A mathematical model describing the sintering rate of a viscous material powder
bed is presented. This model assumes that the powder bed is composed of cubic
packed, equal-size spherical particles. The sintering rate equation is derived in
terms of a unit cell dimension or the relative density of a powder bed. A
mathematical factor, fraction of sintering, is introduced to explain the
phenomena of partial sintering. Key words: model, viscous sintering.Mechanical Engineerin

Barlow, Joel J.

Beaman, Joseph J.

Nelson, Christian

Sun, Ming-shen

English

Texas ScholarWorks

A Model for Partial Viscous SinteringMing-shen M. Sun, J. Christian Nelson, Joseph 1. Beaman, Joel BarlowThe University of Texas at AustinAbstractA mathematical model describing the sintering rate of a viscous material powderbed is presented. This model assumes that the powder bed is composed of cubicpacked, equal-size spherical particles. The sintering rate equation is derived interms of a unit cell dimension or the relative density of a powder bed. Amathematical factor, fraction of sintering, is introduced to explain thephenomena of partial sintering. Key words: model, viscous sintering.Ie IntroductionThe theory of viscous sintering was first explored by Frenkel [1]. primaryconcept of his theory is: energy balance between the work done by surface tension (thesurface energy loss) and viscous dissipation during the particle deformation. Frenkel'smodel was derived to explain the sintering behavior of two spherical particles and it wasproven to be valid only in the initial stage of sintering [2]. Extending Frenkel's concept,Scherer [3-5] has developed a cylinder structure model that can successfully describesintering behavior of a viscous glass preform. However, for some applications that applythe sintering of a randomly packed viscous powder bed such as the Selective LaserSintering [6], Scherer's model does not give a correct geometric description of the powderbed, and it can not explain the observed sintered depth and temperature dependency ofsteady state density. A new model of powder bed sintering, also based on Frenkel'sconcept is presented in this paper. This model assumes a cubic packed powder bedstructure. By introducing a mathematical factor, fraction of sintering, this model canexplain the temperature dependency behavior of sintered bed density. effect ofvariation in particle size can also be considered in calculating the sintering rate.De Structure of the Powder BedIn some of applications employing sintering of powder beds such as the SelectiveLaser Sintering, the powder bed is a structure of randomly packed spherical particles asshown in figure 1. A cubic packed structure is a powder bed composed of equal-sizespherical particles alined in a cubic way (figure 2). The measured relative density (solidfraction) values of several Selective Laser Sintering powders and the calculated relativedensity of the cubic packed structure are listed in Table 1. It is found that the relativedensity of a randomly packed powder bed is not far from the relative density of a cubicpacked structure, therefore we choose the cubic packed structure to represent the structureof a randomly packed powder bed. The advantage of using the cubic packed structure is itsgeometric simplicity. It is also assumed that the powder bed remains in cubic packedstructure during the sintering process. Sintering rate equations derived hereafter are allbased on this geometric assumption. Although for other types of powder bed packing,46similar sintering rate equations can be derived using the same concept. The cubic packedstructure will give the simplest mathematic form.Figure 2. The.·cubic packed·StructureABSTable 1. The relative density of some Selective Laser Sinteringpowdersandthe cubic packed bedPowder47III. The SinteringModelx:: a 0.815a< x < a 0.805a < x < 0.815a x :: 0.805asphere necking channel close full solidFigure 3. The unit cell and the densification process of a cubic pack structure.During the densification process, x and r satisfy the volume conservation equation,3x3 - 9r2x + 4r3 + 2a3 =0 (1)A criticaIpointinthe densification process iswhenr=¥2x, where. the flat circulararea on each side of.thecellconllect, in other words the channels connecting the void spacebetween the particles close, and the voids are. separated. The value of x at this point is0.8156a.AfterxbecQt11eslesstharrO.81S6a,itheisolated·void keeps decreasing in its sizeuntilx;:;().805aatwhichtimethevoidvanishesalld the structure becomesfun solid. Thevolumeconseryation. equation (1) does not· hold in this period since the contact areabetweelltwo·patticle·isnot·an·exact circle.However,in mostofoursint~ritlgexperimentsofpolYlller powder,thesinteringprocess ends atadensityvalue",hic~correspondsto anx value largerth.artO.8156a, inotherword,most.sinteringprocessesend.beforethepores<areseparateCi.•Therefore it isre.asonable for us to derive.the<sinteringrateequation.911lyfortherange 0.8156a<x<awhere the geometry issimplean4thevolumeconservation equation (1) holds.By Frenkel's theory, the driving force of sintering is the surface tension of thesintering material, Le., by decreasing the total surface area of the powder bed structure the48surface energy will reach its minimum. If we assume that sintering occurs between allcontacted particle, then the surface area will be a monotonic decreasing function of relativedensity, and as the powder densifies, the only nontransient resistance force to the surfacetension force is the pressure in the pore when the pore channels close and the air is trappedin the pores. Mackenzie and Shuttleworth[9] have established a model describing thedensification behavior of porous material containing isolated spherical pores. However,the experimental data shows that most sintering stops even before the pore channels close,therefore, there must be some portion of particles that do not sinter with the others, and asthe powder densifies, the surface area of these particles increases, and the surface tensionacts also as a resistance force of densification.For a sintered particle, the contact area vanishes, and the remained surface area ofthe deformed particle in a unit cen isAs = 121trx - 81tr2, (2)where x and r satisfy equation (1). As is (\monotonicdecreasing function of r,which meansthat the surface area decreases as the powder densifies. and the surface tension acts as adrivingforce ofdensification.Foranunsinteredparticle, the contactarea remains as a boundary between particles,surfac.e.a.reaof the deformedpartlcle isAu (3)value increases monotonically with r,whichmeansthat the surface area increase as thepowderdensifies.Thisbehavior ismathematicallyevi<ient because the sphere is the•shapeof minimum surface areafor a given.volume.lJ1.thiscase,·surface tension·acts. as.a resistantforce .ofdensification.Here we define a factor, fraction of sintering~,.as the fractiQll ofsinteredpaxticles.~ has a v~uebetween.OandJ..~=O implies no.sintenng/occurs between particles, and ~=1implies all particles are sinteredtogether.Using the factor ~, the effective surface area ofa partially sintered particleisA==(1- ~)Au- (6~+2)1tr2 - 6(1- ~)1tX2(4)(5)49IP~-~~===i::==+=~~E:} 0.9L=:4==~~:-t::::=<::::~lO.8lO.71---"""""":: ! """''''''''' .. ! ~]°0..65. OL • ..r .. ·"· ..··0· ~·.. ·· ·, .. · · ,· ..· · ·, .1 r" ¢J~~~ ........*'0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98Nonnalized Cell Dimension xlaFigure 4. The surface area during densification processID. The Sintering Rate EquationFrom Frenkel's concept, the· sintering l'.ate equation is derived from the energybalanc.e equation that the surface energy reduction rate is equal to the viscous dissipationrate during the deformation. For each unit cell, this equation is(6)where 'Y(N/m) is the surface tension ofthe material. l1b(Pa'sec) is the bulk viscosity of theporous structure, eis the volumetric strain rate of the structure and V(m3) is the volume ofthe unit cell.The surface area rate Acan be derived from (5) asA=121t(tx + rx) - 2(6~+2)1trt - 12(1- ~)1tXx,where x and r satisfy equation (1), and xand t satisfy the derivative of equation (1):9x2.t - 18rxt - 9r2x 12r2t =0(7)(8)In equation (6), the bulk viscosityl1b is a function of the true viscosity of thematerial· and the relative.. density, various models have been proposed to describe theirrelations [7...8]. Skorohod's model [8] which directly relate l1b to the relative density p isused here,(9)where 11 is the viscosity of the liquid material, and· the relative density can be calculatedfrom the geometry of a unit cell as50(10)Also in (6), the volumetric strain of the unit cell is by definitionX£ == 3(1--),a(11)which gives the volumetric strain rate as£==- .a(12)The last variable in equation (6) is the volume of the unitcell:V==8x3Substituting (7) to (13) into (6), the final sinrering rate equation can bederived as(13)J 9(x2-r2)})r •18rx-12r2 (14)0=0.1Integrating equation (14) using standard numerical techniques,. the densificationcurve x(t) or p(t) can be obtained. The relation of xlaand p versus<thenortnalized timex'Yt/6na isplottedin Figure 5with~ value between .0.1 and 1. Thes~curve are plottedwithin the.rangeofO.8156a<.x<a since the modelis derived only for this range.~!';3j 0.95~ 0.9i 0.851~ 0.80~-""--""--1J.....------J...---.....I..------J.---.....L-------J~ 2 3 4 5 6Dimensionless Tune n;yt/6J,Ul651 2 3 4Dimensionless Time .1tyf/6J.taFigure 5. The densification curves0.5 '''----__.......-A---__-'---_---J. ....l.-_-....------l. -Jo51The densification rate increase with increasing 'Y or ~, and decreases with increasing" or Q. •This means that the powder will densify faster with a higher surface tensionparticle size. Alsoifor~valuelessthanabout.O.3,thedensificationprocessstops before xreachesO.8156a/orbefore the pore channels close. This may explain the observed behaviorof lack. of pressure sensitivity on final sintered density described elsewhere in thisproceedings [9].IV. Physical Significance of the Factor ~For ~ less than about 0.3, the steady state value of x can be determined from thesintering rate equation by setingthe sintering rate in equation (14) to zero:(15)The plateau x and p values satisfying eq.(15) and eq.(I) can be solved and are plotted infigure 6.0.30.20.11/L.:://////"7/0.550.60.850.950.9Q. 0.8'$~ 0.75>I 0.7s:: 0.650.80"----L----I..-----J0.1 0.2 0.31r----,----,-------,0.98 \\: , , ~ ..j0.96 \~ 0.94 \:\ i ~ ..j'$ 0.92 '\., ~ ~i0: \\0.86 \0.84J- , ~\..."'~ ..0.82 J- ; .;. , ~"Figure 6. The relation between' and plateau values of x/a and p.A physical interpretation of the factor ~ is the probability forformingsinteringnecks between particles. Fromastatisncalthermodynanlic.point of view, this probability isdirectlyrela.ted to the energystatus ..• atthecontactarea.Asa consequence, the probabilityshould increase with temperature because the surface will be at a higher energy state. Alsothe strain caused by defonnation near the contactarea will also increase ~. Strong evidenceof the. temperature effect on the steady state. relative density values is observed in thehomogeneous sintering experiment of polymer powder [10]. Shown in figure 7 are some52densification .curves of polycarbonate powder at different temperatures. Although theplateau values of the experimental HlHo curves.do not fit the theoreticalxla·valuesinthe~<O.3 range, the temperature effect is evidenced by the relative plateau values. Possiblereasons for the difference between the experimentaland theoretical values may includesqueeze flow, anfsotropic.sintering.and expansion of gases in the experimental work.1700C_ .....n _25020050 100 150time (min)Figure 7. Experimental sinteringcurves of a polycarbonate powder bedat different sintering temperatureV.. The Effect of Particle Size Distribution(16)If the particle size of the powder is not uniform but has a variation, the sinteringbehavior of these particles becomes very colllplicated.alld<asuitablegeometricpackingmodel.may not be available. However,ifwe~ssullledthat the deformation ofevery unitcell containing onePa.rticle .• isindependentftom others,the/apparent densificati0llbehaviorcan.becalculatedbyvolurnetricanysummingup.thedeformatiQ.ncurve ofeachparticle.Assuming. that· the powder is composed of ni particles of radius ai,· i=1 ,2,3.... Forevery radius, a densificatiOl1curve pi(t) can be.ca!culatedfrom•• the sintering rate equation(14). The total volume of the powder bed can be summed up by:Vt = ~1t!r~3) = ~rt I;ltJl3And the apparent relative density of the powder bed can be calculated by:(17)where Vi =nia,3!'£niai3 is the volume fraction for the particle ai in the powder.53The calculated defonnation curves of two powder with different particle sizedi$tribution are shown in figure 8. Fonn this curve, we can see that powder with smallerparticles will have higher densificationrates.60~ 50~ 40 i powderApowderBf 3020£~ 10010 20 30 40 50 60 70 80 90 100Particle Size {J.Lm)12010080powderB4020viscosity 11 =3xl06 Pa·Sec fraction of sintering ~=o.2surface tension 1= 0.04 N/m...__--2po~w~d~e~rA~ --_=:j60time (min)Figure 8. The particle size distribution and densification curve of two powders10-.~ 0.9! 0.8~ 0.7.~S& 0.60.50VI. SummaryA modeLis presentedwhichestimates.the sintering in a powder bed of viscousmaterial. This model assumes a cubic packed powder bed structure. A· sintering fractionfactor' is introduced to this IDodel· to explained the partial sinterlng behavior observed inmost sinteringexperiments which other sintering models do not consider. A sintering rateequation is derived based on the energy.balance equation. The calculated densificationcurves qualitatively depict the characteristics of a real. sintering process. Incompletesintering is approximated by sintering curves with' value less than about 0.3. A physicalinterpretation of the , factor and it's relation with temperature is discussed. Finally, themodel is augmented with a particle size distribution effect.References1. Frenkel, J., "Viscous Flow ofCrystalline Bodies Under the Action ofSulface Tension,"J.Phys. (Moscow), 9 [5] 385..91 (1945).2. Rosenzweig, N., Narkis, M., "Dimensional Variations of Two Spherical PolymericParticles During Sintering", Pol. Engr. Sci. 21 [10] 582-5 (1981).543. Scherer, G., "Sintering of Low-Density Glasses.' I, Theory," J. Am. Cer. Soc. 60 [5-6] 236..9 (1977).4. Scherer, 0., "Sintering ofLow-Density Glasses: II, Experimental Study," J. Am. Cer.Soc. 60 [5..6].239..45 (1977).5. Scherer, 0., "Sintering ofLow-Density Glasses: [II, Effect of a Distribution of PoreSize," J. Am. Cer. Soc. 60 ••[5...6] 245..8 (1977).6. Dekard, C., "Selective Laser Sintering," PhD. Dissertation, The University of Texas atAustin (1988).7. Bordia, R., Scherer, 0., "OnConstrained Sinteing-I. Constitutive Model for a SinteringBody," Acta.Inatall· 36 [912393-7 (1988).8. Bordia,R., Scherer, 0., "On (;onstrained Sinteing-II. Comparison of ConstitutiveModels," Acta matail. 36[9]2399-409 (1988).9. Nelson, J. C., Vail,N. K., Barlow, J. W., "Post-Processing of Selective LaserSinteredPolycarbonate Parts,"SFF Symposium, Austin, TX (1991).10. Nelson, C., Barlow, J., "SinteringRatesinthe< Selective Laser Sintering," SFFSymposium, 164..70, Austin, TX (1990).55

A Model for Partial Viscous Sintering

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A Model for Partial Viscous Sintering

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