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APPLICATIONS OF TOPOLOGY TO SINTERING By LEON JOSEPH BUTEAU, JR. A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FTI.ORIA ACKNODWLEDGMNTS author wishes express his sincere gratitude Dr. John Kronsbein suggest flang the subject matter and methods of research. Without guidance encouragement this disser station would not have been possible. The author also wishes to express his appreciation Dr. F. Rh inca Head of the Metallurgy Department, II. B. ReedHill, Mathematics, Professor of Metallurgy, and Dr. W. O. Smith, Dr. R. G. Blake, Professor Professor of Mechanical for serving on his supervisory committee. Finally , the author expresses his appreciation to Mr. R. A. Runmel for developing the precise technique required to obtain the serial sections used in the example calculation. ~ Fneer in TABLE OF CONTENTS Page ACKNOWLEDGM4ENTS . . . . . . LIST OF TABLES . . . . LIST OF ILLUSTRATIONS . . . . . NOTATIONS . . . . . . INTRODUCTION . . . . . . CHAPTER DESCRIPTIVE TOPOLOGY AND SINTERING Terms Symbols The Decomposition of the Void Space NETWORK TOPOLOGY The Algebra of Networks The Incidence Matrix The Loop Matrix Relations Between the Fundamental Loop the Incidence Matrix Cut Matri ces SINTERING AND ALGEBRAIC TOPOLOGY The Conjugate Algebra Space Model Modulo Two Knotting Within a Sinter The Linkage Numbers Body The Higher Dimensional Betti Numbers APPLICATIONS TO SINTER METALLURGY Some Relations Between the Betti Numbers of Sinter Body Void Space filmy Piramnnl as r La fsn   III. TABLE OF CONTENTS (Continued) CHAPTER Page CALCULATION OF THE BETTI NUMBERS FOR AN ACTUAL SINTER BODY 4 4 4 S 0 0 5 4 0 0 5 0 5 14 SUMMARY OF THEORETICAL RESULTS FOR APPLICATION TO SINTER METALLURGY . . . . . REFERENCES . . . . BIOGRAPHY . . . . . . . LIST OF TABLES TABLE I. Page TOPOLOGICAL SYMBOLS AND DEFINITIONS TOPOLOGICAL AND SINTERING TERMINOLOGY III. TOPOLOGICAL CHARACTERISTICS AND DIMENSIONS TABULAR DATA OF TOPOLOGICAL ELEMENTS AND INVARIANTS TABULAR DATA OF THE TOPOLOGICAL INVARIANTS UNIT VOLUME SECTION THICKNESSES AND CUMULATIVE VOLUMES LIST OF ILLUSTRATIONS Figure Page Rhines Topological Model a Sinter Body S S 3 Decomposition of the Void SpaceIllustration of 6 Decomposition Decomposition Decomposition Decomposition Decomposi tion Voids at the of the Void of the Void of the Void of the Void of the Void Vertices SpaceIllustration SpaceIllustration SpaceIllustration SpaceIllustration SpaceIllustration of 6 of 6 4 of of 6 of 6 of the Tetrahedron Branchnode Section of Network a Sinter Superimposed on a Body . . Twodimensional The Network N Loops Trees of N of N * S S S SS S S S S S S S The Co trees of N A Set of Fundamental Loops N * A Set of Nonfundamental Loops of N The Cuts The Cuts of N of N with an Arbitrary Orientation Photomicrograph of Linde Copper Powder 170 +200 Mesh, Sintered in Hydrogen for 128 Hours at 1005 150X Pho tomi crograph of Linde Copper Powder, 170 +200 Mesh Sintered in Hvdronen for 32 Hours at 1005 150X t LIST OF ILLUSTRATIONS (Continued) Figure Page Tetrahedron for Constructing a Face branch Incidence Matrix Cubic Knotte Tinker Representation Tinker Representation * S S S S S S S S S * S S S S S S S S S S U Knotted Branchnode Sinter Body Representation Example of Examp 1 e Exainpi Linkage Number Zero of Linkage Number One of Linkage Number Two * S S S S S S S S * S S S S S S S S S S . Illustration of Representative Sinter Body Areas Partial Pho tomi Twodimensional crograph Sinter Linde Copper Body Branchnode Powder 170 +20 Network 0 Mesh Sintered Hypothetical in Hydrogen Serial for 1024 Hours Sections at 1005 C a Sinter , 150X Body . Branchnode Network for the Hypothetical Serial Sections Section 1 Section 2 Section 3 14 Serial 14 Serial 14 Serial Sections, 90X Sections, 90X Sections, 90X * S S S S S S S S S * S S S S S S S S * 5 S 5 4 5 4 S S S Section 4 of 14 Serial Sections, Section 14 Serial Sections U S S S S S 57 . Section 6 of Serial Sections, Section Section 8 Rae t.{ on  14 Serial Serial of ia R iial Sections, Sections, Qor.i nna. 90X 90X QOx *~~~ ~ ~ U * 5 5 5 LIST OF ILLUSTRATIONS (Continued) Figure Page Section Section Section 13 11 of 14 Serial 14 Serial 14 Serial Sections Sections Sections * a S S S * a a S a a a * a S a a a a S Section 14 of 14 Serial Sections Branchnode Network for the 14 Serial Sections Betti Betti Number Number Number Number of Sections of Sections * 5 S S S S S S S * 5 S S S S S S NOTATIONS Area Matrix element fundamental loop matrix Branch Branch connecting nodes k and between sections and J Contact columns of E2 remaining after has been removed Incidence matrix Reduced incidence matrix Loop matrix Fundamental loop matrix Cut matrix Fundamental cut matrix The columns of E1 ii a tree Rhines Modulo genus two incidence matrix Unit matrix of order m Unimodular matrix Dimension Linkage number Node Node on section k Particles Betti Betti Rank of number number the cut matrix singular matrix Tree matrix Tetrahedron Volume Symbol k k dimensional a set of nodes simplex INTRODUCTION A sinter body is produced exposing a large number of small metallic particles (0.001 to 0.3 inch diameter) a temperature that near but below their melting temperature. The particles arranged some type of geometric configuration which allows interparticle contacts. When these particles are maintained at such a temperature for a sufficient length of time, it is found that interparticl e contacts become fused or welded. As time increases, these welds, often referred to as necks, grow. space between the particles constitutes a void and in time, consequence of heating is breaking up of the void into separated parts. If enough time elapses, all the voids may disappear and the metallic mass will have its natural bulk density. It is desirable to have a mathematical apparatus capable describing the characteristics a sinter body at each stage of the sintering process. This was first undertaken F. N. Rhines (a)* in 1958 , when he introduced the idea of applying descriptive topology to sinter progress. Instead of concerning himself with those geometric parameters normally considered in sintering, he decided to represent *ThP ,undrlrl I nod nimbhors in nnPrnrhsin ft .rafr fnnf8 ri.P are or describe the sinter body terms one of its topological invariants. The first and simplest model adopted Rhines is shown in Figure where the vertices of the tetrahedron represent sinter particles the edges the interparticle contacts. advantage of a topological description is invariance with respect continuous deformations. This means that particle sizes and shapes not directly enter the problem, resulting in considerable simplification. The important of particles parameters and the interparticle in the Rhines model contacts. are the number The invariant considered is given Euler's formula for the connectivity two dimensional manifolds embedded in a threedimensional Euclidian space. This G = C  Interparticle contacts  Particles  Genus of the surface bounding the sinter body In topology term genus may be used to mean the number handles on a sphere and thus the number of self reentrant cuts that be made on the manifold without dissecting it into more than one connected surface. It is restricted to the surface and does not apply to the threedimensional manifold enclosed or excluded In the case of sintering, the manifold enclosed the sinter body surface must be defined as the sintered material. can P~ 3 Figure Rhines Topological Model a Sinter Body 4 The expression Pi was used Kicrc hoff describe an associated quantity that will be introduced later subsequently named the cyclomatic number by Maxwell who used a similar connection. Veblen uses this term , but Lefschetz Alexandrov and Cairns refer to it as the t Betti number and denote it by This terminology will be used throughout this dissertation. From a generalized point of view, the sinter body is composed two threedimensional mani folds one twodimensional manifold. These are the void space manifold and the sinter body manifold and they are related inasmuch as they have common same twodimensional manifold, the topological entity first considered Rhines. result of the present research is a mathematical model giving relationship of the two topological spaces. Subsequently DeHoff in collaboration with Rhines introduced a type metrical concept (i.e. units measure such as length, area, volume, etc.) in twodimensional topology of sinter bodies. tion This , surface enabled area them uni t to measure volume, such etc. parameters Thus, as volume a considerable frac advance ment in the interpretation of twodimensional random measurements sintering processes was made and probability combining theory. sinter body and associated void space represent extremely complicated manifolds in that and (C are very large numbers and may be of the order of several million. Hence, visual P e present research to extend these methods so as to make use of other topological invariants besides genus (i8t Betti number) and also to introduce algebraic methods for the study of wintering. The first method to be considered is network topology, since this gives a model of an actual sinter body. Investigations in this field lead to the introduction of matrix algebra. The matrices of importance in the present problem are the incidence, loopbranch, treebranch, cutbranch matrices , respectively. The topological invariants, such as the th as the 0 be defined later) and 1st Betti number, are expressible terms of matrix ranks, which can in principle be obtained counting procedures or alternatively evaluation of linear equations. In addition some reference is made to linkage numbers discussed Veblen and Alexandrov, and a new method classifying stages sintering is introduced, which is related to these and the previously mentioned numbers, CHAPTER I DESCRIPTIVE Terms TOPOLOGY AND SINTERING and Symbols In this concepts are chapter introduced some elementary and briefly topological outlined to enable operations the reader to correlate sintering and topological terminologies. It should observed that the topologist does not in general concern himself with the study their of topological characteristics. manifolds which are On the other hand, t constantly his changing is the principal concern of the metallurgist. Therefore, it is necessary that topological study of a sinter body or its associated manifolds (described below) always be conducted at a stage where at least temporarily none of the topological invariants change. In Table we demonstrate a correlation between topological terminology symbols (and terms symbolism) that will and that be used used in repeatedly sintering. throughout Various this work are presented therein. These terms are used in this disser station without further definition. Two other terms use in sintering are defined below A onecircuit is a onedimensional complex such that each i a i nr idnnt" ui rh onrrtlv twn hranchon. Tn tPrinn of sintprinu nnrlP . 7 TABLE TOPOLOGICAL SYMBOLS AND DEFINITIONS Node A point Branch A segment end points a line excluding its Twodimensional plane area bounded three non simplex (area) collinear points and the branches joining them a triangle without sides and vertices Threedimensional simplex (volume) A threedimensional the four but not faces of including manifold bounded a tetrahedron the faces, edges, or vertices Four dimensional simplex four dimensional manifold bounded five threedimensional simple exes Zerodimensional One node simplex Zerodimensional More than one node complex Onedimensional or more branches complex provided each branch associated tei th two nodes Twodimensional or more areas including their complex boundaries Three dimensional complex or more volumes their including boundaries A twocircuit a set of twodimensional complexes such that each branch is incident with an even number of twosimplexes. It is a set of areas which form a closed surface. In sintering, this a surface in threedimensional space which bounds a volume. Such a twocircuit is referred to as a closed two dimensional manifold. tion of use of the term surface differential geometry the term manifold for differential is also used in topology geometry. in sintering, allows However, terms su reserva since lrface and manifold may occasionally be used interchangeably. parameters of interest in sintering are several of the topological invariants. Invariant, in this case, means parameter is unaffected the manifold continuous is undisturbed. deformation provided The invariants the continuity of of primary interest sintering are defined below the 0th Betti numberis the number of disconnected parts of a manifold. For the sinter body this will be assumed to be 1 hence * while for the void space, It has been mentioned above that network topology will be used initially. We obtain a network for a sinter body defining a particle as a node of a netw and a ork are related contact as a branch. branches. of nodes the following The sinter equation which body two numbers applies is then introduce separately composed d above to the sinter body and the void space, which explains the superscripts have been omitted. 9  n+ P called represents the Ist the number of Betti number onecircuits for the sinter closed loops) body For the void space manifold the corresponding P1 depends upon how far sintering progressed, this will discussed later. Table II indicates the parallelism between topological terminology and sintering terminology TABLE II TOPOLOGICAL AND SINTERING TERMINOLOGY Topological Topological Sintering Sintering Symbol Terminology Symbol Terminology a 0 cell n Node a 1 cell b Branch 2 2 cell A Area a 3 cell V Volume 3 To begin with, the sinter body will be regarded as an assembly of nodes sinter branches. body At a later the void time space will the bounding also surface be considered. between A definite prescription for the mathematical construction of the void space, the basis of the physically given slnter body, will be given At the beginning of the sintering process, both the void space the sinter twodimensional body are completely boundary. connected and possess a formation of interparticle common contacts is equivalent to the existence of branches between particles. a physical observation that a continuous path exists between any particles.) As sintering progresses, particles fuse and the places of fusion are sometimes referred as necks. In the sintering process these necks or branches grow, but from a topological branchnode analysis this growth has no meaning. As a result the topology of the problem remains unchanged until a branch disappears in the void space. This socalled channel closure in the void space causes one of things occur. Either the void space remains connected or segre nation of voids (isolation of parts of the void space manifold) initiated. The fact that this segregation can occur has consequences that will become apparent shortly. As sintering progresses the void space breaks up into more and more separated parts so that in the end the void space consists only of isolated pores. The Decomposition of the Void Space It will now be demonstrated how it is possible for the void space to decompose into isolated parts while the sinter body remains connected. Consider the void space breaking up into distinct separated parts as illustrated in the following six figures. interior of the cube of Figure a representation of a volume of void space. two 11 Figure 3 shows it under the influence of a topological deformation. It t. important that one understands that Figures 2and 3 are topologi call completely equivalent. fact that Figure an additional two lines indicated is on Figure significant fact as it can is that still be mapped onetoone the continuity of the complex has not been changed. Figure 4 is a different topological complex in that there are now two volumes which are still connected. They are connected because it is possible to find a continuous path from one to the other without leaving the manifold. (The boundary is here considered as part of the manifold in view of what has been stated before.) Figure is different again in that the volumes remain connec ted branch. an area. transition Figure 6 illustrates to 7 them connected is significant a single in that the latter illustrates two complexes entirely disconnected. view of the fact that two threedimensional volumes are discon nected, it may be said that the void space is bounded in four dimensions. A question naturally arises as to whether two voids in three dimensional this, space are we proceed as in Euclidian n space, connected follows: is project in fourdimensional an ndimensional s d linearly into an space. implex,  1) answer embedded dimen sional space, the ndimensional simplex fills entire (n1) dimensional space. is immaterial what definition is ascribed the infinite points of this space; e.g., three dimensional projective space with a suitable metric impressed upon it might have been chosen.) irmuateri al from Figures Figure Decomposition Illustration f the of 6 Void Space t i Figure Decomposition Illustration of 2 of Void Space Figure Decomposition Illustration Void Space Figure Decomposition Illustration Void Space of 6 Figure 6. Decomposition of Void Space Illustration of 6 'iNNi~iNNN"flQI''' N: 17"""""" Figure Decomposition Illustration void space Thus, a fourdimensional simplex is projected into threedimensional space, four of the five threedimensional simp 1 exes bounding it may be made to form one tetrahedron The remaining threedimensional simplex occupies the threedimensional space exterior to this tetra hedron. To determine whether the threedimensional voids, referred previously, are connected in four dimensions, consider two isolated voids in three dimensions. topological deformation these voids can be located at the vertices of the tetrahedron referred above. Such a construction is illustrated in Figure In this way two vertices of the tetrahedron are truncated as shown. truncated tetrahedron in addition to the fifth threesimplex, which covers the remainder threedimensional space, when projected into four dimensions forms the boundary a fourdimensional simplex. The fourdimensional simplexes, Assume the abov has been so simplex, a continuum of e topological conducted that bounded points the five on its exterior deformation in the images threedimensional and interior. threedimensional of the two void space threedimensional vertices lie along the boundary of the exterior part mentioned above. Then there exists a path from the image of one of the voids to the other through the continuum not occupied set of five three dimensional dimensional simplexes. space Hence can be connected any two separated voids a continuous lin in three e in four dimensional space, which lies exterior to the threedimensional manifolds and meets them nowhere. I a  void   void Figure Voids Vertices Tetrahedron T1 CHAPTER II NETWORK TOPOLOGY The Algebra of Networks The methods of network topology can be correlated to the topology of a sinter body and its associated void space. Figure represents a twodimensional section of a sinter body. The material in white corresponds to copper while the black corresponds to void space. On the copper a branchnode network has been constructed this indicates clearly that the complicated sinter body can be repre sented, from a topological point of view , in tefln8 of branches and nodes. It should be noted that a network can also be constructed in the black areas of the photograph. Such a network would be representative of the void space manifold. corresponding topological actual invariants construction of can be related this so that is given later In this chapter the algebra of matrices as related to a sinter body with its associated void space will be presented. important matrices are four in number and are named: Incidence matrix Loopbranch matrix Treebranch matrix ^" __J_ ^.   1. __  t j *,_ on. D~~DDDDDDKDDD ~ liiaDDD iggjgggg 'N DDDDDDDDDD DmiE:" iiIIIDDDDErhraDINNADD AEffl N E St. Figure Branchnode Network Superimposed on Twodimensional Section of a Sinter a Body In addition be outlined. to presenting the matrices that their interrelationships will introduced in the previous chapter. Statement  n This introduction equality implies of matrices an advancement in the topology in that of sintering, it allows which follows: All of the following statements apply to both the void space parts the sinter body. this reason the superscripts temporarily omitted. Disconnected networks may be connected common node method, which will be described subsequently. The Incidence Matrix Consider the network of Figure denoted byN p consisting of four nodes and to each b and six branches. ranch an integer Assign to each node and an orientation. Note that number ri ng of the branches and nodes as well as the orientation is completely arbitrary. This does not imply that there may not most expedient numbering or orientation. incidence matrix, denoted is formed as follows: Provide a column for each branch and a row for each node. Atcn ha 4n ai4h a nn^n 4k 4; 4l The notation used is now are an integer number P+ 3 x kranrrk ia *~JA Figure Network node and is oriented away from the node, insert = +1 in the matrix. e= 1 if the orientation is toward node a zero otherwise. The incidence matrix for N is presented below: 1 +1 0 +1 1 1 0 1 +1 +1 0 +1 Statement The rank of the incidence matrix is equal the number of nodes in the network minus the number of separated parts. This statement been proved by mathematicians will not be repeated here The Loop Matrix Loop definition: A loop, of a network is a onedimensional complex of N having precisely two branches incident with each node. The loop matrix, denoted is formed as follows: Provide a column for each branch of the network anda row for each loop. Arbitrarily orient and letter each loop. If branch is contained in loop and is oriented in the same direction as the loop, insert S+1 in the matrix. the orientations are different, write = 1, and if the branch Figure The Loops 0 1 1 0 1 The matrix El obviously has property that rows linearly dependent because the summation of the rows zero. Hence, one may delete row of the matrix and call the resultant matrix, designated the reduced incidence matrix. The deleted row is arbitrary and therefore in specific cases reference should be made the number rows that have been deleted. Tree definition: tree a connected network a one dimensional complex containing all the nodes of N no loops. From this definition and the proofs of (4) and (9), follows that: Every tree of N has a branches. number r trees in N equals the value of the determinant where a prime denotes transpose. An enumeration all the trees of Figure 10 is presented in Figure Every tree of N divides the b branches into two sets.  1, branchh.s in the treeC are Figure The Trees The (b  n + 1 branches are said to form cotree of the network. cotrees of Figure are presented in Figure Statement III. The number of branches in a cotree of N equals the 1st Betti number a network. This statement has been proved mathematicians and will not be repeated here (10). Since a tree contains no loops, there a unique tree path between any two nodes. This together with an appropriate branch, defines extr a loop. 'a branch The loop is introduced. arbitrarily This set o oriented f loops in the is called same direction fundamental they define a submatrix of E2 of order x b, called the funda mental loop matrix E A set of fundamental loops corresponding tree of Figure 12 is presented in Figure 14 and the corresponding 0 1 The term fundamental has an important significance. The number fundamental loops is equal to the number of linearly independent loops. It has been shown by mathematicians (e.q , Veblen, page that a relation exists be tween the incidence and loop matrix. relation is 29 Figure A Set of N of Fundamental Loops Statement rank of E2 equal a the ist Betti number of the network N This been proved by mathematicians A set of three independent loops of the network, which fundamental, are illustrated in Figure The corresponding matrix, denoted 0 1 It should be noted that there is no unit submatrix in This is implied the fact that it is not a fundamental loop matrix. Statement rank of the loop matrix E2 is equal the 1st Betti number of the network N. This has been proved mathematicians These two statements are of importance in the study of a sinter body because when the incidence matrix is known, the fundamental loop matrix can be determined from it and hence the 0th Betti numbers of the network ascertained. Thus a known algebraic apparatus is available to deal with the invariants of the sinter body associated void space. Relations Between the Fundamental Loop the Incidence Matrix For a given loans can tree of a network N, be numbered and oriented the branches and an that Rn. and R. are fundamental ai ven are ~E """" X Figure A Set Loops of Nonfundamental of N 33 then (This is proved elsewhere Cederbaum (W.) where The fundamental loop matrix columns The columns of E2 of E1 remaining after corresponding has been to a set of (b removed  n + 1) branches not in a tree The columns of E1 corresponding a tree The reduced incidence matrix unit matrix of order m A necessary and sufficient condition for a ( 11+ matrix E2 to be a fundamental loop matrix of a connected network that after permuting its columns, we can write ( Im, P F) The numbering of the branches of N must be chosen so that E1 can be written in the form ( T . This is always possible. the network of Figure then (12) tU Hence giving , F) so that I C J. UflL In addition All of these matrices can be used to determine topological properties of the sinter body or associated void space any stage of the 0 0 1 F Cut Matrices Cut Definition: A cut of a network a set of branches whose removal increases the disconnectedness of the network one, provided no proper sub set has this property.* The cuts of the network decpicted in Figure are given Figure Networks that are not connected may be considered connected if they are joined the common node method. This method consists of connecting the Po separate parts at any arbitrary reference node. Such a process has the property leaving the Ist Betti number unchanged but obviously not the 0th Betti number . This procedure may be convenient from time to time. removal a cut divides the network N into two parts Each branch of the cut was joined to a node in N1 and N2 We define cut as an ordering of N1 and N2, either NIN2 N2N1 An oriented as the cut if node in N it i the cuts branch of N N2 oriented away dividing Figure is said from its to have node 10 into N1 the same orientation in N1 and N toward are shown in Figure Incidence a matrix E3 between constructed the branches in accordance wi and cuts may be th the following described convention: sequel this last phrase will not be repeated implied and N2 N Figure Cuts of N I I I Figure Cuts of N with an Arbitrary Orientation Provide a column for each branch and a row for each cut. If branch is contained in cut j and oriented in same direction as cut, write = +1. If the branch is oriented in the opposite direction, write = 1 and a = 0 otherwise. The cut matrix for Figure 10 is given below: 1 1 It is proved elsewhere that = E3B '= A cut is a minimal set of branches so that it contains least one branch every tree of the network. has a rank equal to (n  1) for a connected network. Statement The 1st Betti number a connected network equals the number branches in the network minus the rank (12).  r  . The rank of El ~b nf 39 If the cuts are numbered and oriented so that for a particular network (KIC is called the fundamental cut matrix; a unit matrix of  1) rows and columns; K is a unimodular matrix. This matrix has property that all its sub determinants, including the elements, have the value + or O. in addition , the loops are numbered and oriented so that and E2 are as given and (9) then terms of the cuts previously illustrated 0 O 1 K IL is necessary to pursue further the relations between cut, loop, and incidence matrices. Given the incidence matrix, others can be found solving sets of linear equations conversely the cut matrix also has an interpretation in sintering. When sinter body is broken into two parts, a row of the cut matrix is determined. In orinciole then. the cut matrix of a sinter body can be I ways. might a lengthy procedure, but since most sinter bodies are made with uniformly graded sizes of metal particles, they essentially isotropic. Hence, a small specimen, which must, however, still contain a relatively large number of particles, can be considered to be representative of the entire sinter body any stage. This leads to the idea unit volume, of topologicall " which has been used properties on occasion per unit mass or per . DeHoff (13). are CHAPTER III SINTERING AND ALGEBRAIC TOPOLOGY The Conjugate Model In the model of the sinter body proposed here the particles taken to be nodes remainder and the interparticle of the threedimensional contacts Euclidian space to be branches. e is taken to b ie void space. Thus, the sinter body void space have a common two dimens ional boundary (surface) which is also a topological manifold. These three manifolds must be carefully distinguished. From a topo logical point of view, it is necessary to be quite precise in the definition of the void space. When considering sinter body, there is little doubt, at least in the early stages sintering, apparatus as to what described a node previously and what is then easily a branch. applicable. algebraic This so in the case of the void Bpace, Even cursory considerations examination of Figure show that these e element a are not obvious the void space during later stages of sintering. In addition, branches and nodes in both the s itter body the void space are not obvious in Figure out as follows: therefore, First construction of construct the void space a regular tetrahedron carried in three diusetnsonal space. The corners of this are assumed to be loaded with Space r r 'h I" t...,_,,,,,,,_I..,.. L ., Figure Photomicrograph of Linde Copper Powder, 170 +200 Mesh, 128 Hours at 1C Sintered )05 C, in Hydrogen for 150X "" """"""""""" "ijiEEEE"X(" 43 a SW  Figure Photomicrograph 170 +200 32 Hours Mesh, t 1005 of Linde Sintered Copper Powder, in Hydrogen 150ox Now determine mass center of these particles. From this point, a perpendicular is dropped upon each of the four faces of the tetra hedron. perpendiculars are extended beyond the limits of the tetrahedron within which mass center is situated. extens ion is made arbitrarily far and equal for all four perpendiculars, least so far that the faces of the regular tetrahedron bounded points of these extends ions nowhere intercept the original tetra hedron. This new tetrahedron is the conjugate of the original. masses are actually taken to be sinter particles contact points between particles to be sinter body branches , then the void space tetrahedron and as as nodes branches the vert ices and faces, of the so constructed new the corresponding configura tions that which belonging the sinte assumes to it. body a sinter cannot a particle part icle be constructed at each vertex of happens simplicial the tetrahedro to be missing so subdivision, n. then a sufficient number of fictitious vertices are assumed and the sinter body thus theoretically is simplicially subdivided contain a part of the physical void space. This does not interfere with topology of the problem because the fictitious nodes are subsequently removable without corresponds distrubing to Veblen's the void space procedure of "regular nodebranch manifold. subdivision" This of a sim plicial subdivision of a manifold. The surplus simplices may moved whenever convenient. The number of nodes and branches in the void space may be considerably larger than absolutely necessary re 45 conjugate to one anotherreferred as void space and the associated s inter body, and conversely. This is illustrated in Figure Both can as constructed be represented in the previous terms of the where the branches were assigned an orientation. However, in the sinter body problem, the use of oriented branches not immediately nece ssary. Hence, slightly modified algebra not requiring orientation is used this chapter. AlgebraModu lo In the later stages of sintering, the branchnode identification method becomes more and more difficult and therefore appears use ful to introduce the additional threedimensional manifolds conmonn bounding surface of the sinter body void space. For this purpose the following algebra now introduced. numbers 0 and form a ModuloTwo Algebra when combined according to these rules: 0+0 OxO = 0 x = 1xO Lxi To understand this convention, consider the branchnode complex in Figure set of nodes be denoted the symbol (x1, * xn) where n U is the total number of nodes. Thus, of nodes are given the symbol  U I. these manifolds algebra of matrices, chapter, x_  * 1 * * 46 Figure Conjugate Space Model Df;IIII'lIID r 47 is given Geometrically +1(2 represents set of nodes that are in X1 in K2, not in both X1 Using ModuloTwo Algebra, incidence matrices can be formed. Such matrices, denoted H (the notation is that used Veblen), give the information required of a nonoriented network. matrix gives information concerning the nodes of a network. Figure (24) A second matrix can formed denoting the incidence branches and nodes. This matrix, denoted is termed the branch node incidence matrix. convent ion for forming such an incidence matrix as follows: Provide a column for each branch a row for each node. branch is incident with node then aij =1i, while they are not incident, = 0. maliH matrix for Figure 10 is presented below: (25) Such an arrangement can generalized to higher dimensions. and X2. face opposite to it and designat so on face branch incident e matrix can constructed. Such a matrix , denoted for Figure , is illustrated below: (26) be shown that * H2 (27) as one generalizes to higher dimens ions it is found that Hk+l (28) There is a definite parallelism between this representation and that of the previous chapter From time to time one or the other will used depending upon expediency Extending to higher d dimensions, finds Table can be constructed. Knotting Within a Sinter Body Consider the two tinker toy mode ls shown in Figures 22 and 23. In Figure 22 a Simple Cubic Array is illustrated = 13) In Figure = 13), a complex is illustrated that iseby means siuole. The significant difference is that the latter for n2, can one .. i jjj:EEE 49 * S if 5 Figure Tetrahedron for Constructing a Facebranch Incidence Matrix to. p~igure Cubic Tinker Toy Representation ~""''"' r, :E EE" E ":,": x, C, Figure Knotted Tinker Toy Representation body problem and a branchnode network illustrating entanglement, normally seen in early stage sintering, is photographed in Figure A number of the branches penetrate faces. This property of the sin ter body not a topological invariant and thus will not be considered further. The Linkage Numbers An invariant, which COnnon) to both the sinter body manifold the void space manifold, is termed the linkage number. It follows from the Alexander Duality Theorem (14) "The k dimensional Betti number of an arbitrary polyhedron lying in an Rn is equal to the (nk)dimensional Betti number of its complement ary region R" In Figures , (6), linkage numbers of 0 illustrated respectively. Statement VI I. The linkage number , denoted of the void space manifold summed over all its parts, is equal to the linkage number of the sinter body manifold. Statement VI II. Betti number the sinter body equal to the linkage number L of either manifold. Statement Betti number of the void space, equal to the linkage number only when the void space manifold connected. Algebraic manipulation of these statements yields are Figure Knotted Branchnode Sinter Body Representation I'i";;l Figure Example of Linkage Number Zero Figure Example Linkage Number Figure Example Linkage Number Two P0 jo a  n + 1) This is the basic topological relationship between threedimensional manifolds. The numbers and (b. nj + 1) imply these two essential  n + 1) . invariants and PV of the What system. is important One may is that choose to find for any one invariant analytically , two others must be determined experi mentally. The Higher Dimensional Betti Numbers So far all that has been dealt with are the 0th and 1st order Betti numbers. It is natural to ask if any new information be obtained considering the higher dimensional invariants. example, one considers the surface of the sinter body (i.e., twodimensional manifold that croumot to both the void space the sinter S of P body manifolds) The Alexander it is natural Duality Theorem tells to inquire abou t us directly: k nk' = 1, The twodimensional manifold is embedded in threedimensional space so that n = 3 and k or 2. Hence Thus the branchnode representation gives same information terms of Betti numbers that would have been obtained considering can the value  1. S = P 2 Next consider the 3rd order Betti number. In this case n li , ki. or 3; = P = P 3 Thi s use o Betti type of procedure f the higher numbers generalizes dimensions and their offers and thus no new connection with it is concluded information as f sinter bodies that ar as the and their associated void spaces are concerned. ; ""'E':Eiii;iii"E"E'"" x CHAPTER APPLICATIONS TO SINTER METALLURGY Some Body Relations and Void Between the Betti. Numbers of the Sinter Space A method of characterizing a sinter body is the determination of its incidence matrix. This problem has several aspects: In the early stages of sintering, if individual sinter body layers were removed, say of particle size thickness representative could incidence matrix A typical for the sinter section a sinter body body manifold is presented in Figure The incidence matrix would not be exact as every branch and node possibly might not be found. However if homo geneity and isotropy are assumed, of the sinter body could be evaluated on a unit mass or volume bas is. This is done in practice metallurgists. In Figure 28 the 2 equal areas of 2.09 square inches each enclose approximately 10 particles. The entire photograph con tains particles per square inch and the sub areas have same number of particles per square inch. Therefore, the 2 areas, although small are representative of the entire sinter body. criterion of whether a part of the sinter body is representative 59 Figure Illustration Representative Sinter Body Areas par tial areas 1 and in Figure 28 have approximately 10 particles and whether or not this is sufficient depends upon the purpose which the sinter body is to be used or upon other considerations. A partial incidence matrix for the network constructed Figure 29 is presented approximately a half be low. layer taking lower the next section , the threedimensional 1 network can be constructed. 000 0 0 1 0 000 0000000000 0000000000 0 0 000 0000 0 0000000 00000000000 0000000 0000 00000 00000 0000 00000000 00000 000000000 00000000 0000000 00000000 000000000 0000000000 00000000000 00000000000 0000000000 000 000 000 000000 100000 111010 0000 0000000000 IInl~l I. S n nn nnnnn 000 0 0 (33) I II I I "12 nil '3io "1, ~~''"'" b12 000 000 001 000 000 In late cult stages to distinguish of sintering individual (Figure particles it becomes and interparticle diffi con tacts due to the coalescence of the material. Thus , in studying the topology sintering during late stages may be simpler to consider the topology of the void space. This is known become disconnected as sintering progr esses which means there will a large number parts. A practice al method of estab lishing a branchnode incidence matrix of the void space is out lined in the following: Consider the six sketches shown in Figure 31 and assume the sections to be taken successively so that isolated void s lie between them. validity of this assumption depends upon particle size and section thickness. First number all the voids taking individual voids to be nodes. Subscripts on nodes refer to particular voids and superscripts to the sections on which they appear i.e., is void one on section two. Then on the bas is of the geometry a void and its location on a section decide whether is connected k+1 n. 3 Figure 31 shows the following: is connected is connected is connected 3 to n1 3 and n2 is connected is connected 3 to n3 3 to n4 are connected 4 to n1 and ni respectively is connected is connected is connected is connected n an to n1 to n2 to n5 3 and n4 and n5 are connected are connected is connected ton6 2 to n5 4 to n6 and n6 is connected to n 4 Branches scripts. will Superscripts also be numbered i and indicate with subscripts sections super and subscripts and 1 indicate the nodes at the end of the branch. Reference to the branchnode representation of Figure 32 will clarify this terminology. incidence matrix (35) for Figure 32 follows subsequently. It should be emphasized once again that both the loopbranch 64 Figure Photomicrograph of Linde Copper Powder, 170 +200 Mesh, Sintered in Hydrogen for 1024 Hours at 1005 C, 150X 65 Cl ~ ~ ~ ~ ~ ~ ~ a ra la4.4a nr LT, U1,1 0,,+1 45 bl 11 3 35 34 4 4; 5 56 6j Sn b 5b n 44 4 44 4 4 Figure Branchnode Serial Network Hypothetical Sections 12 bl 11 la teal anc 'On3 in e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f lNe a 0 0 0 0 0 0 0 0 0 .0= as~ (nd *O a~ xi xxxi x CNN A xx s as 0 0 0 0 0 0 0 'i: ';'''''''' 'ii''' '''''i' '' 0   0 0 O O 0 0 0 0 I _ m ~iiiilil~ 'ii'''''//iiii:: Ijj!/iii/ii!!///jj! /////////;; iii//////ii!/i,, iiiii// '"''~''iiiii'" !,,,,.iiii'''''i "::"jj/ii///iii, ,,//ji """"" """""""" ijjj iiii:: E: ,i;; iii,,,,, "" """"j~~~~i~E::EEiiiEBji Eijji"" "iij" ii" """:,%bl ,,, iillll,,,, ii Other Examples The incidence matrices for Figures 22 and 24 respectively are constructed subsequently. The numbering individual branches the related and nodes has been omitted configuration is desired. because only It is the character necessary to assign specific numbers because they are arbitrary and the columns rows are subject to arbitrary permutations. 001 000 000000 000 00000 0000000 000 000 000 0000 000 000 000 0000 000 001 0000 00000 0000000 0000000000 000 00000 000 000 00000 001 000 000 000 000 000 00000 0001 000 000 000 000 0000 000000 000000 00001 000 0000 00000 0000 000000 000 00000 0000000000 001 00000 000 000000000 00000 001 :N~i~ihiEE~ii 69 000 000 00000000000 000 001 00000 000000 000 0000 000 000 110 000 00000 0000 0000 0000 0000 000 000 0 000000000 0000 000000 000 001 000 000 00000 00000 000000 00000000 000 000 0 0 000 000 0000 0000000 000000 0000 000 000000000 00000 000 001 001 000000 0000 000 000 000 000 000 00000000000000 000 0000 In Chapter a sample calculation is presented an actual sinter body to be PV=0.40 The number X 105/cubic of separated inch. parts of void Evaluation of space is computed n+1) yields cubic P=105/ inch. Hence a method of determining the Betti numbers a sinter body has been presented and a numerical example com puted. (37) T(b    n1 V a1 This true because of the method of constructing the conjugate void space and the fact that they have a common twodimens ional manifold. As sintering progresses and the void space decomposes into two parts S S n2  n + 1) I a"2 + 1) where now the subscripts on the right hand side refer to the separated parts. The above equation follows from the fact that each component of the void space has its own Betti number; at the same time the ist numbers body has its own space Betti equal s number; the Ist sum of Be tti number of the sinter body because of the equality of the ranks appropriate matrices. An alternate interpretation of (39) obtained from Figure in the sinter body 20 if there one notes exists that a closed loop every closed in the void loop space Thus * n + 1) over the void space parts must equal Betti number of the sinter body. With further wintering  n en3  n + 1) "2 + 1)  n3 + 1) (40) S Ut (i1)  n + 1) the sinter Betti of the void the ist Y b 1 + (b2 + (bg f(b2 ~ (bi =~(bj U~ superscript v has been omitted in order to avoid confusion but the right hand side refers to the void space components. left hand side constitutes a monotonically decreasing sequence of integers. same is true of each  nji1) + 1) on the right hand side however the number is monotonically increasing thus the rate of decrease the difference II.) must greater. It is of interest to note the change in structure of the sinter body and void space matrices. As sintering progress the elimination of sinter body nodes and branches trans forms the sinter body incidence matrix as shown be low. where the solid lines within the matrix represent the elimination of rows and columns. In the case of the void space matrix two possibilities must be considered: The void space decomposes into several parts. The void space remains connected. A partitioned matrix can always be obtained, one for each component the void space. Experiment 1 Proposals During course of this research several experimental pro posals were forward and are enumerated below: Determination of the 0th Betti number of the void space. procedure is to consist of innmmersing a copper sinter body molten void lead so that space parts the void that space manifold would are located become completely within flooded. the sinter body not accessible would course not be flooded but this be avoided choosing a sufficiently emil1, but still repre sentative, sinter body volume. At liquid helium temperatures lead would become superconducting while the copper would not for practical purposes would an insulator. applying a poten tial difference to probes and then checking current the number of separated parts of the void space is determined. Determination of the Oth Betti number the fountain effect. If one face helium then exhibit the of the sinter the void fountain body space which effect. T is brought int h is connected he number o contact with to that of fountain liquid face will streams dicates the number of connected voids. All void space aperatures except one) be blocked so connectedness to the aperatures the other faces will be exhibited. The rate of growth of the necks or branches as they have been can (15). The maximumflow minimumcut theorem is fundamental in this connection. is proposed to investigate this in a subsequent communicate ion. Experimental methods suitable for these theories which might be considered are the following: Heat conduction enables the measurement of maximum heat flow from particle body with to particle the exception of insulating two separated the entire particles sinter to which probes are attached with an arbitrary temperature difference. Repeated application of this method enables one to compute the maximum flow minimumcut property. Electrical known conduction to provide maximum flow. from one or more Here particles an additional is also natural be brought to bear in the sinter body problem because in any network Thus, i the distribution of t is possible current by measuring causes heat minimum heat generated within generation. a given sinter body (suitably insulated) to obtain information about current distribution within the sinter body. I: :: ::: 39; CHAPTER CALCULATION OF THE BETTI NUMBERS FOR AN ACTUAL SINTER BODY The sinter body used for the following calculation was made with Linde copper powder 170 +200 mesh). was sintered 1024 hours at 1005 in hydrogen and its final density was calculated to be 8 12 gms./cc. In Figures 33 through 46 microstructures the above mentioned are presented. These are serial sections and on the average are approx imately 0.0005 inches apart A close examination of the sections illustrates that specific voids can be followed from one section next considering both the void geome t ry and the surrounding structure. It is to be noted that situations do occasionally arise where one cannot sure what happens a void between particular sections. In such cases where the frequency occurrence is high, thinner sections are required. The boundary illustrated on the figures is arbitrary and in this case has been taken to be inch square. Since the twins occurring within the grain are parallel to each other on the various sections , it is reasonable assume that section not rotated relative to section (i The total vo lume material cons idered in the example tp '/) t Figure Section Serial Sections, 75 >7 Figure Section Serial Sections, 4 ~1: iS* S *i *.rw at~ Figure Section of 14 Serial Sections, K.P Figure Section Serial Sections, 90X I I Iz ': iiEilliiiiii":::: F~ 7;5C" xxxx x7 9 Figure 2' Figure Section Section of 14 10 of Serial Serial Sections, Sections, 90X 90x Figure Section 11 of Serial Sections, 90X cc1F f 1" ~P"~ r, ,r "" "" :::iii~ : "4 ,  Figure Section of 14 Serial Sections, 90o IEE'"""""":C:ii""iE""""" 5, whether other randomly chosen volumes of the sinter body would possess same values and P deferred. The calculation of the Betti numbers proceeds as follows: Consider a set of serial sections and rule an arbitrary area on each. It i advisable to pick an area where sufficient struc ture present so that following the individual voids facilitated. Arbitrarily number each and every void that occurs entirely within the boundary a particular section and continue process for all sections. a network as shown Figure 47 and connect branches between node node 1+1 (where subscripts and superscripts are as defined previously) a continuous path exists be tween them. If a node occurs on section i that connected to the external void space between sections and ( then connect a branch and let this branch terminate between sections + 1) without a node at its other end. Thi s same as con i nesting nj to the common node of the external void space. When sections and i are in contact with the external void space the total number of branches consists of those between sections of sections and i aud i addition to to the void the branches space connecting common node. the nodes In addition it must be remembered that the total number nodes must include  t " ::~ BE" I r _ * It is incorrect over to count all the separated the total parts number and take of branches total  n and nodes total In view of the above the network* is presented in Figure 47 while in Table the numbers of individual branches and nodes listed with the appropriate Table V lists and P1 1 on a uni t volume basis, while Table VI lists the indi vidual section thicknesses with the cumulative volumes over given sections. In Figures 48 and 49 the Betti numbers per unit volume are plotted vs. the number of sections. It is to be noted that curves tend zero slope as more sections are considered. Thus ques tion of how many sections are required can be answered graphing appropriate invariant vs. the number sections determining where the slope tends zero. *Since calculation made the complete from this network network presented and the resultant the appropriate matrices will be constructed. are 1""""EEEEEBEEiii "" QXI n 'r Figure 47. Branchnode Network 85 tA 4 + a O~ 4; 01j C 4S u1 Co inL C 0% 0 CO SI N J N\ N~ NV NN NV C N 01) .0 '0 OW "U ('SP 002r $4 II2 I00~ k' 0 rI r4 .4 (N NJ (' CM ( V Idl 34r 34 o A C O "4 e n 0 i C 4) O~~~~ Cl Cl 01 ~lFl I 04 Fs rut it 0 4 0 .4 C 0 .4rtp  4 a ~ "~14 su OO c ~ a, 0 *1 'A Ur 0 Cs, a a ~ . a a 86 tf S 0j I.'J~193o r JII Ia U, V N NJ J l 4~ 0J 0( r 0.r LA Sl 0 14 U '0 C I C O 1% '00 0' 00% N Li, (N"5 NC ' C 4V t o1 01 0 0000 Iii ~ ~ ~ 9~O H J . O 0I t ~ N NV N~ (' 4 N NJ eN NJ en Vt enS .0J 4 0: a to 0' 4 CC 4O sO N 4Y '0 N 4 sO C Vt 5%.m c N in CC 0 ('4 10 00; 0 10 1 00l 4 .4 4 emI NJ N NJ NZ Vt" (' U1 5 001 U ' '4 1511 o ': EXE, ia , TABLE VI SECTION THICKNESSES AND CUMULATIVE VOLUMES Number of Sections Volume Thickness (Inches) (Inched3 x 105) 0.0007 0.0015 0.0021 0.0024 0.0029 0.0035 0.0038 0.0042 11.4 12.8 15.4 18.6 20.2 22.4 0.0048 0.0053 28.2 0.0057 0.0067 0.0071 30.4 35.6 37.8 88 89 e CHAPTER SUMMARY OF THEORETICAL RESULTS FOR APPLICATION TO SINTER METALLURGY space t il has been s space introduced conjugate t as a topological o the sinter manifold. body which bears a definite quantitative relation to the sinter body has been effected. Network and the associated algebraic r topology have been introduced and a physical interpretation, terms of the sinter body and its associated void space or spaces , of the following matrices has been given: Incidence of the branchnode configuration. Loopbranch. Cutbranch. Treebranch. It has been shown that the introduction of the higher dimensional Betti numbers gives no additional information. A topological relationship between two threedimens ional manifolds was found to be At the beginning of PS 1 s inter ing and P" 0 While in general a' S a  The void The construction of p0 C ** r 'L 