material data, leading to solutions poorly aligned with practical needs. ... Effective computer simulations of compaction and sintering top the wish list for ... economic trends, stock prices, transportation networks, space flight, and even ...... A. Van De Vorst, âNumerical Simulation of Viscous Sintering by a Periodic Lattice of a.
COMPUTER MODELING OF SINTERING PROCESSES Randall M. German Brush Chair Professor in Materials Center for Innovative Sintered Products P/M Lab, 147 Research West Pennsylvania State University University Park, PA 16802-6809 ABSTRACT Computer simulations of sintering first emerged between 1955 and 1965, and the field has expanded rapidly in the last ten years. Yet, in spite of enormous efforts and nearly a thousand publications, sintering practice ignores these simulations. This presentation overviews the history of sintering computer simulations, illustrates some of the current performance benchmarks, and provides suggestions on near-term gains and remaining barriers. Computers are affordable and the conceptual models are in reasonable shape, but the knowledge infrastructure required for accurate computer simulations is weak. For the important engineering attributes, such as final component dimensions, sintering practice is much more accurate than modeling. The difficulties come from simplification assumptions required because of missing or inaccurate material data, leading to solutions poorly aligned with practical needs. This keynote paper examines the successes to date and suggests some near-term opportunities. Immediate gains are possible in sintering process control and early sensitivity testing in research programs. However, considerable research is required to better understand material behavior during sintering to create industrially relevant simulations. The current models are inaccurate, not because of computational limitations, but because of the approximations made concerning microstructurelevel events occurring during sintering, especially for longer times and at higher temperatures. INTRODUCTION Effective computer simulations of compaction and sintering top the wish list for the sintered materials industries. There is much advantage to such efforts, but there are also some unrealistic expectations. This presentation overviews the historical developments in computer simulation of sintering to show the current state of affairs, and projects forward to speculate how the models might satisfy industry needs. Justifications for computer simulations are based on the fact that if you cannot model the process, then you really do not understand nor can you control that process. Although current simulations are not accurate, they are important in forcing the technical community to organize our knowledge, thereby determining where we do understand 1
sintering and where there are pending problems. Sintering simulations have been with us for many years. The attached references, organized in chronological order, are selected from nearly 1000 publications to provide a perspective on the developments over the past 35 years [1-106]. Early powder bonding models date from 1955, but the significant publications on sintering kinetics emerged in the 1960's. Most early interest was on predicting the time-dependent events associated with sintering, including neck size or shrinkage. Today, our focus is on predicting final properties, component size and shape, processing cost, tool motions, furnace set-up, and process sensitivities. About every ten years the field takes a new turn, and in simple form the history of sintering simulations is as follows: 1960 to 1970: numerical simulations of two-particle neck growth in sintering 1970 to 1980: multiple mechanism calculations of one-dimensional shrinkage 1980 to 1990: predictions of density including pressure-assisted sintering 1990 to 2000: continuum mechanics and finite element simulations of shape and size 2000 to date: control programs with hybrids of simulations and artificial intelligence. Along the way, there were various excursions into most of the simulation types - Euler, Monte Carlo, Finite Difference, Discrete Element, Finite Element, Fluid Mechanic, Continuum Mechanic, Neural Network, and Adaptive Learning. None of these satisfactorily succeeded in predicting final component attributes important to manufacturing. Why so much interest in computer simulation of sintering? The answer is simply engineering expediency. The major goal of net-shaping is to produce components with final dimensions and target properties. This must be achieved rapidly with minimal tool and process iterations. In sintering, component size depends on many independent parameters, where the complex interaction between those parameters exceeds simple calculations. The complexity is best handled by simulations. Computers provide a low-cost means to consider many factors and to ponder trade-offs between control parameters such as forming pressure, powder type, particle size, heating rate, peak temperature, hold time, furnace design, and atmosphere. Accordingly, simulations highlight what is understood and effectively show where there are problems in need of experimental attention. Only when a simulation is coded do we realize the gaps in sintering theory, especially with respect to microstructure evolution. So if computer simulations have so much benefit, then why are we not using more of them? Or alternatively phrased, why is sintering still essentially a phenomenological field? The answer lies in several factors. A first realization is that the computer simulation environment is relatively new. Only in the past decade have the tools and protocols become available on a widespread basis. Unfortunately, the input data for the models and even some of the basic relations needed for simulations are not well developed; accurate material data are lacking for most materials at the sintering temperature. As an example, the tabulations of strength, elastic modulus, and Poisson’s ratio for steels rarely extend to at the 1120°C (or higher) sintering temperature; most cut off at the end of a reasonable upper application temperature (which is below the sintering temperature). Further, constitutive models do not exist for the conditions relevant to sintering. For example, there is no verified model for the elastic property changes during sintering densification. Thus, the problem of high-temperature porous-body elastic behavior is simply approximated in the simulations. Hence, the barrier is the weakness of the underling material 2
models and data. In spite of elegant mathematics, there is insufficient attention to model verification. Consequently, the simulations are compromised by either approximate data or simplified relations. However, the near-term prospects are not so dismal. There are some opportunities to implement hybrid simulations that will be relevant to industrial sintering while the science community backfills some of the problem areas. BRIEF HISTORY Sintering simulations are a consequence of long-term interest in predictions. Mankind has always sought to predict events, with much effort over the years on predictions of weather, economic trends, stock prices, transportation networks, space flight, and even net-shaping processes. The first major publication in computer simulation of sintering traces to 1965 [1], and that work was ten years behind the first simulations of related topics. Some of the early simulations were two-dimensional (sintering two wires) with a single diffusion mechanism. Unfortunately, the simulations were slow, requiring ten times more computer time than the actual physical sintering time. Most damaging, these early models lost mass and increased energy [18], so their contribution was to show the eventual possibilities, but not to provide accurate results. Within a few years contributions emerged using variants on the atomistic transport idea [2-6]. Within 20 years the concept was extended to multiple transport mechanisms, multiple sintering stages, and even pressure-assisted sintering [12,13,24,27,29-32,37]. These simulations were able to predict sintered density versus time, temperature, pressure, and particle size, for isothermal sintering of single phase materials. Those successes provide benchmarks for all subsequent efforts. However, significant limitations became evident because of assumptions on isothermal conditions and microstructure coarsening. Dilatometry experiments show most sintering occurs on the way to the peak temperature, so isothermal models poorly reflect actual behavior. And the assumed homogeneous and ideal microstructure limited the models to shorter times and lower temperatures. During sintering it is common for thousands, and even millions, of initial particles to form a final grain in the sintered microstructure. Most models simply approximate microstructure coarsening with average behavior. To model sintering requires much more detail on the pore-grain boundary interactions and possible anisotropic behavior. Accordingly to dodge the problem, many recent efforts have simply ignored microstructure. As a consequence, sintering densification is treated as a viscous flow event. The benefit is that only a few adjustable coefficients are needed in finite element analysis to generate pretty pictures. Unfortunately, simple viscous flow models can lead to considerable error, so the simulations are usually only good approximations. This pendulum swing away from atomistic view to assuming a homogeneous body now ignores the evolving microstructure. Although easier to implement in finite element analysis, such casual treatments are just approximations and become less reliable as sintering progresses. A balance is needed to go beyond atomistic models, while adding more insight than present in finite element analysis. Possibly the solution might come from tempering the atomistic simulations with a means to locally adjust the constitutive equations based on microstructure evolution and then feeding that result into the finite element analysis. 3
SIMULATION BASICS All simulations require elements that are essentially the same, independent of the target [65]: * basic rules and boundary conditions * input data and input interface * questions and simulation goals * process monitors, error detections, and stability parameters * verification and validation * experimentation * output data and output interface. Such elements ensure the simulation process is understood and structured by known or assumed rules. The input data and boundary conditions are then evaluated using these rules to calculate the output - garbage in garbage out; alternatively, good input data and rules give reliable results. I like to call the process “computer experimentation” to reflect the goal of doing experiments with the simulation once we have faith in the outcome. For example, we might examine the implications of setting a different flight schedule for an airline to improve on-time arrival. If the result is not favorable, then we should reject the policy before it is ever put into place and upsets passengers. Questions in the simulation are constructed by the user as a basis for decisions; early knowledge on the character of the questions allows the simulation to handle the intended range of cases. To ensure the simulation is well behaved, it is necessary to incorporate stability monitors to ensure the simulation is not operating in an unstable range. With all computer models, it is important to first verify the simulation is properly coded and then to validate the results by running cases where the solution is known by prior experience or analytic solution. The core activity is the “computer experimentation” where fundamental questions are probed, assuming the simulation has passed muster. Finally, consideration is required for the input and output modes, be they graphical, data files, or animations, making sure the needed output is collected from the desired input file.
Figure 1. The flow process for performing computer experiments in general, including sintering simulations.
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The simulation process then can be captured conceptually as diagramed in Figure 1. Probably the weakest link in most computer simulations is the stability. As the basic model is formed and the simulation performed for the first few times, it is necessary to ensure the results converge to known cases. Early sintering simulations failed to conserve volume and even
increased energy over time. Although the pictures of sinter bonds were attractive, the reality of the resulting kinetic models was never established. Stability checks often involve seemingly simple changes to the simulation details, such as changing the precision of the variables, changing the forward time steps, increasing or decreasing the number of computational cells, Figure 2. Stability tests in computer simulations are and even varying the number of necessary to ensure the output is insensitive to the solution terms in spline fits or polynomial conditions or input data. approximations. It is laborious but necessary work, unfortunately often overlooked in practice. Additionally, it is important that the simulation be constructed to test for metastable situations, or even unstable conditions based on a robust variation in input parameters and run conditions. An appropriate test is to examine the response behavior versus some of the input variables. Example outcomes are sketched in Figure 2, and what is sought are those conditions producing a stable simulation, largely insensitive to slight data or run condition variations. A common problem in sintering simulation is in the forward time step. Most efforts rely on a fixed step size, as small as 10-3 to 10-6 s. However, a fixed step size leads to systematic errors. A more appropriate technique is to compute the time advance based on the forward projection of the rate of change. In periods where little change occurs, then the forward time step is large, while equal accuracy is preserved in periods where there is rapid change by taking smaller time steps. Characteristically, most computer simulations fail to show the estimated error, nor do they perform simulation experiments to minimize error. This process alone gives the best sense of the accuracy. Another overlooked problem relates to the simulation level. For example, to understand the weather it would not be appropriate to model the splash of a rain droplet. With respect to sintering, it is important to consider the simulation can operate at the atom level, small atomic cluster and even at the grain level, and on up to the particle level. However, much of the phenomenological behavior requires attention at the microstructure level, so the simulation should be operating at the scale of the particle agglomerate and larger microstructure scale. The constitutive behavior at this level then impacts on the observations relevant to the component and even operating machine. Finally, for industry planning purposes there is a need for simulations at the plant and even industry levels. But to emphasize a key point, the level at which the simulation focuses must be relevant to the scale of the problem. Simulations rely on known rules. If the rules are unknown, then the simulation is not realistic. Consider the case of an athletic event, such as baseball. The rules are well known - “three strikes and you’re out” - so the outcome of a sequence of events can be predicted. In cases where the 5
underlying physics or chemistry are unknown, it is possible to invoke probability tables or other look-up means to input the discreet events. Consider baseball where we find a batter might have a 0.7% probability of hitting a home run on any pitch. From such raw statistics it is hard to sense the aggregate quality. But since Figure 3. Energy loss in forming processes create there are multiple pitches in each at significant simulation problems, since often idealized bat, we can eventually calculate the behavior is assumed, yet as illustrated here with die compaction, wall friction varies during the process and is outcome of each appearance at the plate, then game, and even for a 162 equipment sensitive. game season. Thus, if the probability table is known, then a batter’s season can be simulated in seconds. In turn, new batting lineups, player combinations, and batting strategies can be evaluated and optimized via simulations; a fact well used in computer and video games such as “John Madden Football.” This is the power of effective simulations - although the game still depends on random events, simulations can be used to enhance the chance of a favorable outcome. PROBLEMS WITH SIMULATIONS Why do computer simulations of net-shape forming lack accuracy? A basic problem traces to the assumed constant material properties. In most cases the “material constants” are changing during the forming process. The material changes with time, stress, strain, strain rate, and temperature phenomena known as stress relaxation, work hardening, adiabatic heating, exothermic reactions, thermal softening, microstructural coarsening, and strain rate hardening. A further complication is energy loss. For example, friction is highly variable, with time, temperature, forming process, devices, surface condition, lubrication, and even age of the equipment or tooling. Frictional energy loss is not constant, but varies between machines and over time in a single machine. In one study of die compaction, the coefficient of Figure 4. Comparison of actual density contours in die friction proved to vary with pressure compacted spray-dried alumina as compared with and with compaction press, giving the simulated assuming constant friction. twofold variation shown in Figure 3. 6
Figure 5. Illustration of the machine setting (commanded), simulated, and experimental (actual) ram velocity profiles versus time, illustrating the basic friction model problem with making accurate simulations.
In one recent computer simulation of die compaction of spray-dried alumina, the simulation assumed constant die wall friction. As a consequence, the bulk density (average over the compact) had a difference from experiment of 2.7%. However, since the friction was assumed constant, the local density was in error by 11.8%. Graphical visualization of this effect is provided in Figure 4, which shows density contours with assumed constant wall friction as compared to the actual contours. When these data are fed into sintering simulations, the predicted final sizes are grossly in error. Skepticism with regard to the simulation is justified.
Other factors that inhibit accuracy in net-shape simulations relate to mass loss, compressibility, phase changes, inhomogeneous materials, and residual effects from upstream processes. Further, industrial processes fluctuate and are under constant adaptive correction by operators. The simplified assumptions of most models do not reflect the variability, leading to prediction errors. For example, Figure 5 plots three views of the ram velocity in a forming operation; the horizontal line indicates the commanded ram velocity (what the operator told the machine), the second is the computer simulation result, and the third is the actual measured velocity. Note the large difference in detail; accordingly, the simulation has an embedded simplification that feeds forward to generate further errors, leading to inaccurate predictions of final component dimensions. In this case, the simulation needs a chaotic stick-slip model to account for the machine friction. Analysis of successful net-shape models shows computer simulations are best applied to isothermal processes, over shorter process times (injection molding, casting, blow molding, stamping, extrusion, or forging - all are characterized by forming steps of just a few seconds), where energy flow into or out of the control volume is well controlled. Volume conservation is another characteristic of the successfully modeled processes. These situations have simple or no phase changes and tend to be volume reshaping (volume conservation) rather than dimensional change situations. Now contrast this with sintering which involves shape and size changes (lacks volume conservation) is not isothermal and is a slow process measured in fractions of a day, but not seconds involves considerable energy flow during the process almost always involves a phase change (ferrite to austenite or liquid phase formation). Consequently, compared to the successes elsewhere, sintering processes are not as well suited to 7
accurate computer simulations. This is not to say that computer simulations cannot be applied to sintering, but compared to forging or molding there will be less accuracy. Because of the push for 100% density, pressure-assisted and hot isostatic pressing simulations for full densification are favorite simulation topics. When we focus on full densification, some of the HIP simulations help determine the effects from nonuniform heating Figure 6. Simulations assume a constant material parameter, and stiff containers. However, the yet the reality is a distribution in parameters, a factor that current simulations are also interferes with accurate simulations. simplistic, lacking good thermal gradient effects (the powder bed thermal conductivity changes with degree of particle bonding and with temperature, and temperature is not uniform in the compact during heating, so deformation is not homogeneous). An example of the underlying materials database problem is given in Figure 6. This plots a histogram of tensile strengths for a typical stainless steel based on repeat measurements - the average is 581 MPa with a standard deviation of 20 MPa. Which value Figure 7. A typical problem with simulations traces to the should be used as the input for the underlying assumed versus actual material behavior. This room temperature strength in the plot shows measured yield strength versus test temperature simulation? To further complicate for 316L stainless steel versus the straight line model used in the problem, note in Figure 7 the the simulations, resulting in a 35% error. assumed thermal softening behavior (linear in most models) for stainless steel is a poor reflection of the actual behavior. The average error in the high temperature strength is 35%. Obviously the temperature dependence is more complicated than a straight line. To summarize, the material data are actually distributed (not single-valued), and the models for changes in these properties are poor. So if the simulation depends on quality input data and accurate rules, then it is no surprise the output is inaccurate - garbage in, garbage out!
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To further illustrate the underlying problems faced by sintering simulations, diffusion data are widely variable. Figure 8 is an example for grain boundary diffusion in tungsten. The models assume a single value straight line (coded as a frequency factor and activation energy) to capture this Arrhenius plot. However, the actual measurements show a difference between high and low angle grain boundaries, further complicated by impurity effects and measurement errors which produce a 30% variation in diffusivity at any temperature. Sintering simulations are sensitive to diffusion data, so these problems are very significant. As a demonstration, consider predictions for sintering an injection molded stainless steel powder compact heated to 1350°C for 1 h in hydrogen. Handbook data for the material properties gives a predicted 92% final density while a 10% decrease in the grain boundary diffusion activation energy gives a 96% density. The consequence is a 1.2% change in sintered dimensions, far larger that a typical tolerance of ±0.2%. Hence, small, but realistic variations in diffusion data produce product predictions errors that are larger than tolerable in practice.
Figure 8. Another illustration of basic material property scatter, showing the grain boundary diffusion data for tungsten as compared with the typical simulation assumed single line.
Sintering simulations are weak with respect to microstructure coarsening. During densification the pores and grains change size, initially the former shrink and the latter grow, but late in sintering the pores grow as well. Thus, during densification there is a varying trajectory to different elements of the microstructure which cannot be approximated by simple relations. Further, most simulations ignore the common problems associated with impurities, particle size distributions, nonuniform packing, phase transformations, and inert phases. But the simulations can predict the time, temperature, particle size, and pressure combinations that provide ful density. This is not necessarily the fastest, best, or most efficient cycle, but gives at least one solution for full densification. The solution is adequate for many situations, hence pressure-assisted sintering simulations are employed at several sites. But do not expect too much accuracy. Current simulations for pressureassisted sintering are poor at predicting dimensional accuracy, with errors that range up to 1.5%; thus, they cannot support net-shape production. SINTERING SIMULATIONS
Success in some net-shape simulations helps to define what is needed for effective sintering simulations. Several fields are ahead of sintering, including extrusion, casting, forging, blow molding, plastic injection molding, and stamping. These are older, traditional forming processes with well-established rules. Further, the boundary conditions are stable with fixed volumes and 9
controlled energy flow during the reshaping process. Probably plastic injection molding is the furthest along, in part reflecting the worldwide commercial importance of injection molding. Although there are several competitive simulations, most are based on modules for mold filling, cooling, process control, and optimization for initial set-up and tool design. In spite of widespread industrial access, recent statistics show that computer modeling of plastic injection molding has only minor impact on practice. The net reductions are about 10% in time or cost by use of mold filling simulations as compared to trial and error processes. Accordingly, less than 28% of the designers use mold filling simulations, and the results are generally considered only somewhat accurate. Thus, even for one of the most advanced net-shape processes we find the reality is that computer simulations do not substantially improve practice. If this is the situation for plastic injection molding, then what is the hope for sintering simulations? Clearly, the near-term prospects are not very good for tool and component design. We will be able to predict general trends and sensitivities, explore research options, but predictions of final sizes and microstructure will be inaccurate. There is little hope that tooling design can be based on sintering simulations. But, not all hope is lost - there will be some nearterm applications in process control using trained simulations in an adaptive learning environment. The first sintering simulations trace back Nichols and Mullins for two sphere sintering by surface diffusion [1]. This initiated several efforts to perform atomistic simulations of sintering. However, many problems became evident. A mainframe computer was required and the simulations required ten times more time than practice - to simulate 60 min of isothermal sintering required 600 Figure 9. The basic plan for a computer simulation with the core model requiring information on many parameters to min of computer time. Further, cumulative errors destroyed mass accurately provide required calculation results. conservation, hence manual intervention was required to renormalize the simulations. Construction of a computer simulation for sintering requires consideration of a large body of information. As diagramed in Figure 9, the core simulation is targeted at prediction of final size, shape, and properties. For such a simulation, the calculation reads input material properties, component specifications (typically both in the green and desired final conditions), various rules or process models, and information on the equipment. Also, relations are required between the material, equipment, processing cycle, and control options.
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My sintering simulation efforts started in the early 1970's. Based on the model of Nichols and Mullins [1], James Lathrop and I reformulated the simulation to avoid mass loss [14]. An example of the resulting two sphere sintering simulation is illustrated in Figure 10. That model was subsequently distilled into an integral neck growth equation for surface diffusion controlled sintering [28]. Following Ashby’s initial model [12], widespread atomistic simulations of solid-state sintering began to appear with access to personal computers. Kuen-Shyang Hwang presented a first model at the Notre Dame sintering meeting [30], Sierra and Lee revised the model [41], and subsequently the working model was published in 1991 [52]. Initially it focused on neck growth and showed excellent agreement with published data as shown in Figure 11. Subsequently the model was extended to predict shrinkage and surface area [60], demonstrating significant sensitivity to small shifts in activation energies. Figure 12 is an example of the sintering shrinkage during constant Figure 10. Neck profiles simulated heating, showing dilatometry data compared to simulation results with different surface diffusion and grain boundary for two spheres sintering by diffusion activation energies, QS and QB. surface diffusion [14]. More recently, John Johnson extended the ideas to multiple phase systems with excellent success as evident in Figure 13 [79]. Finally, the multiple mechanism sintering models were modified to allow for different particle sizes, shapes, green conditions (compacted or loose, agglomerated), and nonisothermal cycles [74,82]. Figure 14 is an example of the agreement between the model and experiments for nonisothermal sintering of agglomerated tantalum flakes. This
Figure 11. Neck size ratio X/D predictions for multiple mechanism sintering of copper as compared to experimental data [52].
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study led to commercialization of new powders doped to shift the sintering behavior to enable higher energy capacitors. Yang Liu converted the model into its current form SintWin 2.3, a personal computer program running in the Windows environment. Computer experiments are fairly easy once such a simulation is running. It is easier to make variations in the computer to assess process options, prior to ever performing the physical experiment. Figure 15 shows one example using Figure 12. Constant heating rate sintering shrinkage SintWin 2.3 as applied to tantalum predictions as compared to dilatometry measured powder sintering, giving the change shrinkage for alumina [60]. The simulation is very in surface area with shifts in the sensitive to the combination of grain boundary and surface particle size for 100 :m diffusion activations energies. agglomerates heated to 1350°C. Such calculations take a few minutes to set-up and execute, yet the alternative of classifying flakes, agglomerating, compacting, sintering, and measuring the response would require weeks. In validation tests for these simulations the experimental results gave 5.1% shrinkage (±0.3%) while the simulation predicted 4.4% shrinkage. Although the simulations are not highly accurate, the cost of experimentation was a major advantage. Substantial progress has taken place in the computational infrastructure. Since 1974 when Ashby [12] proposed multiple mechanism atomistic simulations for predicting density as a function of the processing parameters, the calculations have become thousands of times faster and more accurate [74,79,82,87,94]. Hardware to support modeling is now commonplace and affordable. The new programs make predictions of final dimensions within seconds. And prove to be nominally close to those measured after sintering. Achieving this level of sophistication comes after much work. However, much still needs to be done since there are errors in the predictions.
Figure 13. Density during heating and hold cycle for tungsten-copper as measured and compared to computer simulated results [79].
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As we look forward, some of the advances necessary for improved sintering simulations become evident. The short list for improved accuracy are as follows: Î allow for nonspherical powders Ï include nonisothermal heating and cooling effects Ð allow for inclusion of dopants, impurities, and atmospheres chemistry and flow Ñ include compaction effects and green density gradients Ò allow for range of materials - alloys, mixed powders, and even inert phases Ó include reactions, melting, transformations, homogenization, dissolution events Ô move to three-dimensional microstructure models Õ allow for agglomerated and dispersed powders.
Figure 14. Comparison of simulated and experimental results for sintering shrinkage and surface area loss for agglomerated tantalum powders based on the SintWin simulation [82].
Figure 15. An example of computer experiments using SintWin to examine different particle size effects on the surface area loss for an agglomerated tantalum powder in an industrial heating cycle.
Even with such improvements, the simulations cannot be any more accurate than the input data. For example, consider that even knowing the starting particle size distribution and green density, there is no knowledge on the pore size distribution. Yet we know monosized pores sinter much better than widely dispersed pores [70], large pores coarsen while small pores shrink. This is characteristic of the problem - accurate simulations require new knowledge. Another weakness comes from the time-dependent changes in material properties as temperature and microstructure change. Accuracy is limited by our knowledge on polymer burnout effects (carbon potential and thermal expansion), thermal softening, phase transformations (especially in mixed powder systems where local and global effects differ), grain growth, poregrain boundary interactions, annealing, and recrystallization. Only recent models have taken on these complexities [73,80,83,84,87,91,93,97,98].
In a sense, computer modeling fumbles because basic materials science still must discover several underlying relations. These include diffusion data errors, heterogeneities in the green microstructure, surface emissivity changes during densification, and 13
material property changes with sintering, especially events associated with pore and grain coarsening. Final dimension predictions are no more accurate than the input parameters, hence the simulations struggle because of the simplifications imposed by our current basic knowledge. For example, density gradients and powder size segregation in the green body dominate dimensional warpage in sintering. However, we have poor measures of these attributes. Consider the measured density distribution and simulated green distribution shown earlier in Figure 4. A significant difference exists because the simulation assumes constant die wall friction, which is invalid. Thus, to accurately simulate sintering warpage requires tedious measurement of the actual density distribution. This is an unattractive option. So no matter how good our constitutive models, the reality is that the knowledge infrastructure is inadequate to support accurate simulation of the sintered dimensions, at least at the level required in production. A recent focus in computer simulations has been on the more industrially relevant liquid phase sintering processes, including supersolidus sintering [67-71,78-89,97]. Many of the efforts integrate rheological models into finite element packages to predict final size and shape. A further objective is to integrate compaction and sintering models into an inverse solution package that allows tool design for final product attributes. A parallel effort is focused on sintering simulations in support of powder injection molding; Debby Blaine will present some of that work in this same meeting. Figure 16 provides an example finite element simulation of predicted distortion in a thick-thin injection molded sample. Although visually exciting, the accuracy of these predictions is still questionable. A related development is the prediction of in situ and sintered strength [96]. This follows from early studies on strength evolution during sintering, realizing that knowledge on in situ strength can be derived from SintWin sintering simulations. The success of those efforts is demonstrated in Figure 17. Further, Figure 18 shows the links to prediction of sintered strength for two hold times. The data are within 10% of the predicted strengths. Hence, sintering simulations can make useful predictions of engineering properties, but the accuracy is still low. Since the simulation is no better than our basic models and material property data, work is going into nondestructive measurements to expedite generation of in situ data during sintering. We recognize sintering computer simulations require models that embrace the multiple events of Figure 16. Finite element simulation of surface diffusion, grain boundary diffusion, lattice distortion during sintering, in this case for diffusion, dislocation climb, plastic flow, a four-step injection molded stainless steel evaporation-condensation, liquid phase induced component with thick-thin sections. rearrangement, grain shape accommodation, viscous flow, and other means for moving mass. Sintering is not isothermal, yet many simulations 14
Figure 17. Comparison of experimental (points) and simulated in situ strength for bronze during various sintering cycles [96].
assume constant temperature. As illustrated in Figure 13 most sintering happens on the way to the peak temperature. Many commercial sintering systems start with mixed powders (and these might be as simple as graphite as a carbon source in a steel). If the additive melts or reacts, then there is a corresponding dimensional change, and shift in the transport kinetics and microstructure [87,94]. The particle geometry is not uniform, so inhomogeneous microstructures are inherent to sintering. Further, because of thermal softening during heating, gravity is a significant factor in many situations, especially those with liquid phases sintered to near-full density. Finally, we recognize the important role of the sintering atmosphere with respect to impurities and heat transfer, so realistic simulations need to include the atmosphere composition and flow [76]. A variant is now emerging where the simulation will run in parallel with the process controller, providing guidance on process changes and the implications with respect to product quality [104].
Figure 18. Comparison of the computer simulated sintered room temperature strength for bronze sintered The classic sintering simulation assumes for no hold or 60 min hold at various peak sintering the initial geometry, calculates the temperatures [96]. gradients in that geometry, calculates the motion or rate of change, rearranges mass corresponding to a small time advance, and then repeats the cycle with the new geometry, temperature, and time. Several simulations exist for such processes, but the user must be cautions and closely examine the details. Do the simulations match with the “best practices” in computer models? Some standard tests for sintering simulations include the following: U the simulation must conserve mass and volume as appropriate U the simulation must allow for nonuniform particle packing, including agglomerates U the simulation must allow for nonspherical particles U the gains and pores must be treated with size distributions, not averages U grain growth, pore pinning, and pore coarsening must be included in the model U the simulation must allow for heating, holds, and cooling U compaction and the initial neck size must be adjustable to reflect real powder compacts U material properties must be functionally related to the cycle, including thermal 15
softening U sintering stress must be calculated for the actual microstructure, not an assumed case U the important events associated with polymers, including burnout, should be included U delayed neck formation must be included with respect to densification U mixed powders, inert particles, chemical dissolution, and alloying must be allowed U contamination, atmospheres, inclusions, and many nonideal aspects are desirable. The good news is that we are about 50% of the way through this list in the more advanced computer simulations. MERITS OF COMPUTER SIMULATIONS This presentation is loaded with scepticism. However, to be fair, we must appreciate the gains generated by computer simulations of net-shape processes and sintering specifically. In spite of limited accuracy and poor attention to industry’s priorities, there is still considerable value. Some of the important gains include the following: ” lower development costs in conceptualization of new processes ” avoidance of early mistakes in the planning stage ” compressed start-up time and better initial design of experiments ” help in early focus on critical problems and knowledge gaps ” reduced tooling errors and improved compatibility in equipment and processes ” help in identifying process sensitivities, such as setting pro forma standards. But most important, simulation helps confirm that the process is understood. This is the main goal and justification for continued efforts. A few examples have been given here - confirmation of basic knowledge and knowledge gaps, assessment of options and process sensitivities via the “computer experiments”, and coupling of simulations with adaptive learning schemes to generate closed-loop and feed-forward control systems. Since ancient times, universities have played a role as repositories of human knowledge. In the modern era this role now extends to making contributions to increase our knowledge. Computer are essentially fast means of following rules. If the instructions are wrong, then the results are wrong. Hence, computer simulations of sintering are only reflecting our current knowledge, confirming what we know and do not know. Consequently, computer simulations are an appropriate university activity, and through these frustrations the community is defining what is needed in future studies. Until the simulations catch up with reality, they will remain the providence of the universities. FUTURE DIRECTIONS AND PROSPECTS An assessment of the current situation and future prospects comes from a look at where sintering simulations succeed when compared to industrial requirements. Table 1 provides this contrast by ranking the success of the sintering simulations versus the hierarchy of industry desires in sintering predictions. It is easy to see that the industrially relevant topics have not received much attention.
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Table 1. Comparison of Industry Desires and Sintering Computer Simulations Successes ranking 1 2 3 4 5 6 7
simulation successes microstructure neck size, shrinkage, surface area average density (isotropic) relative dimensions warpage, distortion sintered strength fracture path
industry needs cost final dimensions, sizes, shape tool dimensions, setup, and motions warpage, distortion, precision defect generation and avoidance equipment operation, control sintered properties
Are the industry goals incompatible with current sintering models? Essentially the answer is yes. Tolerances in production are at the ± 0.2% range and sintering models tend to be accurate in the ±2 to 5% range, with a few possible of predictions within ±1% of final dimensions (for selected materials and cycles). This tenfold difference makes use of computer simulations questionable for any process design; we still need to run the part and make corrections. The situation is even more dramatic when looking at tool design, where tolerances are tighter. This is one of the reasons whey rapid prototyping fails to displace classic machining - the required tolerances are just too tight for the existing technologies. Where do the barriers come from and where should we put our attention? One suggestion comes from examination of how the models handle microstructure evolution. Most simulations treat the average properties of grain size, porosity, pore size, and neck size. Yet each of these is a distributed property which requires consideration of the mean and its variance. The simulations use average properties because the alternative is too complex; consider in some systems (tungsten heavy alloys come to mind) every sintered grain consists of thousands of initial particles. For statistical accuracy we need a few hundred final grains to calculate the grain size distribution, but this implies thousands of initial particles. Similar problems are evident elsewhere in simulations. Hence, after 30 years of effort in computer simulation of sintering, I feel we now appreciate the limits. Although the near-term prospects might seem dismal, we also must face reality - as Professor Yee Cheong Lam of Nanyang Technological University likes to say, part of the computer modeler’s job is “expectation management” or do not let the customer have unrealistic goals. Figure 19 plots the relative gains in knowledge on sintering by number of publications per decade and shows the parallel growth in computer simulation knowledge. Sintering knowledge is growing at about 10% per year, but modeling knowledge is growing at nearly twice that rate. The computers are up to the task of quickly simulating sintering behavior, but the knowledge infrastructure is far behind. Until that gap closes, simulations will give rough estimations to help isolate new directions, Figure 19. Knowledge gains on sintering and but will not enable final process or tool sintering simulations are measured by publication design. There are too many uncontrolled rates. 17
variables. Adaptions in manufacturing naturally occur to compensate for the many factors not embedded in the simulations. However, in this problem area simulations are beneficial. They allow quick assessment of alternatives and help establish sensitivities to material or process variations; this is a means to couple specification alternatives with direct economic or technical Figure 20. Schematic overview of how process simulation will impact assessments. link a computer simulation to a net-shape operation, such as sintering. Here the simulation would be constantly corrected, It is possible some of the allowing it to learn and provide intelligent control. current models can be endowed with improved accuracy. Much discussion has taken place on intelligent processing. Here a model will be given equipmentmaterial specific knowledge, allowing constant control of the process to ensure quality. The idea of a closed-loop feedback control system is outlined in Figure 20. One proposal by Nandi and Ruhe [104] is to use the sintering simulations with an overlay of adaptive learning to provide online process control. Here a fast simulation would learn its errors when compared to practice and adjust over time to better reflect reality. This solution is viable for use in industrial process control where the materials, cycles, and products undergo modest changes over time. CONCLUSIONS Sintering computer simulation accuracy is not very good, independent of the constitutive equations, computer and software, and programming sophistication. This is due to an underdeveloped knowledge in the underlying materials science events associated with sintering; the problem ranges from raw material variations to fundamental diffusion data. Many of the events associated with sintering are still treated as simplifications. Accurate simulations require more detail, better models, and improved conceptualizations. The raw materials are not homogeneous, isotropic spheres, so the models must move to include normal material variations and departures from reality. Unfortunately, the simulation sensitivity to such variations is large. Remember the models are not real, just mathematical approximations to what we understand as the basic rules. Computer simulations of sintering require a large body of input data that needs to be far more accurate than currently available. Likewise, normal variations in equipment, tooling, and raw materials induce further statistical variations. Modeling is far behind practice, but the gap is closing. Even so modeling is defensible since it forces the organization of sintering knowledge and over time that makes all of us involved in sintering smarter. For the next several years, computer modeling will remain under the control of the academic community because of the underlying complexity and the fact that industrial needs far exceed our basic knowledge.
18
ACKNOWLEDGMENTS During the past 30 years, funding for various sintering simulation efforts was provided by Sandia National Laboratories, National Aeronautics and Space Administration, National Science Foundation, Department of Energy, U. S. Army Research Office, Cabot, Allegheny Ludlum, Kulite Tungsten, Advanced Powder Processing Consortium, Powder Metallurgy Consortium, BASF, Ben Franklin Technology Center, Honeywell, Defense Advanced Research Projects Administration, and most recently by the Center for Innovative Sintered Products at Penn State. The guidance and financial support by this collection of sponsors is most appreciated. This manuscript was written while serving as the Nanyang Professor in the School of Mechanical and Production Engineering at Nanyang Technological University during 2001. Much credit for the progress goes to my several students, research associates, and faculty partners. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
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