Modeling and simulation of MIM(11.3-11.5)

Table of Contents

• 11.3 Modeling and simulation of the thermal debinding process

• 11.3.1 Theoretical background and governing equations MDC for a single reaction step

• 11.3.2 Applications

• 11.4 Modeling and simulation of the sintering process

• 11.4.1 Theoretical background and governing equations

• 11.4.2 Experimental determination of material properties and simulation verification

• 11.4.3 Applications

• 11.5 Conclusion

11.3 Modeling and simulation of the thermal debinding process

As PIM is a binder-assisted forming technique, removal of the binder without loss of product integrity is a crucial point. The process of binder removal is termed debinding. A multicomponent binder can overcome distortion problems by leaving one component in the compact to hold the particles in place while a lower stability component is removed. Often, the lower molecular weight binder component is dissolved into a fluid in a process termed solvent debinding. Thermal debinding, a conventional heat treatment in a gas atmosphere, is used to remove higher molecular weight components of the binder systems. Previously, debinding cycles were based on "trial and error" until an adequate time-temperature cycle was achieved. With the use of a multicomponent binder system, achieving an economical and effective debinding cycle while maintaining the compact's shape can be quite difficult. To counter this, thermogravimetric analysis (TGA) and differential thermal analysis (DTA) are used to determine the weight percent of the binder component left as a function of time and temperature. This protocol is to show heating at points of rapid weight loss to avoid compact damage. This expedites the debinding process and lowers the overall cost. But still, a large number of TGA and DTA experiments are needed to optimize the percentages of each binder component in a multicomponent binder system. To minimize TGA experiments, a master decomposition curve (MDC) is proposed.To make a part consistently and repeatedly by MIM, every step in the MIM process must be repeatable, from the powders used to the sintering step. In each step, there are several process variables. These variables must be controlled within a window which yields the desired results. In case of debinding, every bit of the binder must be removed, which means there is a minimum processing time and any additional time beyond the minimum has almost no negative effect. In case of sintering, which is controlled by diffusion rates for the different materials, time and temperature are the major variables. The final result after sintering of the parts must produce the correct part dimensions. This should also result in the correct density and shrinkage that produce the correct physical properties. Continuation of the sintering time beyond a certain level has little effect on the density but could have a profound effect on grain growth. Extremely large grains could have a detrimental effect on some physical properties.

Fig. 11.7 Filling time optimization by using a CAE tool for PIM: (A) mesh used for optimization study; (B) optimum filling time finding which minimizes injection pressure.

11.3.1 Theoretical background and governing equations MDC for a single reaction step

Depolymerization of polymeric binders can be described by first-order reaction kinetics. The remaining weight fraction of a polymer, a, is expressed as

where t is the time and K is the rate constant for thermal degradation and follows the Arrhenius equation

in which k0 is the specific rate constant, Q is the apparent activation energy for thermal degradation, R is the gas constant, and T is the absolute temperature. Combining Eqs. (11.22), (11.23), and integrating

where Q is termed the work of decomposition, defined as follows:

The remaining weight fraction of a polymer, α, is related to the work of decomposition Q from Eq. (11.24) as follows:

The previously mentioned equation defines the MDC, wherein we can merge the decomposition curves for a given polymer with any decomposition cycles using the concept of work of decomposition, Q.

Calculation of apparent activation energy

To calculate the work of decomposition, Q from Eq. (11.25), apparent activation energy, Q needs to be determined. The activation energy for polymer decomposition can be determined from TGA data in a method developed by Kissinger (1957). The Kissinger method utilizes the temperature Tmax at which maximum rate of weight loss occurs at various heating rates as follows:

at the maximum rate of weight loss.

Under the condition of constant heating rate r, that is dT/dT¼r, Eq. (11.27) can be expressed as

or

Thus from Eq. (11.29), we can plot a graph between In[r/(Tmax)2]and-1/RTmax from the TGA experiments with several constant heating rates. The slope of this plot gives the apparent activation energy, Q of the reaction.

MDC for multireaction steps

Typically, the TGA curves of polymer degradation follow a single sigmoid path. But multicomponent binder systems may have two or more sigmoids due to the different molecular weights, bonding groups, and degradation paths of various polymer components. Each sigmoid represents a rate-controlling step whose activation energy differs from those of other controlling steps. Powder-binder systems tailored for PIM often consist of several components. These components may be categorized mainly into two groups according to their molecular weight. Low molecular weight ingredients, such as solvents and plasticizers, either evaporate or are decomposed at rather low temperatures. High molecular weight polymers, possessing higher thermal stability, decompose at relatively high temperatures. Consequently, the TGA curve often has morethantwo sigmoids. One sigmoid ofthis curve could be a superposition of several sigmoids that have similar activation energies. Two sigmoids of this curve could mean two groups of reaction steps with sufficiently separate activation energies. Each sigmoid may be described by three kinetic parameters in an Arrhenius-type equation. These parameters are reaction order, activation energy, and frequency factor. Some powders may have catalytic effects onthe pyrolysis rate; however, the shape of the pyrolysis curve with powders is similar to that without powders. Therefore, the mathematical form of polymer pyrolysis can still be applied. Two polymers of the binder system used in this study are decomposed during the thermal debinding process.

where a is the mass ratio of the initial masses of two polymers, a1 is the mass ratio to initial mass of low molecular weight polymer, a2 is the mass ratio to initial mass of high molecular weight polymer, and w is the ratio of initial mass of low molecular weight polymer to initial mass of the two polymers.

The mathematical form that is generally applied to the TGA curve of polymer is modified to describe the TGA curve of organic pyrolysis with powders. In the present analysis, we assume that the kinetics are solely controlled by the temperature.

where Tt is the transition temperature between the first and second sigmoids, subscripts 1 and 2 on α, Q, and k0 denote low- and high-temperature degradation, respectively. A decomposition kinetic curve with more than two sigmoids can be expressed in a similar manner. The Kissinger method is applied to estimate the activation energy, as described in the previous section.

The left-hand side of Eq. (11.31) is a function of only the mass ratio α and material properties except Q, which then becomes

The right side of Eq. (11.31) is denoted by

Table 11.1 Characteristics of binder components used in the binder system

which depends only on Q, Tt, and the time-temperature profile. Note F(a) is a characteristic that quantifies the effects of binder system components on the decomposition kinetics. The relationship between α and F(α) is defined as the MDC, which is unique for a given powder and binder system and is independent of the decomposition path, given the assumptions described earlier.

11.3.2 Applications

Calculation of activation energy

A typical example of wax-polymer binder used in PIM is listed in Table 11.1 with characterization of each binder component. TGA is a mass change method in which the mass of the sample is measured as a function of temperature while the sample is subjected to a controlled temperature program. This can be achieved as a function of the increasing temperature or isothermally as a function of time with a controlled atmosphere. The TGA is performed with results for each binder component at three different heating rates to calculate the activation energy by the Kissinger method. This activation energy is then used to develop the MDC.

Fig. 11.8A shows the TGA of polypropylene at three different heating rates. It is evident that the decomposition range is 350–490°C. Fig. 11.8B gives the peak temperatures at maximum weight loss of polypropylene. These are plugged into Eq. (11.29) to plot a graph between In[r/(Tmax)2] and -1/RTmax(Fig. 11.8C). The slope of this graph gives the apparent activation energy for polypropylene. Table 11.1 gives the material parameters of each binder component for plotting the MDC.

Fig. 11.8 TGA results and Kissinger method for calculating activation energy for polypropylene (Aggarwal et al., 2007). (A) TGA result. (B) Temperatures at maximum rate of weight loss. (C) Temperature dependency.

MDC for single reaction-step decomposition

Eqs. (11.31)-(11.33) are then used to plot the MDC for polypropylene (Fig. 11.9). Polypropylene exhibits a similar decomposition behavior at all heating rates. Similarly, MDCs for other binder components (paraffin wax, polyethylene, and stearic acid) were generated from their respective TGAs (not shown here) based on calculated activation energies by the Kissinger method. The overall MDC for the binder system is plotted from the individual MDCs of its components. Individual decomposition behavior of binder components is important in basic understanding but decomposition behavior of the whole binder system is of more interest in PIM. This behavior is shown in Fig. 11.10 for the binder system. It can also be seen that the model is in close agreement with the experimental results. In addition, the remaining weight of each binder component can be predicted and monitored at any time-temperature combination based on the constructed MSCs.

Fig. 11.9 MDC for polypropylene, showing all TGA.

Fig. 11.10 Synthesis of overall decomposition behavior for the binder system

MDC for multireaction step decomposition

The decomposition behavior of the feedstock during thermal debinding is imperative in deciding the debinding cycle for the injection molded parts. This makes MDC for the feedstock an attractive tool in both cost and effort optimization. To demonstrate this concept and find out the processing parameters for debinding, niobium feedstock (57% solids loading) is used. Fig. 11.11 shows the niobium feedstock MDC. A two-step decomposition behavior is observed in which during the first step, the lower molecular weight binder components (wax and stearic acid) decompose and later the higher molecular weight binder components (polypropylene and polyethylene) go away. It is also observed that the decomposition is rate dependent, i.e., at a particular temperature more binder remains at a higher heating rate. This is not the case for the pure binder. There is an effect of powder on the decomposition behavior.

Fig. 11.11 Multireaction step MDC of Nb feedstock (Aggarwal et al., 2007). (A) TGA for feedstock 1. (B) MDC for feedstock 1.

Effect of metal powders

Fig. 11.12 shows the MDC of solvent debound samples of pure binder (D-Binder), Nb feedstocks (D-Nb), and 316L stainless steel feedstock (D-316L). It can be observed that for a particular binder wt%, a lower temperature (here higher work of decomposition, Q) is required for the feedstocks (D-Nb and D-316L) than just the binder without any powder. This seems reasonable as the presence of powder will enhance the debinding process and a higher work of decomposition (lower temperature) will be needed, which results from the catalytic effect or faster heat transfer due to higher thermal conductivity of the metal powder (54W/m K for Nb, 12W/m K for 316L, versus 0.1W/m K for typical polymer). Another difference can also be considered between MDCs of D-Nb and D-316L. This could also explain the effect of powder shape on the debinding process. D-Nbis a niobium feedstock which hasirregularly shaped Nb particles whereas D-316L is a 316L feedstock having spherical shaped powder particles. Both feedstocks have the same solids loading (57 vol%). Spherical powder particles have a lower surface area as compared to the irregular shaped powders. It is also evident from the MDC (Fig. 11.12) that lower work of decomposition is required for debinding D-Nb (irregular) than D-316L (spherical shape). Finally, it is concluded that the catalytic, thermal conductivity, and particle shape effects are combined in decomposition behavior of the binder system with metal powder. Detailed analyses will be required for more accurate explanation.

Fig. 11.12 An illustration of catalytic effect of metal powders on decomposition behavior 

Weight-temperature-time plot

Fig. 11.13 shows a weight-temperature-time plot for decomposition of solvent debound 316L feedstock in hydrogen atmosphere. Based on this information, the required time hold can be predicted for a given temperature and targeted binder weight loss. This will help in reducing the experimentation required to determine the optimum debinding cycle.

Fig. 11.13 Weight-temperature-time plot for decomposition of solvent debound 316L feedstock

11.4 Modeling and simulation of the sintering process

Effective computer simulations of metal powder sintering are at the top of the powder metallurgy industry's wish list. Since the pressed green body is not homogeneous, backward solutions are desiredto selectthe powder,mixing,injectionmolding, debinding, and sintering attributes required to deliver the target properties with different tool designs, machines, and processing conditions. In building toward this goal, various simulation types have been evaluated: Monte Carlo, finite difference, discrete element, finite element, fluid mechanics, continuum mechanics, neural network, and adaptive learning. Unfortunately,theinput data and some ofthe basic relations are not well developed; accurate data are missing for most materials under the relevant conditions. The simulations help to define the processing window and set initial operating parameters.

11.4.1 Theoretical background and governing equations

The methodologies used to model the sintering include continuum, micromechanical, multiparticle, and molecular dynamics approaches. These differ in length scales. Among the methodologies, continuum models have the benefit of shortest computing time, with an ability to predict relevant attributes such as the component density, grain size, and shape.

Constitutive relation during sintering

Continuum modeling is the most relevant approach to modeling grain growth, densification, and deformation during sintering. Key contributions based on sintering mechanisms such as surface diffusion, grain boundary diffusion, volume diffusion, viscous flow (for amorphous materials), plastic flow (for crystalline materials), evaporationcondensation, and rearrangement. For industrial application, the phenomenological models are used for sintering simulations with the following key physical parameters.

• Sintering stress is a driving force of sintering due to interfacial energy of pores and grain boundaries. Sintering stress depends on the material's surface energy, density, and geometric parameters such as grain size when all pores are closed in the final stage.

• Effective bulk viscosity is a resistance to densification during sintering and is a function of the material, porosity, grain size, and temperature. The model of the effective bulk viscosity has various forms according to the assumed dominant sintering mechanism.

• Effective shear viscosity is a resistance to deformation during sintering and is also a function of the material, porosity, grain size, and temperature. Several rheological models for the effective bulk viscosity are available.

The previously mentioned parameters are a function of grain size. Therefore, a grain growth model is needed for accurate prediction of densification and deformation during sintering.

Typical initial and boundary conditions for the sintering simulations include the following:

• initial condition: mean particle size and grain size of the green compact for grain growth and initial green density distribution for densification obtained from compaction simulations;

• boundary conditions: surface energy condition imposed on the free surface and friction condition of the component depending on its size, shape, and contact with the support substrate.

The initial green density distribution within the pressed body raises the necessity to start the sintering simulation with the output from an accurate compaction simulation, since die compaction induces green density gradients that depend on the material, pressure, rate of pressurization, tool motions, and lubrication. The initial and boundary conditions help determine the shape distortion during sintering from gravity, nonuniform heating, and from the green body density gradients.

Numerical simulation

Even though many numerical methods have been developed, the FEM is most popular for continuum models of the press and sinter process. The FEM approach is a numerical computational method for solving a system of differential equations through approximation functions applied to each element, called domain-wise approximation. This method is very powerful for the typical complex geometries encountered in powder metallurgy. This is one of the earliest techniques applied to materials modeling, and is used throughout industry today. Many powerful commercial software packages are available for calculating 2D and 3D thermomechanical processes such as those found in the sintering process. To increase the accuracy and convergence speed for the sintering simulations, developers of the simulation tools have selected explicit and implicit algorithms for time advancement, as well as numerical contact algorithms for problems such as surface separation, and remeshing algorithms as required for large deformations such as seen in some sintered materials, where up to 25% dimensional contraction is possible.

11.4.2 Experimental determination of material properties and simulation verification

In the development of a constitutive model for the sintering simulation, a wide variety of tests are required, including data on grain growth, densification (or swelling), and distortion. These are approached as follows.

• Grain growth: quenching tests are conducted from various points in the heating cycle and the mounted cross sections are analyzed to obtain grain size data to implement grain growth models. A vertical quench furnace is used to sinter the compacts to various points in the sintering cycle and then to quench those compacts in water. This gives density, chemical dissolution (for example, diffusion of one constituent into another), and grain size as instantaneous functions of temperature and time. The quenched samples are sectioned, mounted, and polished prior to optical or scanning electron microscopy (SEM). Today, automated quantitative image analysis provides rapid determination of density, grain size, and phase content versus location in the compact. Usually during sintering the mean grain size G varies from the starting mean grain size G0 (determined on the green compact). A new master sintering curve concept is applied to fit the experimental grain size data to an integral work of sintering, since actual cycles are a complex combination of heats and holds. The resulting material parameters trace to an apparent activation energy as the only adjustable parameter. Fig. 11.14 shows an SEM micrograph after quenching test and grain growth modeling for a W-8.4wt% Ni-3.6 wt% Fe mixed powder compact during liquid phase sintering (LPS).

• Densification: to obtain material parameters for densification, constant heating rate dilatometry is used for in situ measurement of shrinkage, shrinkage rate, and temperature. By fitting the experimental data to models that include the sintering stress σs and bulk viscosity K as functions of density and grain size, again relying on the master sintering curve concept, the few unknown material parameters are extracted. Fig. 11.15 shows the dilatometry data and model curve-fitting results used to obtain the parameters during sintering of a 316L stainless steel.

• Distortion: powder metallurgy compacts reach very low strength levels during sintering. Accordingly, weak forces such as gravity, substrate friction, and nonuniform heating will induce distortion and even cracking. To obtain the material parameters related to distortion, three-point bending or sinter forging experiments are used for in situ measurement of distortion. By fitting the experimental data with FEM simulations for shear viscosity M with grain growth, the parameters such as apparent activation energy and reference shear viscosity are extracted. Fig. 11.16 shows in situ bending test and FEM results for obtaining material parameters in shear viscosity for a 316L stainless steel powder doped with 0.2% boron to induce improved sintering.

Fig. 11.14 (A) SEM of a liquid phase sintered tungsten heavy alloy after quenching. (B) The grain size model results taken from an integral work of sintering concep

Fig. 11.15 Dilatometry data showing in situ shrinkage data during constant heating rate experiments and the curve-fitting results used to obtain the

Fig. 11.16 (A) A video image taken during the in situ bending test for a 316L stainless steel sample doped with 0.2 wt% boron. (B) The FEM model results used to verify the shear viscosity property as a function of time, temperature, grain size, and density during heating

Such data extraction techniques have been allied to several materials, ranging from tungsten alloys, molybdenum, zirconia, cemented carbides, niobium, steel, stainless steel, and alumina. 

11.4.3 Applications

Gravitational distorting in sintering

The rheological data for the sintering system allow the system to respond to the internal sintering stress that drives densification and any external stress, such as gravity, that drives distortion. When a compact is sintered to high density, it is also necessary to induce a low strength (the material is thermally softened to a point where the internal sintering stress can induce densification). Fig. 11.17 shows sintering simulation results for a tungsten heavy alloy, relying on test data taken on Earth and under microgravity conditions, to then predict the expected shapes for various gravitational conditions-Earth, Moon, Mars, and in space. The results show that gravity affects shape distortion during sintering. Accordingly, the computer simulations can be used to reverse engineer the green component geometry to anticipate the distortion to achieve the desired sintered part design.

(A)                                                                         (B)

Fig. 11.17 Final distorted shape by sintering under various gravitational environments for complicated test geometries (A) T-shape. (B) Joint part

Sintering optimization

Usually a small grain size is desired to improve properties for a given sinter density. In this illustration, the design variable is the sintering cycle. To obtain maximum density and minimum grain size, the following objective function F is proposed

Fig. 11.18 Minimum grain size for a given final sinter density and the corresponding sintering cycle for achieving this goal in a 17-4-PH stainless steel. (A) Sintered density and minimum grain size. (B) Sintered cycle.

where a is an adjustable parameter. Fig. 11.18 shows an example for maximum density and minimum grain size for a 17-4-PH stainless steel powder. For example, the minimum grain size will be 21.9μm if the specified sintered density is 95% or theoretical. Fig. 11.18B shows the corresponding sintering cycle by matching the value of adjustable parameter a.

11.5 Conclusion

Computer simulations of the PIM have advanced considerably, and in combination with standard finite element techniques show a tremendous ability to guide process setup. Illustrated here are the PIM concepts required to perform process optimization. Although the models are only approximations to reality, still they are of value in forcing a careful inspection of what is understood about the PIM process. In this regard, the greatest value of modeling is in the forced organization of process knowledge. There remain several barriers to widespread implementation. The largest is that traditional powder metallurgy is largely dependent on adaptive process control since many of the important factors responsible for dimensional or quality variations are not measured. The variations in particle size, composition, tool wear, furnace location, and other factors such as reactions between particles during heating, all impact the important dimensional control aspects of PIM, although nominal properties, such as strength, hardness, or fatigue life are dominated by the average component density. In this regard, especially with respect to the initial process setup, the computer simulations are of great value. Still, important attributes such as dimensional tolerances and internal cracks or other defects are outside the cost-benefit capabilities of existing simulations. Furthermore, the very large number of materials, processes, tool materials, sintering furnaces, and process cycles makes it difficult to generalize; significant data collection is required to reach the tipping point where the simulations are off-theshelf. Thus, much more research and training is required to move the simulations into a mode where they are widely applied in practice. Even so, commercial software is available and shows great value in the initial process definition to set up a new component.