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Transactions of NAMRI/SME, Vol. 36, 2008, pp. 57-64TRANSACTIONS OF NAMRI/SME Discrete Element Modeling of Micro- Feature Hot Compaction Process authors P. CHEN J. NI University of Michigan Ann Arbor, MI, USA abstract In the forming of porous microfeatures using a hot compaction process, it is costly and time consuming to determine a proper experiment setting (force, temperature, and time) by trial and error. Product qualities, such as mechanical strength and porosity, are significantly affected by the setting of those process variables. To analytically study the effect of force and temperature on particle bonding strength and porosity, a discrete element model for pressure-assisted sintering was developed for the forming of porous microfeatures. The model was first validated with experimental results for a unit problem (two particles). It was then expanded for a 10-particle channel hot pressing problem. With this model, it was feasible to conveniently assess the effects of force and temperature on the particle bonding strength and shrinkage, which then gave insight on deciding a proper process setting before actual operations. terms Porous Microfeatures Pressure-Assisted Sintering Hot Compaction Discrete Element Modeling Society of Manufacturing Engineers • One SME Drive • PO Box 9302008 Dearborn, MI 48121 • Phone (313) 425-3000 • www.sme.org
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DISCRETE ELEMENT MODELING OF MICRO-FEATURE HOT COMPACTION PROCESS Peng Chen and Jun Ni Department of Mechanical Engineering University of Michigan Ann Arbor, MichiganKEYWORDS INTRODUCTION Pressure Assisted Sintering, Network Model, Porous micro-features with high aspect ratioHot Compaction, Porous Micro-Features are becoming more and more important in the modern industry, especially for high efficiency heat transfer applications (Liter and KavianyABSTRACT 2001). As discussed in our previous studies (Chen et al. 2007), hot compaction process is In the forming of porous micro-features using one of the most promising ways to produce suchhot compaction process, it is costly and time- features, and its capabilities have already beenconsuming to determine a proper experiment experimentally demonstrated. However, it is verysetting (force, temperature and time) by trial and costly and time-consuming to determine aerror. Product qualities, such as mechanical proper experiment setting (force, temperaturestrength and porosity, are significantly affected and time) by trial and error. As investigated byby the setting of those process variables. Chen et al., product qualities (such as mechanical strength and porosity) are In order to analytically study the effect of the significantly affected by the setting of theforce and temperature on the particle bonding process variables (Chen et al. 2007). Therefore,strength and porosity, a discrete element model in order to reduce the time and efforts spent onfor pressure assisted sintering was developed trial and error in physical experiments, this studyfor the forming of porous micro-features. The aims to develop a computational model tomodel was first validated with experimental analytically study the effect of process variablesresults for a unit problem (two particles). And on the particle bonding strength and porosity.then it was expanded for a 10-particle channelhot pressing problem. With this model, we could Hot compaction processes combine theconveniently assess the effects of force and simultaneous application of pressure andtemperature on the particle bonding strength temperature, which is also termed as pressureand shrinkage, which then give us insight on assisted sintering. During sintering, particles aredeciding a proper process setting before the bonded together by atomic transport events. Theactual operations. driving force for sintering is a reduction in the Transactions of NAMRI/SME, Vol. 36, 2008, pp. 57-64
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system free energy, which is achieved by sintering (Hwang and German 1984; Parhamireduction of surface curvatures and elimination and McMeeking 1998), which is dedicated toof surface areas (German 1994). Initially, a grain simulate the diffusion and mass transportboundary is formed at the contact between mechanisms near the particle surface withoutneighboring particles. Atoms travel along this grain growth; (2) models for late intermediateboundary and along the particle free surface to and final stage sintering, which is focused on thethe neck regions. modeling of grain growth and pore shrinkage (Hassold et al. 1990; Tomandl and Varkoly Starting from late 1950s, numerous 2001). The model for final stage sintering isresearchers have studied the computer especially important for ceramic sintering sincesimulation of sintering processes (German large shrinkage is often encountered in this2002). More than 1,000 publications can be case. Since this study is only concerned with thefound on this topic. According to the different initial and early intermediate stages of sintering,scales of constitutive modeling, the existing only the first model will be discussed in detail incomputer models for sintering could be divided this work.into three classes: (1) continuum model(Olevsky 1998; Delo et al. 1999; Sanchez et al. During the initial and early intermediate stages2002); (2) discrete model (German and Lathrop of sintering, necks between neighboring1978; Parhami and McMeeking 1994); (3) particles grow up; and no densification occurs.molecular dynamics model (Zavaliangos 2002; Therefore, a mathematical expression of neckRaut et al. 1998). However, most research growth as a function of temperature and time willefforts were on the modeling of free sintering be sufficient to model the free sintering processprocess, where no external mechanical loading (no external load) (Hwang and German 1984). Inwas considered. Continuum models are most the case of hot pressing (pressure assistedsuitable for free sintering process, and they also sintering), to accurately simulate the particlerequire accurate material testing in high behavior under the influence of both elevatedtemperature condition, which is difficult to temperature and external pressure, sinteringperform. In addition, no microstructure stress induced diffusion and external pressureinformation could be obtained from continuum induced diffusion should be integrated together.simulation. Molecular dynamics method is highly An efficient way to achieve this goal is toaccurate but is difficult to implement for our combine the existing neck growth model withproblem due to time and length scale limitations. Discrete Element Model, which is called networkIn addition, a real particle usually has a model by some researchers (Parhami andpolycrystal structure, which imposes another McMeeking 1994; Parhami and McMeekingdifficulty in the MD modeling, that is, how to 1998), or truss model (Jagota and Dawsoneffectively define the grain boundary in a single 1988). In this model, every particle center isparticle. Relatively speaking, discrete models represented by a node and every contactstand out to be a sound candidate for the between neighboring particles by an element.simulation of hot compaction of powders into Figure 1 is a two dimensional representation of amicro-features. pair of particles bonded together at a neck. A relative axial velocity of the particles centers is In this study, a discrete element model for the consequence of atomic flux from thepressure assisted sintering was developed for interparticle grain boundary to the free surface.the forming of porous micro-features. The model This process, coupled to mass transport on thewas first validated with experimental results for a free surface, leads to the development of grainunit problem (two particles). And then it was boundary area at the contact and the generationexpanded for a 10-particle channel hot pressing of thermodynamically induced normal stressesproblem. on the grain boundary.DISCRETE ELEMENT MODELING OF FORMULATION OF THE NUMERICAL MODELSINTERING PROCESS FOR HOT COMPACTION (NETWORK MODEL) Generally speaking, there are two categoriesof particle-level models for sintering: (1) models Based on the network model mentioneddeveloped for initial and early intermediate stage above, a numerical model was developed which Transactions of NAMRI/SME, Vol. 36, 2008, pp. 57-64
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could predict the pressure assisted sintering (hot − Qgcompaction) behavior of a particle system as a 8δ g D g 0 e Rs T Ω ⎧σ γ ψ ψ ⎫ vn = ⎨ 2 − 4 [4 R(1 − cos ) + r sin ]⎬function of temperature, external force and time. kT ⎩ r r 2 2 ⎭ (3) x The second terms on the right hand side of Eq. (1) and Eq. (3) drive free sintering. According to Swinkels and Ashby (1981), the Fn1 ,Vn1 Fn2 ,Vn2 values of the above coefficients for copper are 2r shown in Table 1. R ψ TABLE 1. MATERIAL PROPERTY OF COPPER (SWINKELS AND ASHBY 1981).FIGURE 1. 2D REPRESENTATION OF A TWO-PARTICLE NECK GROWTH MODEL. Material constant Copper 3 δ g Dg 0 (m /s) 5.12 × 10 -15 γ (J/m ) 2 1.72 Only the initial stage sintering was consideredin our study, in which case the dominant mass Q g (J/mole) 105000transport mechanisms are surface diffusion and R s (J/mole) 8.31grain boundary diffusion. As shown in Eq. (1), Ω (m3) 1.18 × 10 -29the neck growth rate equation was derived k (J/Kelvin) 1.38 × 10 -23based on neck growth rate equation proposed ψby Parhami and McMeeking (1998) and the 146°diffusion coefficient equation used by Exner(1979). Before pressure After pressure − Qg assisted sintering assisted sintering 8 Rδ g D g 0 e RsT Ω⎧ σ γ ψ ψ ⎫r= ⎨− 3 + 5 [ 4 R(1 − cos ) + r sin ]⎬ kT ⎩ r r 2 2 ⎭ (1) 1where δ g is the effective grain boundary Fn , v1 , x1 n nthickness, Dg 0 is the maximum grain boundarydiffusion coefficient (at infinite temperature), γ isthe surface energy per unit area, Qg is the Neck 2activation energy of grain boundary diffusion, Fn , v , x 2 2 n nRs is the gas constant, Ω is the atomic volumeand k is Boltzmann’s constant. T denotesabsolute temperature (Kelvin). As shown in FIGURE 2. ILLUSTRATION OF THE TWO-Figure 1, r is the neck radius, ψ is the dihedral PARTICLE PRESSURE ASSISTED SINTERINGangle at the neck and R is particle radius. σ is MODEL.the normal stress on the contact. Fn1 F2 Assuming that the neck growth rate and axialσ= = − n2 πr 2 πr (2) velocity remains the same in a very small time step, the axial displacement of the particle and Similarly, the axial velocity of the particle was neck radius are updated using central finitederived based on the equation used by Parhami difference method (Cundall and Strack 1979).and McMeeking (1998): x N +1 = x N + (v n ) 1 Δt N+ 2 (4) Transactions of NAMRI/SME, Vol. 36, 2008, pp. 57-64
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where Δt is the critical time step and is found to -7be around 10 s for our case (Martin et al.2002).Modeling of Unit Problem For a simplified two-particle model as shownon the left of Figure 2, tangential force andmoment are ignored. A numerical model forpressure assisted sintering was developed usingMATLAB based on Eqs. (1)–(4) and Table 1.The right diagram in Figure 2 is an illustration ofcalculation result for hot pressing in the format ofneck growth.Modeling of Multi-Particle Problem withBoundary Conditions The discrete element model for cold FIGURE 3. ILLUSTRATION OF THE 10-PARTICLE MODEL.compaction developed by Cundall and Strack(1979) is based on the original particle A code was developed for this multi-particledynamics, where contacts between particles are pressure assisted sintering problem usingnot sustained. It is not well-suited, however, for MATLAB. The step-by-step computing structureapplication where the contacts undergo large of the code is shown in Figure 4.deformations and, once made, rarely break. Inour case (hot compaction after pre-press), theparticle assembly may be assumed to be inequilibrium at all stages of the process (Jagotaand Dawson 1988), permitting solution forvelocities implicitly, as discussed below. Basedon the study of Fleck (1995) and Heyliger andMcMeeking (2001), shearing tractions betweenparticles was neglected, which was found to playa minor role in the particle assembly, especiallyafter pre-press. Particle packings were treated as frameworksof links that connect the centers of particlesthrough inter-particle contacts. The behavior ofeach link in the framework was based on unitproblems for the interaction between individualspheres as described in the previous section. Asshown in Figure 3, a network model for thepressure assisted sintering of 10 particles in a V-shape channel was developed. The anglebetween two V-channel walls was 60°. The FIGURE 4. CALCULATION SCHEME FOR MULTI-particle diameter was 200 µm. Each particle was PARTICLE PRESSURE ASSISTED SINTERINGassigned a number as shown in Figure 3. PROBLEM.Identical force was applied on particles 7, 8, 9,and 10 to account for the compression load. The In the initialization step, constants such asinteraction force between each particle pair was material properties and temperature are defined.obtained via frame analysis. Geometry and dimension of the channel and particles are defined in the assembly step. The Transactions of NAMRI/SME, Vol. 36, 2008, pp. 57-64
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coordinates of the particles are first defined in RESULTS AND DISCUSSIONglobal coordinates, and then transformed intolocal coordinates via rotation matrix for the ease Unit Problem and Validationof computational operation. Before the hotcompaction, the pre-pressed particles will have To validate the above numerical model, thean initial neck radius due to elastic or plastic simulation results (r/R and shrinkage) weredeformation, which was solved using the original compared with the experiment results provideddiscrete element model proposed by Cundall by Exner (1979), as shown in Figure 5. Inand Strack (1979). The subsequent five steps Exner’s experiments, 20 large copper spherescompose an iteration loop, which solves the were sintered at 1027°C without any externalpressure assisted sintering process continuously force loading. In Figure 5, the relationshipuntil a pre-defined sintering time is reached. The between neck radius / particle radius ratio (r/R)approaching velocity between every two and the relative center approach ([X0 – XN]/R)contacting particles was calculated using Eq. (which is the ratio between the approaching of(3), which is stored in an approaching velocity two particle centers and their original distancematrix as shown in Eq. (5). and is an indication of the shrinkage of the particle system) were presented. Simulation ⎡Vn11 Vn12 .... Vn1n ⎤ results agreed well with the experiment results, ⎢Vn ⎥ (5) and the predicted trend of the evolution of Vn = ⎢ 21 ⎥ shrinkage as a function of r/R matched well with ⎢ ⎥ ⎢ ⎥ the experimental observations. ⎣Vnn1 Vnnn ⎦where Vnij denotes approaching velocity onparticle i caused by particle j. The matrix wasconstructed this way such that the absolutevelocity of the particle could be assembledconveniently in the velocity summation step withonly on matrix operation as shown in Eq. (6). ′ ⎡Vn11 Vn12 .... Vn1n ⎤ ⎡θ11 θ12 .... θ1n ⎤ ⎢Vn ⎥ ⎢ ⎥ V = ⎢ 21 ⎥ * cos ⎢θ 21 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣Vn n1 Vnnn ⎦ ⎣θ n1 θ nn ⎦ (6) FIGURE 5. COMPARISON BETWEEN SIMULATIONwhere θ ij denotes the angle between local y axis AND EXPERIMENT RESULTS.(orthogonal to the axial direction) and the vectordirection on particle i caused by particle j. The displacement of each particle is updatedusing central finite difference method [Eq. (4)]with forced boundary conditions imposed by theV-channel. At the end of each iteration, the neckradius is updated using Eq. (1). Post-processingstep store and plot out data. Simulations were run for the above problemwith a force of 10 N for eight minutes of pressureassisted sintering at different temperatures.Each simulation took about five hours of FIGURE 6. EFFECTS OF TEMPERATURE ANDcomputational time on a Sun Ultra 20 (1.8 GHz) FORCE ON r/R.workstation. After validation, a further study of the pressure assisted sintering process was performed using the numerical model. Figure 6 shows the effects Transactions of NAMRI/SME, Vol. 36, 2008, pp. 57-64
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of temperature and compaction force on the contributing factor; the interaction force causedpressure assisted sintering process. The by surrounding particles also affected the neckfollowing conclusions could be drawn from this size. For example, the axial force betweenfigure: (1) the rate of neck growth was very low particle 6 and particle 8 was the highest, butat a low temperature (25-150°C), in which case their neck was not the largest. The largest neckneck did not grow much even if a compaction occurred at the particle 2 and 5 interface, whichforce was applied; (2) an external compression was more than twice the size of other necks. Butforce significantly increased the neck growth its growth rate after the first 10 seconds wasrate at a higher temperature range (300- also the lowest comparing to other necks. A1000°C), which was due to the fact that the review of Eq. (1) reveals that the neck growth 3material was softened in this temperature range. rate is proportional to 1/r , which results in aEspecially in the cases of 1 N at 700°C and 0.1 lower growth rate at a larger neck size.N at 850°C, there was a dramatic increase in theneck growth rate. Figure 8 shows the relative center approach of the particles during pressure assisted sintering at 350°C. While most particles wereMulti-Particle Problem with Boundary approaching each other, some particles wereConditions departing from others. However, the general trend was that all the particles were shrinking As shown in Figure 7, at an isothermal into the center of the particle packing. In thistemperature setting (350°C), the neck growth of case, particle 5 became the center ofdifferent particle pairs were different. The growth approaching. Similarly to the neck growth, theof the neck was very rapid in the first 30 lower the interaction force, the slower theseconds, after which the growth slowed down approaching.dramatically and appeared as seemingly linearincrease over the time. FIGURE 8. RELATIVE CENTER APPROACHING DURING PRESSURE ASSISTED SINTERINGFIGURE 7. NECK RADIUS DURING PRESSURE (350°C, 10 N).ASSISTED SINTERING (350°C, 10 N). Figures 9 and 10 show the neck growth and Depending on the axial interaction force relative center approaching of the networkbetween two particles, the size of the formed model at 422°C. As the temperature increased,neck was different. Generally speaking, the the neck size and approaching speed increasedhigher the axial force, the larger the neck is. For as well. But the general growth trend remainedexample, the neck between particle 7 and the same.particle 8 was the smallest, since the axial forcebetween them is the lowest. However, the axialforce between each given pair was not the only Transactions of NAMRI/SME, Vol. 36, 2008, pp. 57-64
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The numerical model developed in this study effectively captures the atomic diffusions caused by both pressure and heat, and provides means to extend this model for more particles with the consideration of boundary conditions, which could be a convenient tool for engineers and scientists to study the effects of force, temperature and time on the quality of the formed micro-features. REFERENCES Chen, P.; G.-Y. Kiml and J. Ni (2007). “Forming of Porous Micro-Features Using Hot Compaction.” Proceedings of the InternationalFIGURE 9. NECK RADIUS DURING PRESSUREASSISTED SINTERING (422°C, 10 N). Conference on Manufacturing Science and Engineering, MSEC 2007. Cundall, P.A. and O.D.L. Strack (1979) “A discrete numerical model for granular assemblies.” Géotechnique (v29), pp. 47-65. Delo, D.P.; R.E. Dutton; S.L. Semiatin; and H.R. Piehler (1999). “Modeling of hot isostatic pressing and hot triaxial compaction of Ti-6Al-4V powder.” Acta Materialia (v47), pp. 3159-3167. Exner, H.E. (1979). “Principles of single phase sintering.” Reviews on Powder Metallurgy and Physical Ceramics (v1), pp. 11-251. Fleck, N.A. (1995). “On the cold compaction of powders.” Journal of the Mechanics and Physics of Solids (v43), pp. 1409-1431.FIGURE 10. RELATIVE CENTER APPROACHINGDURING PRESSURE ASSISTED SINTERING German, R.M. (1994). Powder Metallurgy(422°C, 10 N). Science. Princeton, NJ. German, R.M. (2002). “Computer modeling ofCONCLUDING REMARKS sintering processes.” International Journal of Powder Metallurgy (v38), pp. 48-66. In this study, a discrete element model for hotpressing was proposed, developed and showed German, R.M. and J.F. Lathrop (1978).a good agreement with experiments, which “Simulation of spherical powder sintering bysimulated the pressure assisted sintering surface diffusion.” Journal of Materials Scienceprocess as a function of force, temperature and (v13), pp. 921-929.time. Under an equilibrium condition, the hotpressing process could be characterized as an Hassold, G.N.; I. Chen; and D.J. Srolovitzatomic diffusion process, during which both (1990). “Computer Simulation of Final-Stageheat-induce diffusion and pressure-induced Sintering: I, Model Kinetics, and Microstructure.”diffusion took place. Considering the processing Journal of the American Ceramic Society (v73),time was short (Chen et al. 2007), only the initial pp. 2857-2864.stage sintering was investigated. Heyliger, P.R. and R.M. McMeeking (2001). “Cold plastic compaction of powders by a Transactions of NAMRI/SME, Vol. 36, 2008, pp. 57-64
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network model.” Journal of the Mechanics and Parhami F. and R.M. McMeeking (1994).Physics of Solids (v49), pp. 2031-2054. “Computer simulation of solid state sintering of powders through discrete element method.”Hwang, K-S. and R.M. German (1984). ASME, Applied Mechanics Div. (v194), pp. 203-“Analysis of initial stage sintering by computer 207.simulation.” Sintering and HeterogeneousCatalysis, G.C. Kuczynski, A.E. Miller, and G.A. Parhami, F. and R.M. McMeeking (1998). “ASargent, eds. New York: Plenum Press, pp. 35- network model for initial stage sintering.”47. Mechanics of Materials (v27), pp. 111-124.Jagota, A. and P.R. Dawson (1988). Raut, J.S.; R.B. Bhagat; and K.A. Fichthorn“Micromechanical modeling of powder (1998). “Sintering of aluminum nanoparticles: Acompacts-II. truss formulation of discrete molecular dynamics study.” Nanostructuredpackings.” Acta Metallurgica (v36), pp. 2563- Materials (v10), pp. 837-851.2573. Sanchez, L.; E. Ouedraogo; L. Federzoni; andLiter, S.G. and M. Kaviany, (2001). “Pool-Boiling P. Stutz (2002). “New viscoplastic model toCHF Enhancement by Modulated Porous-Layer simulate hot isostatic pressing.” PowderCoating: Theory and Experiment.” International Metallurgy (v45), pp. 329-334.Journal of Heat and Mass Transfer (v44), pp.4287-4311. Swinkels, F.B. and M.F. Ashby (1981). “A second report on sintering diagrams.” ActaMartin, C.L.; D. Bouvard; and S. Shima (2002). Metallurgica (v29), pp. 259-281.“Study of particle rearrangement during powdercompaction by the discrete element method.” Tomandl, G. and P. Varkoly (2001). “Three-Journal of the Mechanics and Physics of Solids dimensional computer modeling of grain growth(v51), pp. 667-693. and pore shrinkage during sintering.” Materials Chemistry and Physics (v37), pp. 12-16.Olevsky, E.A. (1998). “Theory of sintering: fromdiscrete to continuum.” Materials Science & Zavaliangos, A. (2002). “Constitutive models forEngineering R (Switzerland) (vR23), pp. 41-100. the simulation of P/M processes.” International Journal of Powder Metallurgy (v38), pp. 27-39. Transactions of NAMRI/SME, Vol. 36, 2008, pp. 57-64
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