geometry is such that the motion of the boundary does not increase the boundary area, the pores then do not anchor these boundaries effectively. Thus, the boundary area can be pinned by a tree-like or connected pore only if the branches in the latter intersect the boundary with their axes at small angles to the plane of the boundary, as illustrated in Figure 55. At all other orientations of the connected pore there is no appreciable increase in the boundary area as it migrates. On the other hand, a simple isolated pore, equiaxed in shape, is an effective inhibitor to the grain boundary motion at any orientation. It is therefore suggested that most of the pores anchoring the boundaries are simple isolated pores. The relatively small changes observed in the connected porosity can then be attributed to its inability to associate itself with the grain boundaries. This sequence of microstructural changes can be summarized as follows. In the beginning of the second stage the porosity is mostly inter connected and associated with the grain edges. The decrease in the connectivity proceeds along with an increase in the grain boundary area until a grain boundary network is formed. The subsequent grain growth has decreased the grain boundary area and the migrating boundaries have disassociated themselves with a part of the pore network by the time simple, isolated pores begin to appear. Thus, the boundaries are brought 108 into association with fractions of isolated and connected porosity. Since only simple, isolated pores pin the boundaries effectively, the grain boundary network is stabilized when it finds itself mostly asso ciated with isolated pores. By this time, most of the connected pores are disassociated,as they do not anchor the boundaries. The isolated porosity that is associated with the boundaries is reduced as the boundaries provide vacancy sinks for the necessary material transport. The balance between the number of isolated pores that disappear and the number of pores isolated by channel closures helps maintain a sufficient number of isolated pores that is associated with the boundaries that renders them immobile. Once most of the associated pores have disappeared, the boundaries become free to migrate and are not pinned by the remaining connected porosity. This slows the reduction of connected porosity con siderably. It is apparent from the preceding discussion that the pore phase in the advanced stages of loose stack sintering must be modeled as composed of isolated and connected fractions that vary in a manner IQ oo OC CO described above. Thus, the models that involve only connected 5 33 37 54 55 or isolated porosity * are not appropriate for describing the evolution of microstructure studied in this investigation. mechanis tic model satisfying the abovementioned geometry requirements may be devised, at least in principle, if grain boundary-associated fractions of connected and isolated porosities are measured during the advanced stages of loose stack sintering. These fractions were not measured since the sections studied for topological analysis could not be etched without affecting the pore features. Thus, it was not possible to devise 109 Figure 55. Illustration of a favorable orientation of a connected pore for effective pinning of a grain boundary. no a mechanistic model and hence derive any mechanistic conclusions from the time dependences of geometric properties of the pore microstructure. However, it was possible to construct a variety of geometric models that describe the evolution of microstructure qualitatively. These models incorporate connected and isolated porosity of regular geometry so that the metric properties such as volume, area and integral mean curvature can be calculated. The models are tested by comparing the calculated metric values with the experimentally determined quantities. Since ' these models do not have any parameters that characterize the extent of association with the grain boundary network, properties that depend SSP SSS on this association, such as Ly and Ly(occy could not be calculated and compared with the measured values. These models are described below. Geometric Models All of the geometric models mentioned above describe the porosity as composed of a collection of isolated, spherical pores of the same size and a set of networks of cylindrical pore channels and spherical nodes, as illustrated in Figure 56. The parameters of such a model are listed in Table 13. Here it is assumed that the connected porosity has only three-branch nodes and one-branch nodes; these were measured for a unit volume during the topological characterization in a manner described presently. Each branching or connection event defines a node formed as a result of merging of three pore channels, Figure 57. Thus, the number of three- branch nodes is given by the number of branching and connection events observed per unit volume (Ty). The number of one-branch nodes is the Ill Figure 56. Geometric model of connected and isolated porosity during the advanced stages of loose stack sintering of nickel at 1250C. 112 Figure 57. Illustration of three-branch nodes signified by branching and connection events. 113 Table 13 PARAMETERS OF A TYPICAL GEOMETRIC MODEL Nature Parameter Definition Topological N3b Number of three branch nodes per unit volume Nlb NV Number of one-branch nodes per unit volume bv Number of branches per unit volume nJS0 Number of isolated equiaxed pores per unit volume Metric r Average length of a branch R Radius of an isolated spherical pore, also assumed to be the "radius" of a spherical three-branch node (see Figure 56) 114 number of "ends" (Chapter 2) that are not associated with isolated pores, since an isolated pore has two "ends" denoted by the appearance and disappearance of a new pore. As illustrated in Figure 58, among the pores that cross the boundaries of the volume of analysis, there are some that have an "end" associated with each of these. Thus, there is an uncertainty as to how many of these pores that cross the boundaries are isolated and how many are connected. An attempt was made to intro duce a correction for the sample surface effects. For a collection of convex particles (pores, in the present context), the number per unit volume Ny and the number of features observed on a section are related 26 by the following equation. na = n (15) Where D is the average of the mean caliper diameters of the said parti cles. Thus, for the volume of analysis with surface area S and volume c V, the number of pore features that may cross the surface, N is given by N S = Nv D S (16) The number of isolated pores that cross the boundaries per unit volume, S Ny, is therefore given by S s NV = T = NV D V 07) Since these Ny pores per unit volume are not wholly contained in the volume of analysis, they contribute only half as many pores to the 115 Figure 58. Illustration of pores of analysis and cross that terminate within the volume the surface, or "IS" branches. 116 collection of isolated pores. Thus, the true number of isolated pores, Ny, is given by iy = NyS0(wholly contained) + Ny , 1so V-S Nv Ny + 2v nvo #) = nJS0 ISO NV = DS 2V (18) 09) 1 SO Where Ny is the number of isolated pores observed to be wholly con- S tained in the volume of analysis. Ny is then given by iso ^5 = n = V V V n; i DS 2V DS V (20) S Thus, among the pores that cross the boundaries, Ny gives the number of those that are isolated; the rest can be taken to be connected pores. The treatment presented above therefore also facilitates the estimation of the number of branches that cross the boundaries per unit volume. Thus, the number of one-branch nodes can be estimated from Ty Ny , , S and V. The number, ofbranches per up it volume, by, cam be calcu lated as shown in the Appendix, where the model is described in detail. i so of Ny, Figure 59, is qualitatively the same as that of Ny Figure 51. It should be noted that due to the large surface to volume ratio of the volume of analysis, the surface corrections change the values of by, Nyb significantly. Thus, the previously measured values of the metric 117 Table 14 MEASURED VALUES OF THE NETWORK PARAMETERS No. vv b, cm"3 M3b -3 N\/ $ cm Njb, cm"3 N, cm"3 1 0.906 4.48 x 108 2.1 x 108 2.5 x 108 V 1.07 x 101 2 0.928 2.42 x 108 6.2 x 107 2.63 x 108 3.33 x 101 3 0.944 8.39 x 107 2.04 x 107 1.02 x 108 5.48 x 101 4 0.971 1.8 x 108 3.0 x 107 2.66 x 108 3.95 x 101 5 0.979 1.29 x 108 3.4 x 107 1.23 x 108 1.55 x 101 118 Figure 59. Variation of N\j, surface corrected number of isolated pores per unit volume during loose stack.sintering of nickel at 1250C. Uncorrected values (N^so) are included for comparison. 119 properties of the connected porosity have also to be corrected; these corrected values are listed in Table 15. The remaining two parameters of the model, T and R, were estimated as follows. Since the model has only one-branch and three-branch nodes, the branches can be either one-three of three-three type. (One-one type branches are isolated pores and thus are not included in the connected porosity.) These branches can be classified further, as shown in Table 16 and illustrated in Figure 60. Since the procedure described in Chapter 3 involves recording of all these events, it was possible to measure the number and apparent lengths of each type of branches. As illustrated in Figure 61 for the case of randomly oriented branches, the true length of a branch is twice -IT the average of lengths measured at different orientations. Thus L and 33 L the true average lengths of 1-3 and 3-3 type branches can be esti mated from the separation between the pertinent events. The average lengths of 1-3 and 3-3 branches that cross the boundaries can be taken as twice the value measured, since for randomly oriented branches the average length of the part of the branch that is contained in the volume of analysis is half of its true length. The overall weighted average of all types of branches yields the required parameter, L. For a collection of isolated, equiaxed pores, VyS0, the volume fraction of isolated porosity, is given by (21) 120 Table 15 CORRECTED VALUES OF V^onn, -conn bV and M conn No. vf ..conn vV cconn -1 Sy cm Mconn -2 My cm 1 0.906 0.09 731 -20 x 105 2 0.928 0.053 475 -15.2 x 105 3 0.944 0.017 138 -3.1 x 105 4 0.971 0.017 165 -4 x 105 5 0.979 0.02 237 -10.7 x 105 121: Type One-Three Three-Three Table 16 TYPES OF BRANCHES Order of Events Observed During Serial Sectioning New appearance + Branching New Appearance -* Connection Branching -* End of a branch Connection -* Bottom end Branching Bottom end Connection Branching Branching -* Branching Connection -* Connection Branching - Connection 122 New Appearance Branching (b) Figure 60. Various types of a) one-three and b) three-three branches and the topological events signifying each type of branch. 123 L(measured) = / / L Cos6Sin0d0d<> = L/2 <Â¡>=0 0=0 Unit Radius Figure 61. Illustration of the relation between measured and true lengths of a branch. 124 *1 CQ Where Ny = number of isolated pores per unit volume and ,iso = radius of a spherical pore. Thus, 1/3 3V1S0 Ris ( V 4ttN iso ) (22) 1 C A The radius of a spherical -node;:R> was taken to be equal to R The Al 1 L values of L, R, by, Ny and Ny are listed in Table 17. It can be seen that r increases slowly until Vy s 0.97 when it increases significantly. The reduction in connectivity by way of pinching off of a branch decreases L whereas elemination of a three-branch node, accomplished when a 1-3 type branch merges into the parent network, leads to an increase in "L, Figure 62. Since the number of isolation events goes through a maximum, L should eventually increase significantly once most of the isolation events or channel closures have taken place. The apparent maximum in R can be attributed to simultaneous isolation and shrinkage processes. When a pore shrinks it can be thought of as going from one size class to the lower one. Since all isolated pores shrink, although at different rates, there is a "flux" toward the smallest size class in a given size distribution. Isolation events bring new pores into this collection, such that there is an influx in all the size classes. Since the number of isolation events goes through a maximum, the net "flux" in the size distribution is towards the largest size class when isolation events dominate over shrinkage and towards the smallest size class when very few isolation events occur. This leads to a maximum in the estimated average volume of an isolated pore. In order to test the model, the metric properties of the connected porosity were calculated, since those of the isolated fraction were used, 125 Table 17 VALUES OF THE MODEL PARAMETERS USED IN CALCULATIONS No. Vv R L microns microns cm -3 1 0.906 2 0.928 3 0.944 4 0.971 2.34 6.63 4.48 x 108 2.1 x 108 2.5 x 108 2.58 8.18 2.42 x 108 6.2 x 107 2.63 x 108 2.79 7.1 8.39 x 107 2.04 x 107 1.02 x 108 2.1 9.56 1.8 x 108 3.0 x 107 2.66 x 108 1.07 25.8 1.29 x 108 3.4 x 107 1.23 x 108 5 0.979 126 Figure 62. Effects of channel closure and surface rounding on L, the average length of a branch. 127 in a sense, to estimate L and R. The calculated and estimated (with surface correction) values of Vy0nn, Sy0nn and My0nn are listed in Table 18. The estimation of the above values was made by using two conn P P experimentally determined values, such as Vy /Vy and Vy; the confidence intervals therefore were relatively large. High sample surface to volume ratio led to significant amounts of corrections, both in estimated and measured values. In light of the difficulties and the geometric simpli city of the model the agreement between measured and calculated values seems to indicate that the modeled geometry is qualitatively representa tive of the real microstructure. SP It was said earlier in this section that the linearity between Sy and Vy probably can be attributed to similar area to volume ratios for pore channels and isolated pores. From the Appendix, for a tubular channel with length = T and radius r = (/3/2)R, the area to volume ratio, Ay, is given by a = 2tttL = 2 _4_ = 2J1 V irr2L r n/3R R (23) as illustrated in Figure 63. For an isolated spherical pore, Ay = 3/R which is not far from 2.31/R. However, the curvature to volume ratios, Cy's, are significantly different. For a spherical pore, Cy vis given by Cy (spherical pore) = ^4 = -4> (24) -2TrRJ R and Cy for a tubular channel is given by C V (tubular channel) = ttL 2r irr L 1 2 r 4 = 1.33 3R2 R2 (25) 128 Table 18 CALCULATED AND ESTIMATED VALUES OF V conn V cconn bV and M conn V No. VS Vv ..conn vV (.conn -1 , cm Mconn Mv , _ 2 cm Calc. Est. Calc. Est. Calc. Est., 1 0.906 0.05 0.09 479 731 -12.4 x 105 -20.3 x 105 2 0.928 0.04 0.053 373 474 -10.3 x 105 -15.2 x 105 3 0.944 0.015 0.017 133 138 -3.6 x 105 -3.1 x 105 4 0.971 0.022 0.017 257 165 -8.9 x 105 -4.0 x 105 5 0.979 0.009 0.02 202 237 -11.2 x 105 -10.7 x 105 2r = 3R Figure 63. Illustration of a typical pore channel in the connected porosity. 130 Thus, a spherical pore has more than twice as much Cy as does a tubular channel. The relative fractions of tubular channels and spheri cal pores can be computed from the number of branches (by in Table 17) and the surface-corrected number of isolated, pores (Figure 59). It can SP be seen from the variation of f fraction of spheres, Figure 64, that the curvature would decrease (in magnitude) relatively slowly with the volume until the maximum in the fraction of spheres; beyond this point the curvature would decrease sharply. This is supported by the variation of My, Figure 21. It is suggested that a kinetic model of this type of geometry should be devised and tested in the future, provided the associ ation of isolated and connected porosity with the grain boundary network can be successfully quantified. The preceding discussion on loose stack sintering will now be used to describe hot pressing and conventional sintering, in that order. Hot Pressing It can be seen from Figure 20 that in the early part of the range SP of observation a hot pressed sample has a higher Sy than a loose stack sintered sample with comparable density. The variation of surface area with Vy approaches the linear relationship for LS series as the density increases. Both hot pressed samples and loose stack sintered samples then continue to exhibit comparable values of surface area. Since the pressure is applied for the whole duration of hot pressing, the abovementioned SP variation of Sy with Vy may have less to do with pressure and thereby particular mechanisms than with the geometry of the pore structure. 131 Figure 64. Variation of the fraction of spherical pores in a collection of spherical pores and cylindrical channels with the volume fraction of solid during loose stack sintering of nickel at 1250C. 132 Thus, it is suggested that the initially dissimilar geometries become similar so that the two paths of evolution of microstructure begin to coincide at Vy s 0.90. It should be noted here that the contention is that of simi.lar, not identical (see below) geometries, based solely on the surface area-volume relationships. Thus it is speculated that the hot pressed samples having densities in the range of observation have porosity in the form of a collection of isolated, equiaxed pores and a set of networks made up of tubular channels and spherical pores. As said in the earlier section on loose stack sintering, the model detailed in the Appendix dictates the pore channels and isolated spherical pores have comparable area to volume ratios, regardless of their numbers per unit volume. If the same type of geometry is assumed for porosity in the hot pressed samples the linearity of area with volume can be explained. The variation of My with Vy, Figure 21, indicates that hot pressed samples have more than twice as much integral mean curvature as loose stack sintered samples with comparable densities until very late in the process. It is suggested that the geometries are similar in that they both can be modeled as a collection of spheres and a set of tubular net works. They are, however, not identical in that a hot pressed sample has a considerably higher number of isolated spherical pores than a loose stack sintered sample with the same density. A need for topological analysis of hot pressed samples becomes apparent, since these analyses will test the postulate of higher number of isolated pores in hot pressed samples as compared to loose stack sintered samples. 133 Unlike loose stack sintering, the grain boundary network did not exhibit an arrest during hot pressing, as illustrated in Figures 37 through 40. The arrest observed in the grain boundary network for LS series was attributed to isolated, equiaxed pores pinning the boundaries effectively. If this assertion is to be valid, the presence of a higher number of isolated pores in HP series postulated earlier in this section must also imply that the boundaries are more effectively pinned. The fact that such a pinning effect was not observed at all is taken to indi cate that the boundaries are able to migrate in spite of the resistance offered by isolated and associated porosity. It is speculated that the application of external pressure induces enough amount of grain boundary sliding for them to overcome the drag offered by the associated pores; 56 Pond et al. have found some evidence for coupling between grain boundary migration and grain boundary sliding. ssp SSS The values of Ly and Ly(occy the quantities that increase with an increase in the degree of association of porosity with the grain boun dary network, were observed to be higher than those for LS. Increased plastic flow, as suggested above, can also account for higher degree of pinching off of channels and thereby a higher number of isolation events. The importance of topological study of hot pressing is emphasized again as it will test the postulate of a higher number of channel closure events, as compared to loose stack sintering. The characterization of associated fractions of isolated and connected porosity will shed a great deal of light on the grain boundary-porosity relationship. 134 Conventional Sintering Cold pressing that precedes the high temperature consolidation packs a larger number of powder particles in a unit volume and hence gives rise to higher Sv in a green compact as compared to a loose stack sample sin tered to the same density. It can be seen from Figure 20 that this addi tional surface area is not reduced enough for the two corresponding paths of microstructural evolution to coincide even at Vy = 0.98, although the tendency for the paths to converge is evident. The variation of My with Vy, Figure 21, indicates that a pressed and sintered sample has higher surface area and integral mean curvature than a loose stack sample sin tered to the same density. If the model in the Appendix is examined, it can be seen from a purely geometric point of view that a similar pore structure with lower "L and R but higher by, Ny and Ny can yield comparable values of volume but different area and curvature. In other words, if the porosity in a pressed and sintered body is viewed as finer networks and larger number of isolated pores, the measured values of Sy and My can be explained satisfactorily. The grain boundary network does not exhibit an arrest, as illustrated in Figures 37 through 40. A considerably higher number of interparticle contacts and therefore a larger grain boundary area brings the onset of grain coarsening at a lower density as compared to loose stack sintering. If the boundaries begin to migrate well before a sufficient number of pores are isolated to produce pinning points, there may not be enough hindrance to the boundary migration to arrest grain growth. Thus, only a small fraction of isolated pores manage to associate with the boundaries 135 Once most of these are eliminated the pores isolated thereafter have only a limited opportunity to intersect the moving boundaries. Since SSP SSS Ly ancj l_v(occ) quantify the degree of association of porosity with the grain boundary network, it is apparent that initially a pressed and sintered sample has a higher amount of associated porosity that decreases rapidly. Before the onset of isolation processes, most of the porosity is in the form of an interconnected network mostly associated with grain edges. If grain coarsening begins well before an appreciable number of pores are isolated, this initially high associated porosity would decrease rapidly since the connected pores cannot anchor the boundaries effectively and are consequently disassociated. The significance of a topological study of the advanced stages of conventional sintering is stressed here as these measurements would characterize the associated fractions of iso lated and connected porosity and thus would test the postulates put for ward earlier in this section. Comparison of Loose Stack Sintering with Hot Pressing and Conventional Sintering If the paths of evolution of microstructure during these processes are examined together, a number of general processing parameter-micro- structure relationships become apparent. Since these relationships have a potential as possible strategies to control the microstructure and hence the service properties of a powder-processed component, they are of both theoretical and practical interest; they are listed below. 136 1) As the grain boundaries are anchored effectively only by equiaxed pores, a fine grain structure can be obtained if a sufficient number of equiaxed pores is isolated before grain growth begins. Grain growth requires a connected grain edge network and hence a certain minimum grain boundary area, therefore, an initial powder stack with a relatively coarse grain structure (low grain boundary area) would have grain growth beginning at a higher density (after a sufficient number of pores is isolated); as compared to an initial powder stack with a finer grain structure. Thus, for a sintered body with high density and fine grain structure requirements, an initially coarse grain structure is better than a fine one. 2) If sintering is carried out in such atmosphere that the isolated pores trap a gas of low diffusivity, these pores are relatively stable and hence offer effective grain boundary pinning. Care must be taken to delay the isolation events so that only a few pores are isolated; otherwise coarsening of these pores would 42 lead to an increase in volume. 3) At least for the conditions of the present investigation, it can be said that hot pressing leads to both higher density and a finer grain size in a shorter length of time as compared to loose stack sintering or conventional sintering at the same temperature, up to a certain density. Beyond this, a relatively coarser grain structure is obtained during hot pressing or con ventional sintering. The findings of this investigation are summarized and the course of future research suggested in the next chapter. CHAPTER 4 CONCLUSIONS Introduction The discussion of the results of this investigation is summarized in a number of conclusions; an outline of suggested research is also presented. Conclusions 1) The porosity can be modeled as composed of a set of networks of cylindrical channels and a collection of monosized isolated spherical pores during the advanced stages of loose stack sintering, hot pressing and conventional sintering. 2) During loose stack sintering, a highly interconnected network of branches and nodes disintegrates into simpler subnetworks which sub sequently break up to form the isolated pores. The connected or tree-like pores continue shrinking until Vy r 0.95 when the rate of removal of these pores becomes significantly slow for the rest of the range of observation, up to Vy s 0.98. 3) Isolated porosity, on the other hand, goes through a maximum and diminishes to a very low value when connected porosity is observed to have been more or less stabilized. 4) The onset of stabilization of connected porosity is coincident with an arrest of grain growth and rapid reduction in the isolated porosity. 137 138 5) It is suggested that because of their equiaxed shape, isolated pores anchor the grain boundaries effectively whereas the connected pores do not. Hence, most of the isolated porosity is associated with grain boundaries. 6) The association of grain boundary network and isolated pores facilitates rapid reduction of isolated porosity as the associated boundaries provide immediate sinks of vacancies; this is in contrast 43 45-47 to the traditional viewpoint that once the pores are isolated, it is very difficult to remove them from the system. 7) The connected porosity finds itself disassociated from the grain boundary network which slows the reduction of such pores consid erably. 8) The higher values of curvature, yet comparable values of area and volume of pore phase for the hot pressed samples as compared to loose stack sintered samples, are tentatively attributed to similar geometries but a higher number of isolated, spherical pores. 9) An absence of an arrest of grain growth in hot pressed samples, in spite of a higher number of equiaxed pores, is believed to be due to stress-induced grain boundary sliding that promotes grain boundary migra- ti on. 10) Porosity in pressed and sintered samples is believed to consist of finer networks and a higher number of isolated pores compared to the loose stack sintered samples with the same density; this leads to much higher areas and curvatures in PS than in LS. 11) Due to larger number of interparticle contacts in a green compact as compared to a loose stack sintered to the same density, a pressed and sintered sample has much higher grain boundary area. 139 12) It is suggested that this higher boundary area brings the onset of grain coarsening at a lower density, well before the pores begin to isolate. The early grain growth in PS leaves only a small opportunity for subsequently isolated pores to associate with the moving boundaries. Thus, an absence of an arrest in grain growth is attributed to the onset of grain growth well before that of isolation events. Suggestions for Further Study It should be apparent from the preceding discussions that topologi cal analysis of the advanced stages of hot pressing and conventional sintering would resolve the speculations about a higher number of iso lated pores in the hot pressed and sintered sample. The associated fractions of isolated and connected porosity during all three processes, when char acterized, would facilitate the mechanistic study of the advanced stages. Thus, the course of further research is outlined as follows. T) An etching procedure should be developed that will facilitate the determination of associated fractions of isolated and con nected porosity. 2) These fractions should be measured on sections in the series studied for topological, characterization. 3) The isolated and connected fractions should be determined on a section with calibrated polish. 4) The advanced stages of hot pressing and conventional sintering should be characterized regarding the topological properties of the pore structure. REFERENCES 1. F. N. Rhines and R. T. DeHoff, Modern Developments in Powder Metallurgy, p. 173, Plenum Press, New York (1971). 2. W. Rostoker and S. Y. K. Liu, J. Materials, _5, 605 (1970). 3. R. D. Smith, H. W. Anderson and R. E. Moore, Bull. Amer. Cer. Soc., 55, 979 (1976). 4. R. T. DeHoff, F. N. Rhines and E. D. Whitney, Final Report, AEC Contract AT(40-1), 4212 (1974). 5. G. Arthur, J. Inst. Metals, 83, 329 (1954). 6. R. A. Graham, W. R. Tarr and R. T. DeHoff, unpublished research. 7. G. Ondracek, Radex-Rundschau 3/4 (1971) 8. S. Nazare, G. Ondracek and F. Thummler, Modern Developments in Powder Metallurgy, p. 171, Plenum Press, New York (1971). 9. J. Kozeny, Sitzber. Akad. Wiss. Wien., 136, 271 (1927). 10. M. F. Ashby, Acta Met., 22, 275 (1974). 11. R. T. DeHoff, B. H. Baldwin and F. N. Rhines, Planseeber. Pulvermet., JO, 24 (1962). 12. Metals Research Laboratory, Carnegie Institute of Technology, Final Report, AEC Contract AT(30-1), 1826 (1959). 13. G. C. Kuczynski, Powder Metal1urgy, p. 11, Interscience Publishers, New York-London (1961). 14. T. L. Wilson and P. C. Shewmon, Trans. Met. Soc. AIME, 236, 48 (1966) 15. G. Matsumara, Acta Met., 19, 851 (1971). 16. F. N. Rhines, C. E. Berchenall and L. A. Hughes, J. Metals, 188, 378 (1950). 17. R. T. DeHoff, Proceedings of the Symposium on Statistical and Probabilistic Problems in Metallurgy, Special supplement to Advances in Applied Probability (1972). 18. E. H. Aigeltinger and R. T. DeHoff, Met. Trans., 6A, 1853 (1975). 140 141 19. R. A. Gregg and F. N. Rhines, Met. Trans., 4, 1365 (1973). 20. R. T. DeHoff, R. A. Rummel, H. P. LaBuff and F. N. Rhines, Modern Developments in Powder Metallurgy, p. 310, Plenum Press, New York (1966). 21. W. D. Tuohig, Doctoral Dissertation, University of Florida (1972). 22. S. M. Gehl, Doctoral Dissertation, University of Florida (1977). 23. A. S. Watwe and R. T. DeHoff, unpublished research. 24. J. S. Adams and D. Glover, Metal Progress, August (1977). 25. E. H. Aigeltinger, Doctoral Dissertation, University of Florida (1969). 26. R. T. DeHoff and F. N. Rhines, eds., Quantitative Microscopy, McGraw Hill Book Co., New York (1967). 27. L. K. Barrett and C. S. Yust, ORNL Report, No. 4411 (1969). 28. L. K. Barrett and C. S. Yust, Metallography, 3^, 1 (1970). 29. S. S. Cairns, Introductory Topology, The Ronald Press Company, New York (1961T! 30. F. N. Rhines, Powder Met. Bull., 3^, 28 (1948). 31. P. Schwarzkopf, Powder Met. Bull., 2 74 (1948). 32. F. Thummler and N. Thomma, Met. Review, 12, 69 (1967). 33. R. L. Coble, J. Appl. Physics, 32, 787 (1961). 34. L. K. Barrett and C. S. Yust, Trans. Met. Soc. AIME, 239, 1172 (1967). 35. R. T. DeHoff and F. N. Rhines, Final Report, AEC Contract AT(40-1), 2581 (1969). 36. F. N. Rhines, University of Florida, private communication. 37. G. C. Kuczynski, Acta Met., 4, 58 (1956). 38. P. J. Wray, Acta Met., 24, 125 (1976). 39. W. D. Kingery and B. Francois, Sintering and Related Phenomena, p. 471, Gordon and Breach Publishers, New York (1965). 40. A. J. Markworth, Met. Trans., Â£, 2651 (1973). 41. W. Trzebiatowski, Zhurnal Physik Chem., B24, 75 (1934). 142 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. T. K. Gupta and R. L. Coble, J. Amer. Cer. Soc., 52, 293 (1968). R. C. Lowrie, Jr. and I. B. Cutler, Sintering and Related Phenomena, p. 527, Gordon and Breach Publishers, New York (1965). G. C. Kuczynski, Powder Met., 12, 1 (1963). M. Paul us, Sintering and Related Phenomena, p. 225, Plenum Press, New York (1973). C. S. Morgan and K. H. McCorkle, Sintering and Related Phenomena, p. 293, Plenum Press, New York (1973). J. E. Burke, Ceramic Microstructures, p. 681, John Wiley and Sons, New York (1968^ E. H. Aigeltinger and H. E. Exner, Met. Trans., 8A, 421 (1977). J. Kronsbein, L. J. Buteau, Jr. and R. T. DeHoff, Trans. Met. Soc. AIME, 233, 1961 (1965). Metals Handbook, 8th Edition, Volume 1. B. R. Patterson, Doctoral Dissertation, University of Florida (1978). M. Hillert, Acta Met., 12, 227 (1965). G. C. Kuczynski, Sintering and Related Phenomena, p. 325, Plenum Press, New York (1976). A. R. Hingorany and J. S. Hirschhorn, Inti. J. Powder Met., 2, 5 (1966). A. J. Markworth, Scripta Met., j>, 957 (1972). R. C. Pond, D. A. Smith and P. W. J. Southerdon, Phil. Mag., A37, 27 (1978). APPENDIX THE GEOMETRIC MODEL OF THE PORE PHASE Introduction Main features of the proposed model are stated and followed by the derivation of equations relating metric properties and the parameters of the model. The corrections for surface effects are also outlined. Parameters of the Model As said in Chapter 3, the pore phase is modeled as a collection of spherical pores of the same size and a set of networks made up of cylin drical branches and spherical nodes. Only two kinds of nodes and branches are assumed to exist in the connected pores; a node is either a one-branch node or a three-branch node, as illustrated in Figure 65. A branch either terminates in three-branch nodes at both ends, or in a one-branch node at . one and a three-branch node at the other, Figure 66. A branch that termi nates in one-branch nodes at both ends is considered an isolated part and hence is not included in the connected porosity. A one-branch node is assumed to be a semi-spherical cap, the radius of which is the same as that of a cylindrical branch, r, as shown in Figure 66. A three-branch node is considered as a sphere of radius R which*is connected to three cylindrical branches of radius r, Figure 67. An isolated pore is assumed to be a sphere of radius R. All branches have the same length, "L. 143 144 Figure 65. Illustration of three-branch and one-branch nodes in a pore network. 145 Figure 66. Illustration of a) a one-three branch and b) a three-three branch. 146 2r Figure 67. Dihedral angle of the edge at the intersection of a cylindrical branch and a spherical node. 147 The abovementioned parameters of the model are discussed in terms of the notations listed in Table 19. Since each one-branch node is associated with a branch and each three-branch node with three branches, by, the total number of branches 1 h and Ny and Ny are related by the following equation + 3Nyb 2 (26) since each branch is counted twice. Alternatively, each one-three branch is associated with one one-branch node and one three-branch node, whereas each three-three branch is associated with two three- branch nodes. Thus ,3b Nr h13 .33 by + 2by (27) since each three-branch node is counted thrice. The number of one-branch nodes, Nyb, is given by Njb = bj3 (28) Equations (26) and (27) give wlb .33 3b NV + 2bV V or 3N?b NJb (29) Thus by is given by K h13 4. h33 bV bv + bV 3|\j3^ N^b My> + 3_Ni_Ny. bv = Nb + 3Nf or 2 (30) 148 Table 19 PARAMETERS OF THE MODEL Feature Notation Definition Branches J3 bV Number of one-three branches per unit volume h33 bV Number of three-three branches per unit vol ume Nodes Number of one-branch nodes per unit volume nf Number of three-branch nodes per unit volume Isolated Pores nJS0 Number of isolated pores observed to be wholly contained, per unit volume 149 which is the same as equation (26). Thus, the number of branches can be calculated from the number of nodes, and vice versa. Metric Properties of the Connected Porosity Three basic properties, namely, volume fraction, area per unit volume and integral mean curvatre per unit volume are discussed. The conn cconn Mconn i S\ cl net Mt; notations used are listed in Table 20; V are made V dna V up of contributions from branches, one-branch nodes and three-branch nodes; they will be discussed in terms of these individual contributions, listed in Table 21. Three-Branch Node It can be seen from Figure 67 that r, the radius of a cylindrical branch, and R, the radius of a three-branch node, are related by the equation r = RCos(ir/6) or r = '^-R (31) Thus, h, the height of the spherical cap, illustrated in Figure 68, is given by h = R/2 (32) C The volume of this spherical cap, V is given by (33) 150 Table 20 CALCULATED METRIC PROPERTIES Property Notation Definition Volume ..conn vV Volume fraction of connected porosity Area -conn Surface area of connected porosity per unit volume Curvature Mconn Integral mean curvature of connected porosity per unit volume 151 Table 21 METRIC PROPERTIES OF NODES AND BRANCHES Property Volume Area Notation Definition Volume of a three-branch node Volume of a one-branch node Volume of a branch Area of a three-branch node Area of a one-branch node Area of a branch Curvature Integral mean curvature of a three-branch node Me Integral mean curvature of edges in a three-branch node (Figure 67) Mlb Integral mean curvature of a one-branch node M b Integral mean curvature of a branch 152 Figure 68. Height of the spherical cap in an extended, spherical, three-branch node. 153 JL The volume of a three-branch node, V is then the volume of a sphere of radius R minus the volumes of three spherical caps. Thus V = tt R 3( 24 ) or V = -gq ttR (34) C The area of a spherical cap, A is given by AS = 2irRh or AS = ttR2 (35) Area of a three-branch node, S is thus S3b = 4ttR2 3(ttR2) or S3b = uR2 (36) There are three edges formed on a three-branch node as a result of intersection of three cylindrical surfaces with a spherical surface. Since integral mean curvature of an edge (26) is given by M of an edge = -^ X 1 (37) Where X = dihedral angle or the angle between the surface normals and 1 = the length of an edge. It can be seen from Figure 67 that X = tt/6 and 1 = 2irr or 1 = /3rrR. Thus, Medge, integral mean curvature of edges, is given by Medge = 3X|xÂ£x(V3)ttR or Medge = ^ tt2R (38) Integral mean curvature of the spherical surface of a three-branch node, c M is simply S 1 M = spherical surface area X(-p-) or MS = (ttR2) (-!) or MS = -ttR (39) 154 OL Thus the integral mean curvature of a three-branch node, M is given by M3b = MS + Medge = -ttR + (^pk2R or M3b = tt (1 ^L)R (40) One-Branch Node Since such a node is a semi-spherical cap of radius r = (*/3/2)R that is connected to a cylinder of radius r, there are no edges involved. Thus V1b=|trr3 or Vlb=^R3 (41) Area of a one-branch node, S^b, is given by Sb = 2Trr2 or Sb = 2tt(^-) or Sb = ^R2 (42) Integral mean curvature of a one-branch node, M^b, is given by Mlb = 27rr2(-l) or Mlb = -2irr = -/3ttR (43) Cylindrical Branch Since all branches have length = T and radius r = C^-)R, Vb = -nr2!: = : ' (44) Sb = 2TrrI = ^3ttRI (45) Mb = -ttL (46) 155 The Networks Thus the properties of networks are given by ,1b conn V conn = b. Vb + N3b V3b + Nyb V conn = b. Sb + N3b + S3b + Njb Mb + Nyb M3b + Jb M lb (47) (48) ly uy II ny n iiy ri (49) Substitution of the definition of individual contributions leads to (50) (51) (52) conn u 3irn2r 3b 177Tr,3 Mlb /3 n3 V = bV TR L NV 24"R NV 41tR Sj0nn = by ^RL + N3b ttR2 + Njb ^R2 M ,conn t .,3b //3-it lb m -byTrL + Ny TT^-IJR Ny /3ttR Computation of coefficients of by, N3b and Nyb yield the equations in the convenient form: V S M y0nn = 2.36byLR-+2.23N3bR3 + 1.36NjbR3 (53) 50nn = 5.44byLR + 3.14N3bR2 + 4.72NjbR2 (54) |Cnn = _3.i4byL + 1.13Nyb R 5.44Nyb R (55) Surface Corrections It was assumed during the measurements of metric properties of connected porosity that all the pores crossing the boundaries of the volume of analysis are of a network character. A fraction of these pore features, in fact, belong to the set of isolated pores. A (a) (b) Figure 69. Illustration of a) "IS11 pore branches and b) "SS" pore branches. 157 Figure 70. Caliper diameters for a convex body. D, the mean caliper diameter, is the average of D's over all possible orientations of measuring planes. 158 treatment is outlined below that attempts to take the surface effects into consideration. 1 s The parameter by is defined as the number of pores per unit volume that are observed to cross the boundaries and terminate inside the volume ss of analysis ("1-S") and by as those that traverse this volume without terminating inside ("S-S"), illustrated in Figure 69. The fraction of these "IS" and "SS" pores that belong to the set of isolated pores is estimated as follows. For a collection of convex pores, the number per unit volume, Ny, and the number of pore features observed on a section of unit area, N^, are related by the equation NA = Ny D (15) where 1) is the average of the mean caliper diameters of the pores, as illustrated in Figure 70. Thus, for a sample of surface area S and volume V, the number of pores observed to cross the boundaries per unit c volume, Ny, is given by Nv = na f Nv X '17> Since these pores are not wholly contained in the volume of analysis, they contribute only half as many pores to the set of isolated pores. If i so Ny is the number of wholly contained, isolated pores per unit volume, the true measure of isolated pores per unit volume, Ny is given by Nis0 NV = NVS0+fV-| Nv*-V 09) i DS "2V Thus, equations (17) and (19) give 159 NV = Nv "V DS NV iso 1- DS 2V DS S (56) The number of "IS" and "SS" pores that contribute to the isolated porosity, Ny, can be estimated from the measurements of NyS0, , S and V. Estimated ;.D is the average of physical separations between sections where the isolated pores appear and those where they disappear. The pores that cross the boundaries but do not contribute to the set of isolated pores are assumed to terminate in the three-branch nodes. Thus, a fraction of "SS" pores belong to the class of "3-3" branches, whereas a subset of "l-S" pores belongs to the class of "1-3" branches. These fractions are estimated as follows. Each "SS" pore contributes two pore features observed on the surface whereas each "IS" pore contributes one pore feature. Thus, the total number of pore features arising from these pores per unit volume, Ny, is given by Nj = bJS + 2b*jS (57) e Since the number of pore features that come from the isolated pores, Ny, is given by equation (55), the number of features that arise from connected r pores per unit volume, Ny, is given by Niso - mc mt ms v. mc JS ..SS INV DS Nv Ny Ny or Ny by + 2by y (58) . DS 2V r It is assumed that these Ny features are distributed according to the relative fractions of "IS" and "SS" type pores. Thus, byS(C), the number of "IS" pores that belong to the connected porosity per unit 160 SS, volume, and by (C), the number of "SS" pores that belong to the connected porosity per unit volume are given by .IS JS = JS,OLSS by +2by JS ISO or and or bJS(C) = .IS.,. SS by +2by rKlS.9KSS {by +2by 1- DS 2V DS, V 1 (59) SS/p\ 1 bV ^ 2 JS 2b SS SS NV by +2bv bSS Niso uSS,^ DV ri_ 1S, olSS NV bV JSjo.SS {bV +2bV ^ by +2by i US^ V V |-2y DS, (60) Since for every two features contributed by "SS" pores there is one "SS" pore that is counted as an additional "3-3" type branch. Thus, the total number of additional branches, by(add), is given by by(add) = \ [bJS(C) + b*S(C)] or JS,uSS by +by ISO u v ruiS,olSS V by(add) hlS+9hSSx ibV 2bV 2(by +2by ) DS DS, V * (61) 1- 2 V since all these branches are considered to contribute half as many to the volume of analysis. Each additional 1-3 branch contributes an additional one-branch node, N^b(add), the additional number of one-branch nodes per unit volume, is given by blS Jso DV r JS.ouSS NV JS.-SS {bV +2bV c by +2by ^DS DS} V 1 Nyb(add) = byS(C) = (62) 161 If by(corr) is defined as the corrected number of branches per unit volume, T(corr) as the corrected average length of a branch and TT(add) as the weighted average of the length of additional branches, then by(corr)L(corr) = by*L + by(add)*L(add) (63) where by and T refer to the previously measured values of branches per unit volume and their average length. The values of by(corr), TT(corr), Nyb(corr), Ny^ and R1S0 are listed in Table 22; these were used to cal culate the metric properties of the connected porosity. The surface corrections used to modify the measured values of the metric properties of the connected porosity are described presently. It can be seen from equation (56) that the additional number of isolated pores per unit volume, Ny (add), is given by ,iso (64) N Miso, 1 S _V DS Ny (add) 2 Ny 2V . . 1_ 2V Since VyS0, SyS0 and MyS0, the metric properties of isolated poro sity, are directly proportional to the number of isolated pores, the additions to these properties are given by .ISO Vjs0(add) = 4 N SO N\/S(add) or VwS0(add) = vjso {^4 i DS 1 2V (65) Similarly, sjS0(add) = sjS0 v v 1-DS W (66) 162 and Mi50(add) = Miso {DSm} V DS 1 2V (67) Here it is assumed that the true D of the isolated pores is the same as that measured for wholly contained isolated pores. The initial measurements of properties of connected porosity were made assuming all pores not contained inside are connected; the abovementioned addi tions to the isolated fraction have therefore to be subtracted from Vj0nn, Sy0nn and My0nn to correct for surface effects. These corrected values, termed Vy0nn, Sy0nn and My0nn by the following equations. Vvconn(C) = vÂ§onn {DS/2V} 1 - DS 2V s5onn(c) + sJonn ciso { } DS/2V , DS 1 2V Mf,nn(C) = M: conn - M iso {DS/2V} 1 - DS 2 V (68) (69) (70) The geometric model, discussed in detail in the preceding section of the Appendix, is tested by comparing the corrected values and the calculated values. BIOGRAPHICAL SKETCH Arunkumar Shamrao Watwe was born on June 16, 1952, in Poona, India. In June, 1967, he finished high school in Bombay, India. In June, 1974, he received the degree of Bachelor of Technology in Metallurgical Engineering from the Indian Institute of Technology, Bombay, India. In August, 1976, he received the degree of Master of Science in Materials Science and Engineering from Washington State University, Pullman, Washington. In September, 1976, he came to the University of Florida to pursue the degree of Doctor of Philosophy in the Department of Materials Science and Engineering. He is a fellow member of Alpha Sigma Mu and is a joint student member of the American Society for Metals and the Metallurgical Society of AIME. 163 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. R. T. DeHoff, Chairmar Professor of Material'Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. L n Â¡ip V .. R. E. Reed-Hill Professor of Materials Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Professor of Materials Science atar Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Inoda, Jr. Professor of Materials Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the' degree of Doctor of Philosophy. / ,// s Rc_L< Scheaffer' v-^Professor of Statistics This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1983 Dean, College of Engineering Dean for Graduate Studies and Research Windows Live Hotmail Page 2 of 3 Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: Watwe, Arunkumar TITLE: Metric and topological characterization of the advanced stages of sintering / (record number: 506243' PUBLICATION DATE: 1983 It v5- 'as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees of the University of Florida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be limited to.those specifically allowed by "Fair Use" as prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and preservation of a digital archive copy. Digitization allows the University of Florida to generate image- and text-base'd Versionsas appropriate and to provide and enhance access using search software. This grant of permissions prohibits use of the digitized versions for commercial use or profit. Signature of Copyright Holder /\aJ K "5. Vv/ A ~f A3 Â£= Printed or Typed Name of Copyright Holder/Licensec Personal information blurred 6 Date of Signature Please print, sign and return to: Cathleen Martyniak UF Dissertation Project Preservation Department University of Florida Libraries P.O. Box 117007 Gainesville, FL 32611-7007 http://bll44w.blul44.mail.live.com/mail/ReadMessageLight.aspx?Action=MarkAsNotJun... 6/10/2008 55 Figure 23. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vy = 0.906, etched (approx. 400 X). 50 dealt with relatively small amounts of porosity (10 percent or lower) P the error in Vy introduced by the polishing technique approached that of the density measurements, namely, about 0.005, as the sintered den sity approached the bulk density. Thus, the polishing was accepted for p further characterization if the metal!ographically determined Vy was p within 15 percent of Vy obtained from the water immersion method, except for the samples 97 percent dense and higher for which the limit had to be relaxed to 30 percent of Vy. Since Vy values range from 0.15 to 0.02, the abovementioned limits translate into a few percent of the sample den sity as measured metallographically. Typically, the samples 97 percent p dense and lower exhibited a precision of 0.05 of the Vy value obtained SP SP from the density measurements. Manual measurements of Sy and My were made on the accepted polished surfaces using standard stereological techni- ques. The measurements of Vy, Sy and My were made with at least 30 different fields and at magnifications that allowed at least 15 pores to be viewed in a single field. As a result, the estimates of the properties were within 5 percent with 95 percent confidence, as illustrated in Figures SP SP 20 and 21. Plots of Sy and My contained metallographically measured values of Vy to yield the paths of evolution of microstructure in order to partially compensate for the polishing errors. Measurement of these metric properties was followed by etching the specimens to reveal the grain boundaries. Each sample was immersed in a solution made from equal parts of nitric acid, glacial acetic acid and acetone for about 30 seconds. The grain boundaries were brought out clearly with some evidence of facetting of the initially smooth contours of pore features. Samples in the lower part of the density range exhibited 134 Conventional Sintering Cold pressing that precedes the high temperature consolidation packs a larger number of powder particles in a unit volume and hence gives rise to higher Sv in a green compact as compared to a loose stack sample sin tered to the same density. It can be seen from Figure 20 that this addi tional surface area is not reduced enough for the two corresponding paths of microstructural evolution to coincide even at Vy = 0.98, although the tendency for the paths to converge is evident. The variation of My with Vy, Figure 21, indicates that a pressed and sintered sample has higher surface area and integral mean curvature than a loose stack sample sin tered to the same density. If the model in the Appendix is examined, it can be seen from a purely geometric point of view that a similar pore structure with lower "L and R but higher by, Ny and Ny can yield comparable values of volume but different area and curvature. In other words, if the porosity in a pressed and sintered body is viewed as finer networks and larger number of isolated pores, the measured values of Sy and My can be explained satisfactorily. The grain boundary network does not exhibit an arrest, as illustrated in Figures 37 through 40. A considerably higher number of interparticle contacts and therefore a larger grain boundary area brings the onset of grain coarsening at a lower density as compared to loose stack sintering. If the boundaries begin to migrate well before a sufficient number of pores are isolated to produce pinning points, there may not be enough hindrance to the boundary migration to arrest grain growth. Thus, only a small fraction of isolated pores manage to associate with the boundaries 132 Thus, it is suggested that the initially dissimilar geometries become similar so that the two paths of evolution of microstructure begin to coincide at Vy s 0.90. It should be noted here that the contention is that of simi.lar, not identical (see below) geometries, based solely on the surface area-volume relationships. Thus it is speculated that the hot pressed samples having densities in the range of observation have porosity in the form of a collection of isolated, equiaxed pores and a set of networks made up of tubular channels and spherical pores. As said in the earlier section on loose stack sintering, the model detailed in the Appendix dictates the pore channels and isolated spherical pores have comparable area to volume ratios, regardless of their numbers per unit volume. If the same type of geometry is assumed for porosity in the hot pressed samples the linearity of area with volume can be explained. The variation of My with Vy, Figure 21, indicates that hot pressed samples have more than twice as much integral mean curvature as loose stack sintered samples with comparable densities until very late in the process. It is suggested that the geometries are similar in that they both can be modeled as a collection of spheres and a set of tubular net works. They are, however, not identical in that a hot pressed sample has a considerably higher number of isolated spherical pores than a loose stack sintered sample with the same density. A need for topological analysis of hot pressed samples becomes apparent, since these analyses will test the postulate of higher number of isolated pores in hot pressed samples as compared to loose stack sintered samples. CHAPTER 1 EVOLUTION OF MICROSTRUCTURE DURING SINTERING Introduction A microstructure is characterized by its metric and topological properties and therefore the following discussion will be carried out in terms of variation of these quantities as the sintering proceeds. These microstructural properties will be defined and the previous investigations of this type will be discussed in detail; a review of metric studies will be followed by topological analyses. The principles of quantitative stereology employed in the estimation of microstructural properties will be described in the next chapter on experimental procedure and results. Metric Properties of the Microstructure These quantities are estimated in terms of geometric properties of 26 lines, surfaces and regions of space averaged over the whole structure. The basic properties are listed in Table 1 and illustrated in Figure 1. Among the properties listed, Vy, Sy and My are used to yield two impor tant global averages of the microstructural properties. These are listed in Table 2. In a sintered structure, there are two regions of space or phases, namely, pore and solid, and two surfaces, pore-solid interface and grain boundaries. Two main linear features of interest are the grain edges and the lines formed as a result of intersection of pore-solid interface and grain boundaries. Superscripts are used to identify the properties that 5 131 Figure 64. Variation of the fraction of spherical pores in a collection of spherical pores and cylindrical channels with the volume fraction of solid during loose stack sintering of nickel at 1250C. 97 porosity, designated respectively as Vy0nn, Sy0nn and My0nn can be measured unambiguously. In the present context, "connected" porosity includes all pores that were not wholly contained in the volume of analysis. The possibility of measuring the properties of the connected porosity was not foreseen before all the series of parallel sections were obtained. Thus, the measurements had to be made on sections with an uncalibrated polish; the ratios Vy0nn/Vy^a^, Sy0nn/Sy0^ and My0nn/My0tal, were estimated. These dimensionless ratios, when multi plied respectively by Vy, Sy and My values determined previously on calibrated polished surfaces, yield the estimated values of Vy0nn, Sy0nn and My0nn. It is thought that the errors introduced by noncali- brated polished surfaces are at least partially compensated for by measuring the abovementioned ratios rather than the absolute values. These are illustrated in Figures 51, 52 and 53. These quantities were estimated from the measurements made on uncalibrated sections; also the amount of isolated porosity in the 98 percent dense sample was too low to yield a sufficient number of measurements for unbiased estimates of metric properties of isolated porosity. Therefore, only the trends, not the actual numbers (at least for the 98 percent dense sample) are significant. It can be seen that after the porosity is reduced to about 0.06, the connected porosity changes very little, whereas the isolated porosity continues to decrease. The results of the present investigation, briefly mentioned in this chapter, are discussed in detail in the next chapter. 56 Figure 24. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vy = 0.928, etched (approx. 400 X). 36 The present investigation that dealt with the microstructural characterization of the advanced stages of sintering has a potential for enhancing and quantifying the existing sketchy picture of the late stages of sintering. The theoretical and practical importance of this work can be appreciated from the following discussion. It has been theorized161835,37-39 that the spheroidization of pores proceeds along with coarsening during the advanced stages. It is necessary to couple topological analysis with the metric measurements to study the spheroidization and coarsening of isolated pores. To date, there has been no such direct observation of the behavior of isolated porosity. If a pore of higher than average size is surrounded by a shell of higher than average density with finer pore channels, then early clo sure of these channels pulls the solid shell away from the large pore so 34 that the continuity of the solid phase is maintained, as illustrated in 34 Figure 18. According to Barrett and Yust, most of the reports of coarsening are in fact the observed removal of smaller channels before the larger ones. Another disputed contention is that of deceleration of densification due to separation of grain boundaries from isolated 43 45-47 pores. A pore that is observed to be isolated on a two dimen sional section may or may not be so in the third dimension, whether associated with the grain boundaries or not. The topological analysis of grain boundary-porosity association alone can determine the true extent of association of isolated porosity with the boundaries. A detailed geometric study of porosity in the advanced stages will clarify some aspects of microstructural evolution mentioned above. 151 Table 21 METRIC PROPERTIES OF NODES AND BRANCHES Property Volume Area Notation Definition Volume of a three-branch node Volume of a one-branch node Volume of a branch Area of a three-branch node Area of a one-branch node Area of a branch Curvature Integral mean curvature of a three-branch node Me Integral mean curvature of edges in a three-branch node (Figure 67) Mlb Integral mean curvature of a one-branch node M b Integral mean curvature of a branch 114 number of "ends" (Chapter 2) that are not associated with isolated pores, since an isolated pore has two "ends" denoted by the appearance and disappearance of a new pore. As illustrated in Figure 58, among the pores that cross the boundaries of the volume of analysis, there are some that have an "end" associated with each of these. Thus, there is an uncertainty as to how many of these pores that cross the boundaries are isolated and how many are connected. An attempt was made to intro duce a correction for the sample surface effects. For a collection of convex particles (pores, in the present context), the number per unit volume Ny and the number of features observed on a section are related 26 by the following equation. na = n (15) Where D is the average of the mean caliper diameters of the said parti cles. Thus, for the volume of analysis with surface area S and volume c V, the number of pore features that may cross the surface, N is given by N S = Nv D S (16) The number of isolated pores that cross the boundaries per unit volume, S Ny, is therefore given by S s NV = T = NV D V 07) Since these Ny pores per unit volume are not wholly contained in the volume of analysis, they contribute only half as many pores to the 106 Occupying the Grain Boundary Figure 54. Illustration of an increase in the grain boundary area that follows the motion of the boundary occupied by a second phase particle. 85 [(j+l)th section] Figure 45. Two typical consecutive sections studied during serial sectioning that illustrate the topological events listed in Table 9. 153 JL The volume of a three-branch node, V is then the volume of a sphere of radius R minus the volumes of three spherical caps. Thus V = tt R 3( 24 ) or V = -gq ttR (34) C The area of a spherical cap, A is given by AS = 2irRh or AS = ttR2 (35) Area of a three-branch node, S is thus S3b = 4ttR2 3(ttR2) or S3b = uR2 (36) There are three edges formed on a three-branch node as a result of intersection of three cylindrical surfaces with a spherical surface. Since integral mean curvature of an edge (26) is given by M of an edge = -^ X 1 (37) Where X = dihedral angle or the angle between the surface normals and 1 = the length of an edge. It can be seen from Figure 67 that X = tt/6 and 1 = 2irr or 1 = /3rrR. Thus, Medge, integral mean curvature of edges, is given by Medge = 3X|xÂ£x(V3)ttR or Medge = ^ tt2R (38) Integral mean curvature of the spherical surface of a three-branch node, c M is simply S 1 M = spherical surface area X(-p-) or MS = (ttR2) (-!) or MS = -ttR (39) It is suggested that isolated, equiaxed pores pin the boundaries more effectively than do the connected pores. Increase in the boundary area accompanies the boundary migration for all orientations of an equiaxed pore whereas this is true only for a limited number of orientations of a connected pore. Consequently, isolated pores are removed via transport of vacancies to the occupied boundaries; subsequent resumption of grain growth slows the reduction of residual connected porosity. Porosity in loose stack sintered samples is modeled as a set of tubular networks and a collection of monodispersed spheres. Comparison of metric properties of loose stack sintered samples with those of conventionally sintered and hot pressed samples led to the speculations that a higher number of isolated pores exist during hot pressing and that the porosity in con ventionally sintered samples is composed of finer networks and smaller isolated pores. Absence of an arrest in grain growth during hot pressing is believed to be due to boundary migration that is induced by grain boundary sliding. A similar absence of an arrest in grain growth during conventional sin tering is attributed to the onset of grain growth well before that of isolation events. viii I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the' degree of Doctor of Philosophy. / ,// s Rc_L< Scheaffer' v-^Professor of Statistics This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1983 Dean, College of Engineering Dean for Graduate Studies and Research 66 Figure 34. Photomicrograph of INCO 123 nickel powder cold pressed at 60,000 psi and sintered at 1250C to Vy = 0.962, etched (approx. 400 X). v 58 Figure 26. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vy = 0.971, etched (approx. 400 X). 38 Mechanical and physical properties of conmercial porous components are influenced by the geometry of the porosity. Thermal conductivity is P influenced by Vy, pore shapes and the relative fractions of connected and 7 8 4 isolated porosity. Permeability to fluids depends on the connectivity, p cp g 3 Vy and Sy Mechanical strength and thermal shock resistance depend on 2 pore shapes whereas ductility is influenced by pore shapes and spacings. Thus geometric characterization of porous structures as a function of adjustible process parameters would suggest a number of potential strate gies to control the final service properties. It is apparent from the review of previous microstructural studies of sintering that the present investigation is expected to offer a much needed general and quantitative picture of the advanced stages of sin tering. The experimental procedure employed in the present research is described in detail in the next chapter. 155 The Networks Thus the properties of networks are given by ,1b conn V conn = b. Vb + N3b V3b + Nyb V conn = b. Sb + N3b + S3b + Njb Mb + Nyb M3b + Jb M lb (47) (48) ly uy II ny n iiy ri (49) Substitution of the definition of individual contributions leads to (50) (51) (52) conn u 3irn2r 3b 177Tr,3 Mlb /3 n3 V = bV TR L NV 24"R NV 41tR Sj0nn = by ^RL + N3b ttR2 + Njb ^R2 M ,conn t .,3b //3-it lb m -byTrL + Ny TT^-IJR Ny /3ttR Computation of coefficients of by, N3b and Nyb yield the equations in the convenient form: V S M y0nn = 2.36byLR-+2.23N3bR3 + 1.36NjbR3 (53) 50nn = 5.44byLR + 3.14N3bR2 + 4.72NjbR2 (54) |Cnn = _3.i4byL + 1.13Nyb R 5.44Nyb R (55) Surface Corrections It was assumed during the measurements of metric properties of connected porosity that all the pores crossing the boundaries of the volume of analysis are of a network character. A fraction of these pore features, in fact, belong to the set of isolated pores. A 159 NV = Nv "V DS NV iso 1- DS 2V DS S (56) The number of "IS" and "SS" pores that contribute to the isolated porosity, Ny, can be estimated from the measurements of NyS0, , S and V. Estimated ;.D is the average of physical separations between sections where the isolated pores appear and those where they disappear. The pores that cross the boundaries but do not contribute to the set of isolated pores are assumed to terminate in the three-branch nodes. Thus, a fraction of "SS" pores belong to the class of "3-3" branches, whereas a subset of "l-S" pores belongs to the class of "1-3" branches. These fractions are estimated as follows. Each "SS" pore contributes two pore features observed on the surface whereas each "IS" pore contributes one pore feature. Thus, the total number of pore features arising from these pores per unit volume, Ny, is given by Nj = bJS + 2b*jS (57) e Since the number of pore features that come from the isolated pores, Ny, is given by equation (55), the number of features that arise from connected r pores per unit volume, Ny, is given by Niso - mc mt ms v. mc JS ..SS INV DS Nv Ny Ny or Ny by + 2by y (58) . DS 2V r It is assumed that these Ny features are distributed according to the relative fractions of "IS" and "SS" type pores. Thus, byS(C), the number of "IS" pores that belong to the connected porosity per unit 158 treatment is outlined below that attempts to take the surface effects into consideration. 1 s The parameter by is defined as the number of pores per unit volume that are observed to cross the boundaries and terminate inside the volume ss of analysis ("1-S") and by as those that traverse this volume without terminating inside ("S-S"), illustrated in Figure 69. The fraction of these "IS" and "SS" pores that belong to the set of isolated pores is estimated as follows. For a collection of convex pores, the number per unit volume, Ny, and the number of pore features observed on a section of unit area, N^, are related by the equation NA = Ny D (15) where 1) is the average of the mean caliper diameters of the pores, as illustrated in Figure 70. Thus, for a sample of surface area S and volume V, the number of pores observed to cross the boundaries per unit c volume, Ny, is given by Nv = na f Nv X '17> Since these pores are not wholly contained in the volume of analysis, they contribute only half as many pores to the set of isolated pores. If i so Ny is the number of wholly contained, isolated pores per unit volume, the true measure of isolated pores per unit volume, Ny is given by Nis0 NV = NVS0+fV-| Nv*-V 09) i DS "2V Thus, equations (17) and (19) give 14 Figure 3 Some closed surfaces and their deformation retracts (dotted lines). TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS iv ABSTRACT vii INTRODUCTION 1 CHAPTER ONE EVOLUTION OF MICROSTRUCTURE DURING SINTERING 5 Introduction 5 Metric Properties of the Microstructure 5 Fundamentals of Topology 6 Sintering from a Geometric Viewpoint 12 Importance of the Present Research 35 TWO EXPERIMENTAL PROCEDURE AND RESULTS 39 Introduction 39 Sample Preparation 39 Metallography 46 Topological Measurements 74 THREE DISCUSSION 101 Introduction 101 Loose Stack Sintering 101 Hot Pressing 130 Conventional Sintering 134 Comparison of Loose Stack Sintering with Hot Pressing and Conventional Sintering 135 FOUR CONCLUSIONS 137 Introduction 137 Conclusions 137 Suggestions for Further Study 139 REFERENCES 140 APPENDIX THE GEOMETRIC MODEL OF THE PORE PHASE 143 Introduction 143 v 77 82 Figure 44. Schematic diagram of a typical specimen used in serial sectioning. Dedicated To My Parents, Mr. Shamrao Vasudeo Watwe and Mrs. Sharada Shamrao Watwe Fraction of pores Figure 41. Variation of fractions of pores on the triple edges (filled), on the boundaries (half-filled) and within the grains (open) for a) loose stack sintered, b) hot pressed and c) pressed and sintered nickel powder at 1250C. 40 Figure 19. INCO 123 nickel powder used in the present investigation (1000 X). 13 begin to impinge each other, as illustrated in Figure 5. Due to differ ent crystallographic: orientations of adjacent particles, grain boundaries form in the interparticle contact regions. In this stage, the area of 32 pore-solid interface decreases with a moderate amount of shrinkage. Throughout this stage, the pore-solid interface has many redundant con nections J During the second stage, the distinguishing features are not the interparticle contacts or "necks" but the pore channels formed as a result of the impingement of neighboring necks. Virtually all of the 1 6 *3*3 porosity is in the form of an interconnected network of channels * that delineate the solid grain edges. The continued reduction in the volume and the area of porosity is accompanied by a decrease in the 1 35 connectivity of the pore structure. The decrease in the connec tivity can be explained by either removal of solid branches or closure 36 of pore channels. According to Rhines, the connected pore network coarsens, analogous to a grain edge network in a single phase polycrystal (driven by excess surface energy) as illustrated in Figure 6. In this scenario, a fraction of solid branches (necks) are pinched off and no new pores are isolated. Although a finite number of isolated pores observed during the late second stage can be explained only by channel closure events, a closer scrutiny is needed to resolve this issue. The isolated pores may be irregular in shape.16,34 30 31 The third stage has begun by the time most of the pores are isolated. The connectivity of a pore network is now a very small number.1 Coarsening 16 18 35 37-39 proceeds along with the spheroidization of pores * so that the volume of porosity, the number of pores and pore-solid interface area 102 Metric Properties of the Pore Structure SP It is evident from Figure that Sv is linear with V,, during the v v entire range of observation. Prior to this investigation, it was antici pated that this linear relationship is a manifestation of continued removal of pore sections of constant surface to volume ratio. In this scenario, a pore section disappears as soon as it attains a certain surface area to volume ratio, because gradual shrinkage would increase the surface to volume ratio which is inversely related to the size of the pore. Since the porosity consists of connected and isolated fractions during the late second and third stages, this hypothesis must be taken to mean that the channels that pinch off, and isolated pores that disappear, do so as soon as they attain certain area to volume ratio. In this "reverse popcorn" or "instantaeous removal of pore sections" model the continued removal of identical pore sections would also mean a linearity between My and Vy, a speculation not supported by the variation of M.,, Figure T-9. It is suggested here that this apparent contradiction can be explained if it is assumed that the pore phase consists of tubular channels and spher ical isolated pores that have comparable area to volume ratios but signifi cantly different curvature to volume ratios. If the relative fractions of these channels and isolated pores vary during the sintering process, the area would still be linear to the volume but the curvature may not be. This is discussed further in the section on the geometric model of the porosity. 116 collection of isolated pores. Thus, the true number of isolated pores, Ny, is given by iy = NyS0(wholly contained) + Ny , 1so V-S Nv Ny + 2v nvo #) = nJS0 ISO NV = DS 2V (18) 09) 1 SO Where Ny is the number of isolated pores observed to be wholly con- S tained in the volume of analysis. Ny is then given by iso ^5 = n = V V V n; i DS 2V DS V (20) S Thus, among the pores that cross the boundaries, Ny gives the number of those that are isolated; the rest can be taken to be connected pores. The treatment presented above therefore also facilitates the estimation of the number of branches that cross the boundaries per unit volume. Thus, the number of one-branch nodes can be estimated from Ty Ny , , S and V. The number, ofbranches per up it volume, by, cam be calcu lated as shown in the Appendix, where the model is described in detail. i so of Ny, Figure 59, is qualitatively the same as that of Ny Figure 51. It should be noted that due to the large surface to volume ratio of the volume of analysis, the surface corrections change the values of by, Nyb significantly. Thus, the previously measured values of the metric 113 Table 13 PARAMETERS OF A TYPICAL GEOMETRIC MODEL Nature Parameter Definition Topological N3b Number of three branch nodes per unit volume Nlb NV Number of one-branch nodes per unit volume bv Number of branches per unit volume nJS0 Number of isolated equiaxed pores per unit volume Metric r Average length of a branch R Radius of an isolated spherical pore, also assumed to be the "radius" of a spherical three-branch node (see Figure 56) 47 of the ring will be discussed later in this section. Rough polishing was done on wet silicon carbide papers of increasing fineness from 180 grit through 600 grit. Fine polishing was done by using 6 micron diamond paste, followed by 1 micron diamond paste, 0.3 micron alumina and finally 0.05 micron alumina. Quantitative Stereo!ogy Metric properties are estimated by making measurements on a two dimensional plane of polish with the help of standard relations pc of stereology. A set of test lines, arranged in a grid pattern, also provide a set of test points and a test area to characterize the plane section; these are usually used to make the measurements listed in Table 6. The relationships between these measurements and the globally averaged properties of the three dimensional microstruc ture are listed in Table 7. The relations yield estimates of popula- pc tion or structure properties provided the structure is sampled uniformly. Stereological counting procedure and the estimated properties will be discussed presently. Each metallographically prepared surface was calibrated by measuring the volume fraction of porosity by quantitative stereology and comparing the result with the value obtained from density measurements. A definite amount of plastic deformation by the polishing abrasive media leads to a smearing effect that introduces some error in quantifying the information on a polished section. This effect can be viewed as local movements of traces of the pore-solid interface; all the counted events (number, inter cept, etc.) are therefore error-prone to some extent. As this investigation 30 Figure 15. The variation of grain contiguity with solid volume fraction for loose stack sintered copper and hot pressed U02.22 BIOGRAPHICAL SKETCH Arunkumar Shamrao Watwe was born on June 16, 1952, in Poona, India. In June, 1967, he finished high school in Bombay, India. In June, 1974, he received the degree of Bachelor of Technology in Metallurgical Engineering from the Indian Institute of Technology, Bombay, India. In August, 1976, he received the degree of Master of Science in Materials Science and Engineering from Washington State University, Pullman, Washington. In September, 1976, he came to the University of Florida to pursue the degree of Doctor of Philosophy in the Department of Materials Science and Engineering. He is a fellow member of Alpha Sigma Mu and is a joint student member of the American Society for Metals and the Metallurgical Society of AIME. 163 109 Figure 55. Illustration of a favorable orientation of a connected pore for effective pinning of a grain boundary. 104 topological properties of the pore structure will be used to offer an explanation for the observed interplay between grain boundary network and the pore phase. Topological Properties Samples with Vy in the range from 0.90 to 0.98 were character ized to obtain the connectivity and the number of isolated pores per unit volume, illustrated respectively in Figures 49 and 50. The connectivity (Gyin) monotonically decreases to a very small value, but remains finite even at 98 percent density. The numbers of iso lated pores first increases, Figure 50, as the pore network present at the end of second stage disintegrates. A large part of the porosity present at this point consists of rather highly interconnected but isolated separate parts as opposed to isolated, equiaxed parts. Sub sequent channel closure rapidly reduces these connected pores to smaller separate parts with increasing density since, in the absence of any significant amount of redundant connections, each channel closure event produces a new separate part. As the sintering proceeds, more and more parts are isolated; at the same time, some of the simple isolated pores shrink and disappear. The maximum in the number of isolated pores seems to indicate that beyond a certain point, very few pores are isolated and hence from there on the number of isolated pores continues to decrease. It can be seen from Figure 51 that the volume fraction of isolated poro sity also goes through a maximum and then decreases to a very small value, whereas the volume fraction of connected porosity changes only a little 23 Figure 10. Integral mean curvature versus volume fraction of solid for 48 micron spherical and dendritic copper powder.25 160 SS, volume, and by (C), the number of "SS" pores that belong to the connected porosity per unit volume are given by .IS JS = JS,OLSS by +2by JS ISO or and or bJS(C) = .IS.,. SS by +2by rKlS.9KSS {by +2by 1- DS 2V DS, V 1 (59) SS/p\ 1 bV ^ 2 JS 2b SS SS NV by +2bv bSS Niso uSS,^ DV ri_ 1S, olSS NV bV JSjo.SS {bV +2bV ^ by +2by i US^ V V |-2y DS, (60) Since for every two features contributed by "SS" pores there is one "SS" pore that is counted as an additional "3-3" type branch. Thus, the total number of additional branches, by(add), is given by by(add) = \ [bJS(C) + b*S(C)] or JS,uSS by +by ISO u v ruiS,olSS V by(add) hlS+9hSSx ibV 2bV 2(by +2by ) DS DS, V * (61) 1- 2 V since all these branches are considered to contribute half as many to the volume of analysis. Each additional 1-3 branch contributes an additional one-branch node, N^b(add), the additional number of one-branch nodes per unit volume, is given by blS Jso DV r JS.ouSS NV JS.-SS {bV +2bV c by +2by ^DS DS} V 1 Nyb(add) = byS(C) = (62) 99 t-conn (cm-1) Vy (Metallographic) Figure 52. Dependence of surface area of the connected porosity per unit volume on the volume fraction of solid during loose stack sintering of nickel at 1250C. 118 Figure 59. Variation of N\j, surface corrected number of isolated pores per unit volume during loose stack.sintering of nickel at 1250C. Uncorrected values (N^so) are included for comparison. 18 Figure 6. Two basic topological events that occur in the network coarsening scenario proposed by Rhines.36 The dotted lines indicate the occupied grain edges. 64 Figure 32. Photomicrograph of INCO 123 nickel powder hot pressed at 1250C to Vy = 0.984, etched (approx. 400 X). 93 0-90 0-95 10 Figure 50. Variation of the number of isolated pores per unit volume (not corrected for surface effects) with the volume fraction of solid during loose stack sintering of nickel at 1250C. Data for the connectivity is included for comparison. 133 Unlike loose stack sintering, the grain boundary network did not exhibit an arrest during hot pressing, as illustrated in Figures 37 through 40. The arrest observed in the grain boundary network for LS series was attributed to isolated, equiaxed pores pinning the boundaries effectively. If this assertion is to be valid, the presence of a higher number of isolated pores in HP series postulated earlier in this section must also imply that the boundaries are more effectively pinned. The fact that such a pinning effect was not observed at all is taken to indi cate that the boundaries are able to migrate in spite of the resistance offered by isolated and associated porosity. It is speculated that the application of external pressure induces enough amount of grain boundary sliding for them to overcome the drag offered by the associated pores; 56 Pond et al. have found some evidence for coupling between grain boundary migration and grain boundary sliding. ssp SSS The values of Ly and Ly(occy the quantities that increase with an increase in the degree of association of porosity with the grain boun dary network, were observed to be higher than those for LS. Increased plastic flow, as suggested above, can also account for higher degree of pinching off of channels and thereby a higher number of isolation events. The importance of topological study of hot pressing is emphasized again as it will test the postulate of a higher number of channel closure events, as compared to loose stack sintering. The characterization of associated fractions of isolated and connected porosity will shed a great deal of light on the grain boundary-porosity relationship. 27 SyP (cm-1) Figure 13. Grain boundary area per unit volume versus volume fraction of solid_for 48 micron spherical and dendritic copper . powder.25 19 There is evidence to suggest that this path of evolution of microstruc- 21 ture for loose stack sintering is insensitive to temperature. SP Data for hot pressed samples indicate that the Sy -Vy relationship 22 23 is only approximately linear even in the late second stage. The path of microstructural change was also found to be insensitive to tem- 22 23 perature. The effect of pressure on the path was significant; increasing pressure delayed the approach to linearity until a lower p value of Vy, as shown in Figure 8. Integral mean curvature per unit volume, My, has been measured for loose stack sintering, conventional sintering (cold pressing followed by sintering) and hot pressing in the density range characteristic of late second stage. A convex particle has a positive curvature whereas a con- 18 21 22 vex pore has a negative curvature. There is a miniimum in.*.My; * 35 this minimum occurs at lower Vy for finer particle size, as illustrated in Figure 9. According to the convention used, most of the "SP" surface has positive curvature in the initial stages. Due to decreasing surface area and increasing negative curvatures there occurs a minimum in My in the second stage. As the sintered density approaches the theoretical density, My must approach zero and hence the initially high positive value of My that becomes negative must go through a minimum. For an p initial stack of irregularly shaped particles, My varies with Vy at a slower rate and has a minimum earlier in the process, compared to an 25 initial stack of spherical powders. This is illustrated in Figure 10. In all the cases studied the paths were insensitive to temperature. In the case of hot pressing, the minimum in My is much more negative and P 23 occurs at a lower value of Vy, compared to a loose stack sintered sample; 44 2. Pressed and Sintered Series A CARVER hydraulic hand press was used to prepare cylindrical pellets about 15 mm in diameter and typically 3 mm in height. Cold pressing at 60,000 psi followed by sintering at 1250C yielded the desired series of samples. Due to the small size of these pellets and their patterns of potential inhomogeneity it was not feasible to prepare the series of samples from a single initial compact, as in the loose stack case. Instead, samples in this series were pre pared individually by sintering the green compacts in an alumina boat under a flowing dry hydrogen atmosphere for preselected sintering times at 1250C 10C. 3. Hot Pressed Series The third series was prepared by hot pressing at 1250C and under a pressure of 2000 psi in a CENT0RR high vacuum hot press. A loose stack of powder was placed in a cylindrical boron nitride die 2.54 cm in diameter and tapped; the die with the top punch resting on the powder -5 was placed in the vacuum chamber. After a vacuum of 10 Torr was reached the induction coil was switched on. The attainment of sintering temper ature which nominally required one hour was followed by the application of a pressure of 2000 psi. The pressure was maintained and the tempera ture controlled to 5C for the specified sintering times; the pressure was then released and the induction coil turned off. After the sample was allowed to cool overnight, air was admitted and the die assembly removed. As in the case of PS series, samples in this series were made individually. 107 motion of these boundaries away from the occupying particles requires that additional boundary area be created. Since this increases the surface energy of the boundary network, there is a hindrance to the 52 boundary motion by associated, particles. However, if the pore geometry is such that the motion of the boundary does not increase the boundary area, the pores then do not anchor these boundaries effectively. Thus, the boundary area can be pinned by a tree-like or connected pore only if the branches in the latter intersect the boundary with their axes at small angles to the plane of the boundary, as illustrated in Figure 55. At all other orientations of the connected pore there is no appreciable increase in the boundary area as it migrates. On the other hand, a simple isolated pore, equiaxed in shape, is an effective inhibitor to the grain boundary motion at any orientation. It is therefore suggested that most of the pores anchoring the boundaries are simple isolated pores. The relatively small changes observed in the connected porosity can then be attributed to its inability to associate itself with the grain boundaries. This sequence of microstructural changes can be summarized as follows. In the beginning of the second stage the porosity is mostly inter connected and associated with the grain edges. The decrease in the connectivity proceeds along with an increase in the grain boundary area until a grain boundary network is formed. The subsequent grain growth has decreased the grain boundary area and the migrating boundaries have disassociated themselves with a part of the pore network by the time simple, isolated pores begin to appear. Thus, the boundaries are brought 142 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. T. K. Gupta and R. L. Coble, J. Amer. Cer. Soc., 52, 293 (1968). R. C. Lowrie, Jr. and I. B. Cutler, Sintering and Related Phenomena, p. 527, Gordon and Breach Publishers, New York (1965). G. C. Kuczynski, Powder Met., 12, 1 (1963). M. Paul us, Sintering and Related Phenomena, p. 225, Plenum Press, New York (1973). C. S. Morgan and K. H. McCorkle, Sintering and Related Phenomena, p. 293, Plenum Press, New York (1973). J. E. Burke, Ceramic Microstructures, p. 681, John Wiley and Sons, New York (1968^ E. H. Aigeltinger and H. E. Exner, Met. Trans., 8A, 421 (1977). J. Kronsbein, L. J. Buteau, Jr. and R. T. DeHoff, Trans. Met. Soc. AIME, 233, 1961 (1965). Metals Handbook, 8th Edition, Volume 1. B. R. Patterson, Doctoral Dissertation, University of Florida (1978). M. Hillert, Acta Met., 12, 227 (1965). G. C. Kuczynski, Sintering and Related Phenomena, p. 325, Plenum Press, New York (1976). A. R. Hingorany and J. S. Hirschhorn, Inti. J. Powder Met., 2, 5 (1966). A. J. Markworth, Scripta Met., j>, 957 (1972). R. C. Pond, D. A. Smith and P. W. J. Southerdon, Phil. Mag., A37, 27 (1978). 162 and Mi50(add) = Miso {DSm} V DS 1 2V (67) Here it is assumed that the true D of the isolated pores is the same as that measured for wholly contained isolated pores. The initial measurements of properties of connected porosity were made assuming all pores not contained inside are connected; the abovementioned addi tions to the isolated fraction have therefore to be subtracted from Vj0nn, Sy0nn and My0nn to correct for surface effects. These corrected values, termed Vy0nn, Sy0nn and My0nn by the following equations. Vvconn(C) = vÂ§onn {DS/2V} 1 - DS 2V s5onn(c) + sJonn ciso { } DS/2V , DS 1 2V Mf,nn(C) = M: conn - M iso {DS/2V} 1 - DS 2 V (68) (69) (70) The geometric model, discussed in detail in the preceding section of the Appendix, is tested by comparing the corrected values and the calculated values. CHAPTER 3 DISCUSSION Introduction Sintering of a loose stack of powder is the simplest consolidation process and hence forms the basis for investigation of more complex and involved techniques. Loose stack sintering was therefore studied in the greatest detail possible in this investigation. It will be discussed at length in the beginning of this section to establish a framework on which the descriptions of hot pressing and pressing and sintering (also called conventional sintering) will be based. This discussion will be concluded with a number of speculations regarding potential strategies to control the microstructure of a powder-processed component. Loose Stack Sintering The discussion of metric properties will be followed by that of topological properties. It is expected that this study will indicate a likely scenario of various geometric events that lead to the observed paths of evolution of microstructure. Usually such a detailed study also suggests a variety of plausible geometric models of the structure; this study is no exception. The model that is in the best agreement with the data will be described and followed by suggestions for further research necessary to complete an understanding of the process. 101 154 OL Thus the integral mean curvature of a three-branch node, M is given by M3b = MS + Medge = -ttR + (^pk2R or M3b = tt (1 ^L)R (40) One-Branch Node Since such a node is a semi-spherical cap of radius r = (*/3/2)R that is connected to a cylinder of radius r, there are no edges involved. Thus V1b=|trr3 or Vlb=^R3 (41) Area of a one-branch node, S^b, is given by Sb = 2Trr2 or Sb = 2tt(^-) or Sb = ^R2 (42) Integral mean curvature of a one-branch node, M^b, is given by Mlb = 27rr2(-l) or Mlb = -2irr = -/3ttR (43) Cylindrical Branch Since all branches have length = T and radius r = C^-)R, Vb = -nr2!: = : ' (44) Sb = 2TrrI = ^3ttRI (45) Mb = -ttL (46) 103 Metric Properties of the Grain Boundary Structure SS In addition to the four parameters mentioned earlier, namely, Sy , SSS SSS SSP Lv Ly^occj and Ly ; the association of pores with the grain boundaries was also characterized. An etched section of the sample was examined to measure the number fraction of pores observed to reside within the grains, on the boundaries and on the grain edges. The variations of these frac tions are illustrated in Figure 41. These numbers indicate the fractions of porosity associated with and not associated with the grain boundary network. The twin boundaries do not participate in grain coarsening and hence were not included in the characterization of the associated porosity. The grain boundary area per unit volume, Sy illustrated in Figure 37, can be seen to change only a little in the density range from Vy = 0.93 to Vy = 0.97. Thus, the grain growth or decrease in grain boundary area appears to have been appreciably inhibited in this density range. It is suggested that the pore structure is changing in such a manner that the associated porosity is able to pin the boundaries during this phase of the process. This aspect of the grain boundary structure will be discussed and explored further in the course of the description of the other grain boundary properties. Variations of LySS, LySP and -vfoCC) 1",lustrated in Figures 38, 39 SS and 40, repsectively, also exhibit the arrest observed for Sy in the same density range. The fraction of porosity associated with the grain boun dary network decrease in this range of Vy, Figure 41. Thus, the more or less stable grain boundary network seems to facilitate the reduction of associated porosity. The following discussion of REFERENCES 1. F. N. Rhines and R. T. DeHoff, Modern Developments in Powder Metallurgy, p. 173, Plenum Press, New York (1971). 2. W. Rostoker and S. Y. K. Liu, J. Materials, _5, 605 (1970). 3. R. D. Smith, H. W. Anderson and R. E. Moore, Bull. Amer. Cer. Soc., 55, 979 (1976). 4. R. T. DeHoff, F. N. Rhines and E. D. Whitney, Final Report, AEC Contract AT(40-1), 4212 (1974). 5. G. Arthur, J. Inst. Metals, 83, 329 (1954). 6. R. A. Graham, W. R. Tarr and R. T. DeHoff, unpublished research. 7. G. Ondracek, Radex-Rundschau 3/4 (1971) 8. S. Nazare, G. Ondracek and F. Thummler, Modern Developments in Powder Metallurgy, p. 171, Plenum Press, New York (1971). 9. J. Kozeny, Sitzber. Akad. Wiss. Wien., 136, 271 (1927). 10. M. F. Ashby, Acta Met., 22, 275 (1974). 11. R. T. DeHoff, B. H. Baldwin and F. N. Rhines, Planseeber. Pulvermet., JO, 24 (1962). 12. Metals Research Laboratory, Carnegie Institute of Technology, Final Report, AEC Contract AT(30-1), 1826 (1959). 13. G. C. Kuczynski, Powder Metal1urgy, p. 11, Interscience Publishers, New York-London (1961). 14. T. L. Wilson and P. C. Shewmon, Trans. Met. Soc. AIME, 236, 48 (1966) 15. G. Matsumara, Acta Met., 19, 851 (1971). 16. F. N. Rhines, C. E. Berchenall and L. A. Hughes, J. Metals, 188, 378 (1950). 17. R. T. DeHoff, Proceedings of the Symposium on Statistical and Probabilistic Problems in Metallurgy, Special supplement to Advances in Applied Probability (1972). 18. E. H. Aigeltinger and R. T. DeHoff, Met. Trans., 6A, 1853 (1975). 140 6 are associated with a particular feature. These notations are listed in Table 3 and illustrated in Figure 2. In addition to the metric properties listed above, the microstruc ture of a porous body is also characterized by its topological properties. A brief discussion of the fundamentals of topology will precede the sur vey of microstructural studies of sintering. Fundametals of Topology The subset of topological geometry of present interest is that of 24 closed surfaces, that is to say, surfaces that may enclose a region of space. In a sintered body the regions of space are the pore and the solid phases; the pore-solid interface is a closed surface of interest. Such a surface may enclose several regions and have multiple connectivity. A surface is said to be multiply connected if there exist one or more redundant connections that can be severed without separating the surface in two. The genus of such a surface is defined as the number of redundant connections. For complex geometries it becomes difficult to visualize the topological aspects of surfaces. It has been found very convenient ;to represent surfaces by equivalent networks of nodes and branches. Such an equivalent network is called the deformation retract of a particular region of space. It is obtained by shrinking the surface without closing any 27 openings or creating new openings, until it collapses into the said 28 network that can be represented in the form of a simple line drawing. A number of closed surfaces and their equivalent networks are illustrated in Figure 3. The connectivity, P, of a network is equal to the number of 54 Figure 22. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vy = 0.871, etched (approx. 400 X). 96 Table 12 TOPOLOGICAL PARAMETERS lin,La ,,S rmax -3 rmin -3 ..iso -3 Number Vy Gy 9 cm Gy 9 cm Ny 9 cm 0.906 1.71 x 108 4.63 x 107 9.55 x 10 0.928 1.39 x 108 1.7 x 107 2.51 x 10 0.944 2.17 x 107 7.23 x 106 4.63 x 10 0.971 3.23 x 107 5.77 x 106 3.33 x 101 0.979 7.6 x 107 3.18 x 106 1.39 x 10( 5 88 When a connection is observed between different subnetworks, the said subnetworks have to be renumbered to keep track of such connections. ATI the subnetworks involved in such connections are marked with the lowest of the numbers of these connecting subnetworks. If an internal subnetwork is observed to be connected to an external one, the said internal subnetwork is marked with the number designating the external subnetwork. A connection between previously unconnected subnetworks, those with different numbers, does not change any of the three parameters. A connection between two or more subnetworks with the same number signi fies a complete loop observed entirely inside the sample volume, and thus increases the count of Gmin by one. Since Gmax includes such internal loops, it is also increased by one. After each comparison of consecutive sections, the counts of Gmax, Gmin and N1S0 were updated and tabulated as shown in Table 10. The values of interest are the unit volume quantities, G!J!ax, Gyin i so and Ny If the features that give rise to these quantities are randomly 25 and uniformly distributed in the sample, then it can be expected that there exists a quantity (Qy) characteristic of the structure and equal to the unit volume value. Thus AQ (the change in quantity Q) = Qy x AV. Dividing both sides by aV leads to "AV = The slope of AQ versus AV plot therefore should be equal to Qy provided AV, the volume covered is large enough for a meaningful sampling. In 25 51 the previous investigations of this kind the analyses were continued 53 grain structures that were too fine to be studied optically; these samples were not included in the measurement of grain structure prop erties. A number of etched microstructures are illustrated in Figures 22 through 36. Typically, the scale of the grain structure was such that the information contained in a single plane section was not enough to yield estimates with the desired precision of 10 percent. Conse quently, SyS, LySS, LySP and Ly^ccj, defined earlier in this report, were measured by repeating the polishing, etching and counting steps a number of times to obtain at least 100 different fields of view. The grain structure properties are illustrated in Figures 37 through 40. The apparent local movements of the traces of the pore-solid interface, mentioned earlier in this section, are likely to introduce some errors in the estimation of grain structure properties whenever the pores are associated with the boundary network. For example, an enlargement.of pore features residing on grain boundaries would underestimate the value of Sy, the grain boundary area per unit volume, as measured metal!o- graphically. However, it was found that these errors are small compared CD CD . to those in Sy and My ; the trends of grain structure properties remain unaffected whether plotted versus Archimedes density or the stereo!ogical density. The quantities in Figures 37 through 40 are thus plotted versus the Archimedes density. The pores observed on a polished and etched surface can be classi fied as to their association with the grain boundaries, that is, according to whether they appear to be inside a grain, on the grain boundary or on a grain edge. The relative fractions of pore features regarding their association with the boundary network were measured. These are illustrated in Figure 41. 48 Table 6 QUANTITIES MEASURED ON POLISHED SECTION Test Feature Quantity Points Pp Lines Ll Area NA ta Definition Fraction of points of a grid that fall in a phase of interest Fraction of length of test lines that lie in a phase of interest Number of intercepts that a test line of unit length makes with the trace of a surface on a plane section Number of points of emergence of linear feature per unit area of plane section Number of full features that appear on a section of unit area Net number of times a sweeping test line is tangential to the convex and concave traces of surface per unit area of a plane section 79 Figure 43. Illustration of contributions of subnetworks crossing the surface towards the estimate of Gmax. 65 Figure 33. Photomicrograph of INCO 123 nickel powder cold pressed at 60,000 psi and sintered at 1250C to Vy = 0.942, etched (approx. 400 X). 138 5) It is suggested that because of their equiaxed shape, isolated pores anchor the grain boundaries effectively whereas the connected pores do not. Hence, most of the isolated porosity is associated with grain boundaries. 6) The association of grain boundary network and isolated pores facilitates rapid reduction of isolated porosity as the associated boundaries provide immediate sinks of vacancies; this is in contrast 43 45-47 to the traditional viewpoint that once the pores are isolated, it is very difficult to remove them from the system. 7) The connected porosity finds itself disassociated from the grain boundary network which slows the reduction of such pores consid erably. 8) The higher values of curvature, yet comparable values of area and volume of pore phase for the hot pressed samples as compared to loose stack sintered samples, are tentatively attributed to similar geometries but a higher number of isolated, spherical pores. 9) An absence of an arrest of grain growth in hot pressed samples, in spite of a higher number of equiaxed pores, is believed to be due to stress-induced grain boundary sliding that promotes grain boundary migra- ti on. 10) Porosity in pressed and sintered samples is believed to consist of finer networks and a higher number of isolated pores compared to the loose stack sintered samples with the same density; this leads to much higher areas and curvatures in PS than in LS. 11) Due to larger number of interparticle contacts in a green compact as compared to a loose stack sintered to the same density, a pressed and sintered sample has much higher grain boundary area. Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy METRIC AND TOPOLOGICAL CHARACTERIZATION OF THE ADVANCED STAGES OF SINTERING By Arunkumar Shamrao Watwe' August, 1983 Chairman: Dr. R. T. DeHoff Major Department: Materials Science and Engineering Measurements of the metric properties of porosity and the grain boundary network during the advanced stages of loose stack sintering, conventional sintering and hot pressing of spherical nickel powder (average size 5.5 microns) were followed by topological analysis of the loose stack sintered samples. Linearity between area and volume of the pore phase for the loose stack sintered series was approached by the conventionally sintered and hot pressed series whereas the curvature values for these series remained significantly different. An arrest in grain growth during loose stack sintering was concurrent with the removal of most of the isolated porosity. Subsequent resumption of grain growth coincided with the stabilization of connected porosity. 46 /, _\ Weight of sample p^9/ I Volume of sample Weight of sample . (Volume of sample + immersed part of pan) - (Volume of immersed part of pan) M1 (W2 + weight of pan in air Wg) - (Weight of pan in air W4) Thus W1 p (g/cc) = w2 + w4 w3 The densities thus measured were reproducible within 0.2 percent of the mean of ten values with 95 percent confidence. The density of a 50 piece of pure nickel, known to have a density of 8.902 g/cc, was measured and found to be within 0.5 percent of the abovementioned value. Metallography The polishing procedure will be described and followed by a brief discussion of principles of quantitative stereology involved in the estimation of metric properties. The estimated microstructural properties will be presented thereafter. Polishing Procedure The wax coating on the samples was dissolved in hexane and the samples were sectioned; a vacuum impregnation method was used to mount the samples, surrounded by a nickel ring, in epoxy. The purpose 59 Figure 27. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vy = 0.979, etched (approx. 400 X). 17 continue to decrease. If the pores are filled with a gas of low solu bility or very slow diffusivity, then coarsening leads to an increase in volume of porosity.^^0 gas Â¡ias en0Ugh pressure to stabilize 16 41 42 the pore-solid interface, the densification rates can be very low. Since exaggerated or secondary grain growth that results from boundaries 43 44 breaking away from pores has been observed to be accompanied by slow rates of shrinkage,43,45-47 has been theorized that the grain boundaries that can act as efficient vacancy sinks are far away from a large number of 43 45-47 pores. 5 The end of the third stage is of course the disappearance of all pores, although that is rarely accomplished in practice. The three stages described above provide a common framework for the discussion of microstructural studies that are reviewed presently. This review is expected to demonstrate the potential that the present research has for providing a perspective of sintering that is more profound than the current one. Metric Investigations SP It has been observed that in loose stack sintered samples Sy decreased linearly with the decrease in Vy^2*1820) (jur-jng second stage. Surface area may be reduced both by densification and surface SP rounding or by surface rounding alone; the linearity between Sy and Vy is believed to arise from a balance between surface rounding and densifi cation. Support for this hypothesis comes from the observation that sur face rounding dominates in pressed and sintered samples until the balance has been reached, as shown schematically in Figure 7. The slope of the SP 20 Sy versus Vy line is inversely proportional to the initial particle size. 57 Figure 25. Photomicrograph of INCO 123 nickel powder loose stack sintered at 1250C to Vy = 0.944, etched (approx. 400 X). 15 Figure 4. Illustration of a one-to-one correspondence between a spurious node and a spurious branch in a deformation retract. 120 Table 15 CORRECTED VALUES OF V^onn, -conn bV and M conn No. vf ..conn vV cconn -1 Sy cm Mconn -2 My cm 1 0.906 0.09 731 -20 x 105 2 0.928 0.053 475 -15.2 x 105 3 0.944 0.017 138 -3.1 x 105 4 0.971 0.017 165 -4 x 105 5 0.979 0.02 237 -10.7 x 105 |