Vol.13 No. 5Trans. Nonferrous Met. Soc. ChinaOct.2003Article ID : 1003 - 6326( 2003 )05 - 1112 - 07Fractal phenomena in powder injection molding processZHENG Zhou-shun(鄭洲順)2 ,QU Xuan-hui(曲選輝)3 ,LI Yun-ping(李云平)',LEI Chang-ming(雷長明)' , DUAN Bo-hud(段柏華)( 1. State Key Laboratory for Powder Metallurgy , Central South University , Changsha 410083 , China ;2. School of Mathematical Science and Computing Technology ,Central South University , Changsha 410083 , China ;3. School of Materials Science and Engineering , University of Science and Technology Bejing ,Beijing 100083 , China )Abstract :The complicated characteristics of the powder were studied by fractal theory. It is ilustrated that powdershape , binder structure , feedstock and mol-filling flow in powder injection molding process possess obvious fractal charac-teristics. Based on the result of SEM , the fractal dimensions of the projected boundary of carbonylie iron and carbonylicnickel particles were determined to be 1. 074 +0. 006 and 1. 230土0.005 respectively by box counting measurement. Theresults show that the fractal dimension of the projected boundary of carbonylic iron particles is close to smooth curve of one-dimension , while the fractal dimension of the projected boundary of carbonylic nickel particle is close to that of trisectionKoch curve , indicating that the shape characteristics of carbonylic nickel particles can be described and analyzed by the .characteristics of trisection Koch curve. It is also proposed that the fractal theory can be applied in the research of powderinjection molding in four aspects.Key words : powder injection molding ; fractal theory ; chaos ; Navier Stokes equationCLC number :TF 12Document code :A1 INTRODUCTIONRACTERISTICS OF POWDERGenerally , The flling process of powder injectionChaos is a major class of order phenomena withoutmolding( PIM ) is analyzed by rheological theory.Rheological model is made up of basic conservation e-periodic. The important characteristics of chaos motionquations( mass , momentum , energy conservation equa-are sensitively dependent on the variation of initial val-tions ) that must be satisfied by melt feedstock in PIM ,ues , that is , in nonlinear dynamics system , tiny varia-constitutive equation that is used to describe the pro-tion of initial values will lead to totally different typcessing parameters of materials , and corresponding re-motion. Fractal theory is one of the geometry languagesstrictive conditions. It is a group of partial differentialused to describe chaos kinetics. Fractal theory andequations and is also a nonlinear dynamics system ofchaos theory are closely correlative. Fractal theory remany influential factors. The variation of material ther-mophyical and rheological properties caused by theveals the unity of order and disorder and that of cer-fluctuation of process parameters may result in multi-tainty and randomness in nonlinear system. Fractalvaluedness of stress- -strain curve , that is , the fluctua-theory was first proposed in 1970s. But through moretion of process parameters result in the bifurcation. Inthan 20 years development , it has been an importantPIM , mutations( defects arise ) of flow state caused bynew branch of studying and extensively applied in al-the small variation of process parameters may correlatemost every field of the natural science and social sci-to the chaos of nonlinear dynamics system'frartal thenry has been one of the2 FRACTAL THEORY AND FRACTAL CHA-front中國煤化工any sciences. Mandel- .TYHCNMHG①Foundation item : Proje( G2000067203 ) supported by the National Key Foundamental Research and Development Program of China ; project( 50025412 ) supported by the National Natural Science Foundation of China and project( 99053310 ) supported by the Education Ministry of ChinaReceived date : 2002 - 12 -02 ; Accepted date :2003 -03 - 10CorrespondeMezHENG Zhou-shun , PhD ;Tel : + 86-731 -8879361 ; E-mail : zszheng@ mail. csu. edu. cnVol. 13 No. 5Fractal phenomenon in powder injection molding process1113-brot 21 proposed fractal geometry to describe and inves-tigate these extremely irregular and fragmental geometryobjects in nature. The research objects of fractal theoryhave two common characteristics. One is not havingcharacteristic length ; the other is having self-similaritybetween the part and the whole. Take natural shorelineand Koch curve for example , no matter how to amplifyit ,the part is as intricate as before , and the intricateshapes contain organizational structure. The mathemat-ics essence of fractal theory lies in that it alters mea-sure view of observing and researching objects , such asthe replacement of Euclidean measure with Hausdorffmeasure. Because of this change of measure view , wecan reveal the self-similarity or self-affine rules that ex-dist in complex phenomena and be more powerful tolearn and deal with complex natural phenomena.There are many refined and complex fractal phe-nomena in PIM' 3. Through observing PIM processingand analyzing rules of defect occurrence during greenpart forming , QU et al 1 pointed out that phenomenasuch as defect occurrence of PIM may be correlative tochaos phenomena in nonlinear dynamics system , andproposed that fractal theory and chaos can be applied tothe research on PIM. Micrographs of projected bounda-ry of powder particle , surface , pores of apparent andtap powders can be obtained by scanning electron mi-(0crograph( SEM ) or optical microscope( OM ). Fig. 1shows the agglomeration , size ,and shape of some typi-cal powders used in PIM at low and high magnifica-tions , showing that agglomeration , size , shape , sur-face and spatial structure of powders are complex. Theparticle surface is tortuous and fragmental with thepresence of tiny isolated islands. Actually there is a lotof information of powder particle in these phenomena.In the past , in both numerical computation and experi-mental research , these phenomena are described or as-000sumed as continuous , smooth curves( plane ) , which infact neglects much important detail information. Thedetail information may be exactly the important factorsFig. 1 SEM photographs of powders used in PIM( a )-Carbonylic iron at low magnification ;of influencing the process of PIM. The research which( b )-Carbonylic iron at high magnification ;fractal theory and chaos are applied in PIM is helpful to( c )-Carbonylic nickel at low magnification ;describe and analyze the important and complicated de-( d )-Carbonyliec nickel at high magnifcation ;tail information.( e )-Reduced cobalt at low magnification ;( f)-Reduced cobalt at high magnifcation .3 FRACTAL PHENOMENA IN PIM PROCESSdensity , with a typical level near 0.6. The optimal sol-There are many tiny and complicated fractal phe-ids content for molding depends on the powder charac-nomena in PIM，such as shape，surface，cross sectionteristics , particle size distribution , particle shape , in-of powder particle , and spatial structure formed by theterparticle friction and agglomeration' 4. Particle sizepores of apparent and tap powder etc.can中國煤化工ng the dimensions of apartitthe measurement tech-3.1 Fractal phenomena of powder particleMYHCNMHG.... .... measured , and particleA particle is defined as the smallest unit of a pow-shape. It can be achieved by several techniques whichder that can not be subdivided by simple mechanicalusually do not give equivalent determinations , due tomeans. In PIM , the particles generally are below 20differences in the measured parameters. One geometricμum in size. The, powders used in PIM have fractionalparameter is used and the assumption of a sphericalpacking denity攪etween 0.3 and 0.8 of theoretical114.Trans. Nonferrous Met. Soc. ChinaOct.2003particle shape is made du-ring measuring process. Theniform feedstock for molding. It has an influence onanalysis parameters are surface area , projected area ,particle packing ，agglomeration ，mixing , rheologymaximum dimension , minimum cross sectional area ormolding , debinding , dimensional accuracy , defects ,volume. This analysis is based on Euclidean measure.and final chemical property of PIM compacts. ThereAs the shape becomes less regular , the numbersare many types of binders , and most of which are poly-of posible size parameters increase and the determina-mers. Polymers are long-chain molecules with carbontion of particle size becomes more diffcult. Thus ,it isbackbones and various side groups and branches.common to find disagreement between the particle sizeThermoplastic and thermosetting compounds are the twodistributions obtained by different techniques and in-general forms of polymers. The longer-chain polymersstruments , and the different techniques or instrumentshave the amorphous characteristics ，that is , fractalare applicable to the different particle sizes. Almost allcharacteristics. Beside thermoplastic component , bind-the techniques and instruments based on Euclideaner contains additives used to lubricate and control vmeasure are not applicable to the very small particlescosity , humidification and debinding. lacocca et alwhose Euclidean yardstick is below 0. 1 μm. Thisused deterministic chaos theory to evaluate the functiondemonstrates that the Euclidean view can not be usedof lubricant in flling process. This demonstrates thatto reasonably describe the characteristics of particlesbinder has fractal characteristics. The properties ofwhich have the" infinitely refined” fractal structurefeedstock vary dramatically with the increase of powderand yardstick size less than 0.1 μm. This is just likecontent in feedstock ，such as the increase of elasticthat Koch curve or shoreline can not be described rea-modulus , strength and viscosity , the decrease of ther-sonably by Euclidean measure , because the self- simi-mal expansion coefficient and ductility. It is diffcult tolarity dimension of Koch curve is not an integer ,but aprecisely describe the influence of particle characteris-fraction between 1 and 2. For example , the self-simi-ics , temperature and binder characteristics applied onlarity dimension of trisection Koch curve D=In 4/ln 3powder-binder system. The results obtained at present≈1.262. It can be inferred that the fractal dimensionsare mostly empirical formulas. Because the particleof particle surface area , projected area , and minimumshape and binder have fractal characteristics , and PIMcross sectional area are not integer. The boundaryfeedstock has corresponding fractal characteristicscurve of projected area of particle is similar to that oftoo. These fractal characteristics will be investigatedKoch curve. The surface powder particle is rather intri-and analyzed with fractal viewpoint and concepts. .cate ,and its fractal dimension may be between 2 and3. Germant41 pointed out that very small particles ,3.3 Fractal phenomena in PIM illing processwhose Euclidean yardstick is below 0.1 μum , could be .The process of PIM flling is a viscoelastic , un-characterized by analysis of their Brownain motion in asteady , non-isothermal complex physical process. Addi-fluid. The nonuniform molecular collisions on the parti-tion to the complicated flow geometry shape of the diecle surface cause the particles to move randomly in in-cavities ,it is very hard to describe the process of feed-verse proportion to the particle mass. The measurementstock flow precisely. At present , common mathematicalof the Brownian motion provides particle size informa-models used to describe the process of PIM flling aretion through the particle velocity detected by a Dopplermainly based on the following assumptions' .shift in scattered laser-light. In fact ，the track of1 ) Powder and binder are mixed well without anyBrown motion has typical fractal property , only that itsgas hole , and the mixture never separates du-ring thefractal dimension is 2 ,still an integer. These phe-flow. The feedstock melt fluid flow is considered as e-nomena and facts demonstrate that PIM powder parti-ven continuous medium non-Newtonian fluid flow , andcles have the outstanding fractal characteristics. To de-the effects of heat expansion and the latent heat can bescribe particle size and shape that embody powder par-neglected.ticle property sufficiently ,it is necessary to apply the2 ) Heat conduction plays an important role inviews and concepts of fractal geometry.cavity wall , and the convective transmit heat in cavitythickness direction is neglected. While the convective3.2 Fractal phenomena of binder and PIM feed-transmit heat plays the greatest role in cavity , and heatstock mixtureconduction in the stream direction in cavity is neglec-During PIM process , the binder is a temporary ve-hicle for homogeneously packing the powder into the中國煤化工is considered only ,anddesired shape and then holding the particles in thatinertEivity are neglected. Theshape until the beginning of sintering. Although thepresMYHC N M H Gant in the thickness di-binder does not dictate the final composition , it has arection.major influence on the success of processing. BinderAccording to these assumptions ,a group of Navi-compositions and debinding techniques are the mainer-Stokes equations( N-S equations ) based on continu-differences between various PIM processes' 43”.ous medium model and used to describe PIM processThe bhma&據(jù)mixed with the powder to form a u-are obtained. Models based on other assumptions ,Vol. 13 No. 5Fractal phenomenon in powder injection molding process1115.such as granular model ，two-phase flow model , arein fractal theory. For the reason that it is difficult tomore complex N-S equations. Except some classicaldefine the fractal dimension applied to all aspects ,differential equations , it is difficult to certify whetherthere are many methods for defining fractal dimensionthe solution of equations leads to chaos phenomenonin documents and books , such as relevant dimensionfrom differential equations. Many chaos phenomena ofinformative dimension , self-similar dimension ，Haus-differential equation solutions are discovered and dem-dorff dimension , counting-box dimension and Kolomo-onstrated by numerical solutions. For the flowing ofgrov capacitive dimension etc. Different definitionPIM flling process , many numerical solutions based ormethods applied to different studying objects , and thecontinuous medium indicate that the flowing is dramat-fractal concept and its dimension are kept on develo-ically influenced by processing parameters , and itping.pressure distribution , temperature distribution etc havefractal characteristics. N-S equations are correlative to4.1Hausdorff dimensionfractal and chaotic phenomena , and can be explainedAssume D > 0 , and cover the set S with ballsby the classical Lorenz equation. Lorenz studied thewhose diameter is ε and quantity is N( ε ) and so theatmospheric turbulence and obtained a group of ordina-D-dimension measure Mp of set S is defined asry differential equations called classical Lorenz equa-M。=limn( D)M( ε )e”,tion by simplifying N-S equations vastly. Chaotic bhavior and Lorenz attractor have been discoveredwhere r D ) is geometrical factor. For straight line ,square and cube ,1( D )=1. For disk and sphere , γthrough solving the numerical solution of the equation.( D )=π/4 and π/6 , respectively ; D is called Haus-The attractor is a fractal set , and its fractal dimensionis 2. 0612]. Scheffer studied the status of solutions ofdorff dimension or Hausdorff-Besicovitch dimension ofN-S equations in three-dimensional space , and foundset S. For self-similarity set , Hausdorff- Besicovitch di-that the oddness of equation solution( if exist ) must bemension equals self-similarity dimension. Here theball”and" diameter" are abstract concepts , whichin the set whose dimension is less than 3l 13].It is il-lustrated by experiment study and production practicemay represent straight line , square , cube , disk andfor years that the same defects of PIM products havesphere etcself-similarity and the occurrence of the defects of PIM4.2 Counting-box dimensionproducts is correlative to the small variation of process-Assume SCR". In the Euelidean distance , covering parameters.These facts sufficiently demonstratethat PIM flling have the outstanding fractal characteris-the set S with lttle boxes whose side length equalstics and chaotic phenomena.Although N-S equations are correlative to fractal ，and let N,( S ) be the minimum number of box coveringthere is no method that can be used to deduce the frac-S ,thenlnN,( S)tal dimension from these basic equations directly. Es-D= limpecially in PIM flowing process , the occurrence of theη→∞ln2”chaotic phenomena has relation to the processing pa-where D is the counting-box dimension of set S.rameters , the characteristics of feedstock etc. The frac-tal characteristics and chaotic phenomena of PIM flling4.3 Experimental measure methods of fractalprocess can be studied by numerical computation anddimension in PIMcomputer simulation. .There are many practical measure methods of frac-tal dimension , such as perpendicular section , island4FRACTAL DIMENSION USED TO RE-surface area direct measurement ,power spectrum ,sec-SEARCH FRACTAL CHARACTERISTICSondary electron scatter etc ，Different methods areOF PIM AND EXPERIMENTAL MEASUREused depending on different research objects. One sim-METHODSple and practical method is the rough visual methodwhich gets the value of the dimension by changing theFractal dimension is one of the important parame-degree of the rough visibility. Its important thought isters of fractal structure. Theoretical researchers devotebased on that the details of fractal set is reflected byto analyzing all kinds of fractal structure and theirchoosing the yardstick , and the smaller the yardstickforming processes , computing fractal dimensions whichis，中國煤化relected. Box-couningexpress their characteristics ; while the experimental re-metheYHmethods.searchers measure the fractal structure and fractal di-CNMHGbyquadratemeshwithmension of its process by experimental measure meth-ε in mesh width , as illustrated in Fig. 2. This step isods , and find out the relationship between fractal dicalled space quantization. Then count out the quadratemension and material property or even further discussmesh number N( ε ) contained in the research scope.the physical reasons for the formation of fractal struc-In the images processing , this is to calculate the num-ture. Fractar thitEhsion is also a fundamental quantityber of all meshes containing image element of fractal1116.Trans. Nonferrous Met. Soc. ChinaOct.2003graphics , and the fractal dimension D can be calculat-edby N( ε)<ε" ".Fig. 2 Similar islands measured byrelative yardstick measureFRACTAL DIMENSIONS OF BOUNDARYCURVE OF CARBONYLIC IRON ANDCARBONYLIC NICKEL POWDER PARTI-CLESSEM is one of the best tools available for observ-ing the discrete characteristics of a particle. Fig. 1shows the scanning electron micrographs of carbonylieFig. 3 Projections of two powder particlesiron and carbonylic nickle powders used in PIM pro-(a)- -Carbonylie iron ;( b )-Carbonylic nickelcessing at low and high magnifications , showing the ag-glomeration , size , shape and the complexity of surface: ,count out the quadrate mesh number N( ε ) con-and spatial structure of powder particle. From the SEMtained in the projection of carbo-nylic iron particle.images of carbonylic iron and carbonylie nickel pow-The relationofε- -M( ε)is listed in Table 1. There-ders ,the approximate qualitative description of particlefore the double logarithmic relationofg- -N( ε ) can becharacteristics of the two powders can be obtained. Al-obtained shown in Table 2 ).so,the aspect ratio and the sphericity index can beMake curve ftting for data about the double loga-used to simply quantitativly describe them. But obvi-rithmie relation of ε-N( ε )of a great deal of carbon-ously it is difficult to describe the difference of particleylie iron particles with straight lines ,the fractal dimen-shape , complication of aglomeration , surface and spa-sions of the boundary of carbonylic iron particles aretial strueture complexity of the two powders in detail ,calculatedtcbe1. 074resulting in difculty in exactly contrlling the process-0.006, which is close to 1.1. The results showing of PIM. .Particle shape relates to the characteristics of theTable 1 ε- _N( ε ) relation of carbonylic iron powdermaterial and powder fabrication approach etc. Carbon-_g34/80 34/200 34/1 000 34/10000ylie iron and carbonylic nickel powders are separatedK(e)116911 075by ultrasonice wave , and through their SEM images ,itis found that the projected boundary of the same kindTable2 ln8- -lnN( ε ) relation ofof powder particles has the similarity ,and it has verycarbonylic iron powdercomplicated fractal structure. Fig. 3 shows the typicallnε (-0.856 -1.772-3.381- 5.684projections of carbo-nylie iron powders and carbonyliclnN( ε )01. 7922. 7734.5116. 980nickel powders by SEM. The fractal dimensions of theprojected boundary of carbonylic iron and carbonyliethat the boundary of carbonylic iron particle is asnickel powders are calculated by the counting-boxnear as smooth curve of one-dimension , but it still hasmethod.中國煤化工5.1 Fractal dimension of projected boundary of5.2TYHCN M H Grojected boundary ofcarbonylic iron particleearoonyie mcker parucleThe projection of the carbonylie iron particle isThe projection of the carbonylie nickel particle isparted by quadrate mesh with ε in mesh width ，asparted by quadrate mesh with ε in mesh width , asshown in Fig. 4. The length of the square in Fig.4( a)shown in Fig. 5. Use the same way above , the absoluteis considered as an unit length( the absolute length is εlength 12 μum or so is defined as the first relative yard-= 100/367取地ith the decrease of relative yardstickVol. 13 No. 5Fractal phenomenon in powder injection molding process1117.Table3 ε- -M( ε ) relation of carbonylic nickel powder811/310/811/81Me)26469Table4 lnε- -lnM( ε ) relation ofcarbonylic nickel powder-1.099-2.092-4. 395InN( ε) 02.0793. 2586. 150! (6)Make curve ftting for data about the double loga-rithmic relation of ε- -N( ε )of a great deal of carbon-ylic nickel particles with straight lines , the fractal di-mensions of the boundary of carbonylic nickel particles .are calculated to be D= 1.230 +0.005 , which is closeto that of trisection Koch curve( 1. 262 ). The results .show that the complexity degree of the boundary of car-bonylic nickel particle is as near as that of trisectionKoch curve. And from the SEM photographs of carbon-ylic nickel particles , it can be seen directly that theprojections of carbonylic nickel particles are very closeto the trisection Koch island in shape. Therefore theshape characteristics of carbonylic nickel particles canbe described and analyzed by the characteristics of tri-section Koch curve.6 APPLICATION PROSPECT ON FRACTALFig.4 Projections of carbonylic ironTHEORY AND CHAOS IN PIMparticle by quadrate mesh( a)- -With ε= 100/36 μm in mesh width ;( b)- -With ε=85/72 μm in mesh width ;The application of fractal theory and chaos in PIM( c)- -With ε= 17/36 μum in mesh widthcan roughly be divided into four aspects :the first is tomeasure the fractal dimension of binder and particlestick e.shape by experimentation , to analyze the physical rea-son for the fractal structure forming of particle , and todescribe the characteristics of particle shape more pre-cisely ithe second is to analyze the fractal characteris-tics of feedstock and its filling processing turbulence ;the third is to approach the fractal chaos phenomenonof Navier-Stockes equations and the mechanism for theoccurrence of defects of the product of PIM ; and thefourth is to study the effects of the fractal structure ofparticle shape and binder on debinding and sintering.In conclusion , the fractal structure and chaos phenom-ena may arise in the whole process of PIM. The studyof PIM using fractal and chaos theory provides the newFig.5 Projection of carbonylic nickel particleview and method to describe the material characteristicsby quadrate mesh with ε in mesh widthand process parameters , to more precisely control theWith the decrease of relative yardstick ε , countprocess of PIM ,and to achieve the goal for high qualityout the quadrate mesh number N( ε ) contained in theand中國煤化工projection of carbonylic nickel particle. The relation ofε-N( ε )is listed in Table 3. And the double logarith-PYHCNMHGCESmic relation of ε一-N( ε ) can be obtained( listed in Ta-[1] QU Xuan-hui , WEN Hong-yu ,AO Hui ,et al. Computerble4 ).simulation of powder injection molding proces[ J] J Ma-ter Eng ,2001( 6):33 - 36.( in Chinese )[2] Mandelbrot B B. The Fractal Geometry of Nature[ M ]New York : W H Freeman and Co , 1982.1118.Trans. Nonferrous Met. Soc. 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