An Experimental Investigation of the Solid State Sintering of Cemented Carbides Aiming for Mechanical Constitutive Modelling

Abstract

The densification of cemented carbides during sintering was studied using an existing constitutive model based on powder particle size and material composition. In the present analysis, we study how well the constitutive model can capture the experimental results of a dilatometer test. Three experiments were performed, where the only difference was the transition between the debinding and sintering process. From magnetic measurements, it is concluded that the carbon level in the specimen is affected by changes to the experimental setup. It is shown, using parameter adjustments, that the constitutive model is more suited for a certain experimental setup and carbon level, which is a limitation of the model. In order to capture the mechanical behaviour under different experimental conditions, further constitutive modelling relevant to the carbon level is recommended.

1. Introduction

Tungsten carbide-cobalt (WC-Co), a cemented carbide, is an exceptionally hard and tough crystalline material used, for example, in drilling and cutting tools. The powder consists of tungsten carbide, cobalt, alloys and a pressing agent. The raw materials are milled and spray-dried to a powder with good flow properties. After spray drying, the powder is compacted to roughly half its sintered volume. Because of friction between the powder and the pressing tool, the density after compaction is uneven, leading to uneven shrinkage during sintering.

After pressing, the powder blank is sintered with liquid-phase sintering. However, a significant part of the densification previously occurs in the solid state, before the cobalt melts, when the surface energy is reduced as particles come into contact and merge, forming necks between each other.

To obtain the right shape after pressing and sintering, the pressing tool must, in most situations, be compensated for. This is a both expensive and time-consuming procedure, but by performing computer simulations of the manufacturing process, the shape after sintering can be predicted and the pressing tool can be compensated before manufacture. Here, the modelling of sintering is considered.

Different continuum models have been presented in order to capture the sintering behaviour of cemented carbides [1,2,3,4,5,6]. Here, the model developed by Brandt et al. [1] is chosen. An important reason is its availability in the commercial finite-element program LS-Dyna [7]. The model describes both compaction and sintering in a coupled manner. Consequently, it is a complete model aiming in full at the behaviour of cemented carbides. The compaction model has already been evaluated and improved by both Andersson et al. [8] and Staf et al. [9]; however, the sintering part still requires improvement.

The sintering model can be divided into a volumetric part and a deviatoric part. To simplify the model in the present study, only the volumetric part is studied. Pertinent results derived using such an approach were presented in [10]. There, it was concluded that the initial densification starts earlier than that which is possible to simulate with the chosen constitutive model. This was believed to be due to experimental differences between the experiment used to develop the constitutive model by Brandt et al. [1] and the experiment developed to represent an industrial situation.

Notably, the debinding step, in which the pressing agent is evaporated, was performed separately by Brandt et al. [1] and not included for the development of the constitutive model. In the work by Rosenblad et al. [10], however, the debinding step is included. The difference in behaviour indicates that the difference between the two experimental methods will affect the densification. Performing a separate debinding step will also lead to increased manufacturing time and costs. The oxygen can also interact with the powder, changing the material properties. The main purpose of having a sintering model is to predict the actual process; hence, there is a need to understand why the experimental data differ and to determine which experimental setup is preferrable in future experiments.

A dilatometer is commonly used to perform in situ strain measurements for a sintering process [11]. The dilatometer differs from the industrial furnace used, which limits the applicability of the results gained using the dilatometer. For example, in the industrial furnace, elements such as carbon can be added into the closed chamber by means of a suitably chosen sintering atmosphere to balance losses during the sintering process. In the dilatometer machine, carbon is not added during the sintering process. The carbon level was previously studied in WC-Co powder, where it was shown to affect the wetting between tungsten and cobalt, as well as the liquidus temperature [12].

With the above as a background, the implemented constitutive model is investigated, aiming towards improvements relying on the present experimental results. Obviously, such improvements must be based on the identification of important physical features not included in the present model. A further issue is to identify which experimental setup is most suitable to use in order to obtain the material parameters needed to accurately simulate the sintering of cemented carbides. The resulting carbon level is, in this case, a quantity to account for in a detailed manner in mechanical modelling. Obviously, an understanding of the importance of different features impacting on the mechanical behaviour is essential when building relevant constitutive models and, again, this is the aim of the present study.

As a final remark, it should be mentioned that the modelling of powder compaction and sintering is often carried through a micromechanical approach relying on analytical [13,14] or numerical methods, such as the discrete element method (DEM) [15,16,17,18]. Such models are based on mechanical behaviour during particle–particle contact, most often assuming that each particle contact is mechanically independent of the neighbouring contacts. At the moment, however, DEM modelling for the simulation of whole and complicated insert geometries requires extremely large computer capacity (if it is possible at all). Consequently, a certain type of macroscopic approach has to be relied upon in this situation in order to describe the powder compaction and sintering process completely in an accurate (and practical) manner. As is hopefully obvious from the discussion above, such an approach is also relied upon here, and the macroscopic constitutive model is described in detail below.

2. Constitutive Model

The constitutive model used was developed by Brandt et al. [1] and is implemented in the commercial finite-element program LS-DYNA [7]. It was improved in the work conducted by Rosenblad et al. [10]. The model is based on the concept of a sintering stress ${\sigma }^s$, which is the driving force for densification. The densification rate is dependent on the hydrostatic plastic strain rate, ${\dot{\epsilon }}_p^m$, as:

$$\dot{d}=-3{\dot{\epsilon }}_p^{m}d,$$

(1)

where $d$ is the relative density. The plastic strain rate is, in turn, dependent on the hydrostatic stress ${\sigma }^m$, the deviatoric stress matrix ${\mathit{\sigma }}^{\mathit{d}}$, and the sintering stress ${\sigma }^s$, according to:

$${\dot{\mathit{\epsilon }}}_{\mathit{p}}=\frac{1}{3K^v}\left(\left({\sigma }^m-{\sigma }^s\right)\mathit{I}+{\mathit{\sigma }}^{\mathit{d}}\frac{1+{\vartheta }^v}{1-2{\vartheta }^v}\right),$$

(2)

where $K^v$ is the viscous bulk modulus, ${\vartheta }^v$ is the viscous Poisson’s ratio, and I is the identity matrix. The viscous Poisson’s ratio is dependent on the relative density, and different formulations of the dependency vary between studies [19,20,21]. Regardless of the formulation of the viscous Poisson’s ratio that is chosen, it should be noted that when external loading, leading to both hydrostatic and deviatoric behaviours, is present, the plastic strain will be path-dependent with respect to the densification. This means that when simulating the sintering of a cutting tool, for example, with a high-density gradient after compaction, it is important to evaluate the full process, not only the final result. In this study, the external load is assumed to be negligible in the experimental setup and, consequently, in (2); ${\sigma }^m=0$ and ${\mathit{\sigma }}^{\mathit{d}}=\mathbf{0}$. Therefore, the only stress affecting densification is the sintering stress:

$${\sigma }^s=f_p\frac{{\gamma }^{Co/p}·\cos \theta }{L}.$$

(3)

In Equation (3), $f_p$ is the volume fraction of the pores, ${\gamma }^{Co/p}$ is the surface energy between the cobalt and the pore, $L$ is a characteristic length which represents the average grain size, and $\theta$ is the wetting angle between the Co and WC particles.

The inverse of the viscous bulk modulus is viscous bulk compliance. Brandt et al. [1] argued that, since the temperature is high and the stress quite low, the dominating deformation mechanism should be diffusional creep. The compliance should thus contain a diffusion coefficient D. Furthermore, it should approach zero as the porosity is eliminated. Brandt et al. [1] also observed a dependence upon the heating rate, which was incorporated into the expression for the viscous bulk compliance by means of an internal parameter, the mobility factor ξ.

Thus, the viscous bulk compliance is expressed as:

$$\frac{1}{K^{\vartheta }}=C{f}_p^{m}D(1+a(T)·\xi ),$$

(4)

with the mobility factor $\xi$, which is determined using the differential equation:

$$\dot{\xi }=-\frac{\mathsf{\Delta }{V}_{m}d{f}^{\alpha }}{V_{m}dT}\dot{T}-\frac{\xi }{\tau \left(T\right)}.$$

(5)

In Equations (4) and (5), $T$ is the temperature given in °C. Furthermore, $\mathsf{\Delta }{V}_m/V_m$ is the relative volume change from the WC dissolution, where $V_m$ is the molar volume and $f^{\alpha }$ is the fraction of WC. The $\frac{\mathsf{\Delta }{V}_{m}d{f}^{\alpha }}{V_{m}dT}$ term is independent of both time and the mobility factor, as justified by the experimental results discussed in [1]. Accordingly, the effect from temperature can be included in the adjustable mobility expression $a\left(T\right)=\frac{a_0}{T^2}{e}^{\left(-a_1/T\right)}$, with constants $a_0$ and $a_1$. The relaxation parameter is described as $\tau \left(T\right)={\tau }_0{e}^{\left(-{\tau }_1/T\right)}$, with constants ${\tau }_0$ and ${\tau }_1$. This relaxation description makes it possible to simulate the densification occurring at a constant temperature.

Furthermore, in (4), $C$ and the exponent $m$ are constants, and $D$ is an effective diffusion coefficient that, in this formulation, should be considered as a proportionality coefficient of the viscous bulk compliance of the Cobalt-based binder phase [1]. Accordingly, it is expressed as:

$$D=d_{V0}{e}^{\left(-d_{V1}/R{T}_{abs}\right)}+{\omega d}_{S0}{e}^{\left(-d_{S1}/R{T}_{abs}\right)}.$$

(6)

where $d_{V0}$ and $d_{V1}$ are the pre-factor and activation energy for volume diffusion, respectively, while $d_{S0}$ and $d_{S1}$ are the pre-factor and activation energy for surface diffusion. The volume diffusion activation energy, $d_{V1}$, represents the self-diffusion of cobalt [15]. In order to reflect the dominating diffusion process, the blending parameter $\omega$ is introduced. $R$ is the universal gas constant, and the absolute temperature is denoted $T_{abs}$ (given in K).

In a standard manner, the thermal strain rate is:

$${\dot{\epsilon }}_T=\alpha \dot{T}$$

(7)

where $\alpha$ is the thermal expansion coefficient and $\dot{T}$ is the temperature rate.

Here, a situation with no external loading is investigated. A so-called free-sintering strain rate can then be determined from Equation (2), with ${\sigma }^m=0$ and ${\mathit{\sigma }}^{\mathit{d}}=\mathbf{0},$ being equal to ${\dot{\epsilon }}_p^m$ (at no external load). The total strain rate, in this situation, is:

$${\dot{\epsilon }}_{tot}={\dot{\epsilon }}^s+{\dot{\epsilon }}_T.$$

(8)

Obviously, the constitutive model briefly outlined above includes quite a large number of parameters. These parameters are defined in [7,10] but are also listed in Appendix A here for clarity.

3. Experimental Procedures

The powder composition used included tungsten carbide (88 wt%), cobalt (10 wt%), alloys and a pressing agent, PEG (PolyEthylene Glycol, 2 wt%). The tungsten carbide and cobalt particles had a size of around 1 µm and are mixed and spray-dried to granules with an average diameter of 100 µm. The same powder configuration was used in [8,9,10]. The dilatometer experiments were performed within a short time frame (weeks) after compaction, and the same batch of powder mixture was used for all the experiments. This was to ensure that the dilatometer results were all comparable. The specimens were stored in air after production. The average relative density was 55% after compaction. Note that the average relative density is defined as the measured density relative to the sintered density.

As already mentioned above, a dilatometer was used to perform the measurements of the WC-Co powder compact. The dilatometer test is a standard experimental procedure aiming towards a mechanical and/or physical understanding of the sintering process. A rectangular specimen was compacted to a relative density of 55% and had the measurements of 6 mm × 6 mm × 25 mm before sintering. The compaction was performed in the width direction to minimise density gradients. The specimen was mounted in the dilatometer, as seen in Figure 1. The shrinkage was then measured every 0.65 s in the length direction, where an axial compressive load of 0.45 N was applied. The test cycle mimicked a typical real manufacturing sintering process, but with the ramp speed and holding time adapted to the smaller volume in the dilatometer compared to the sintering furnace used in industry, meaning that less time was needed for the specimen to reach the desired temperature.

Three dilatometer experiments were performed, which all started with a debinding step. In the debinding process, the specimen was heated to a temperature of 450 °C, where it was held for 15 min. The heating rate was around 10 °C/min (again, to mimic an industrial testing cycle) during heating, and a hydrogen flow was applied continuously to ensure that the pressing agent could be extracted as it melted and evaporated. After the debinding step, the specimen was held in a vacuum to ensure that no gas could be trapped in the specimen.

Three experimental setups were considered for the dilatometer, as illustrated in Figure 2. The first setup represents a normal sintering process used in the industry and is denoted as the ‘Reference’. In this setup, the sintering process was started immediately after the debinding step, and the temperature was raised from 450 °C to 1415 °C at a continuous heating rate of 10 °C/min. In the second experimental setup, the debinding step was separated from the sintering process, and it is denoted as the ‘Separated debinding step’. After the debinding step, the specimen was cooled down to room temperature, still kept in a vacuum. After the specimen had rested in the vacuum for two days, the heating started at a rate of 10 °C/min until it reached 1415 °C. In the last experimental setup, the specimen was brought into contact with air, and this is denoted as the ‘Separated debinding step—aired’. After the debinding step, the specimen was cooled down to room temperature, still in a vacuum. The specimen was then taken out of the dilatometer machine and exposed to air for two days, after which it was inserted back into the dilatometer. The specimen was then heated at a rate of 10 °C/min until it reached 1415 °C. After the dilatometer experiments had been performed, the magnetic properties of the sintered specimens were measured.

4. Results

The results of the dilatometer experiments and the magnetic measurements are presented below. The constitutive model is evaluated with respect to its capability to capture shrinkage during the experiments.

4.1. Dilatometer

From the strain measured in the dilatometer, shown in Figure 3 (the corresponding strain rates are shown in Figure 4 for completeness), a deviation can initially be seen between the reference specimen results and the results for the two specimens with separated debinding curves. In the reference results, the first indications of shrinkage are noted around 450 °C, while in both separate debinding results, this is seen at 800 °C. The reference experiment also shows a slightly faster densification than the second type of experiment. Otherwise, the strain measured in the reference and separate debinding experiments after the initial sintering started are very similar. The aired specimen shows a larger densification in the initial stage of the sintering process than the others. Later in the process, the densification is slower than that for the specimen not exposed to air, but all three reach the same strain by the end of the sintering process. The early densification of the aired specimen indicates that the wetting capability between the cobalt and tungsten carbide increased. It should be emphasized that the large densification occurring at 80 min is not due to melting, which occurs later in the process, but due to the exponential characteristics of the sintering process. While the transition from solid to liquid sintering is not distinct, the melting point will affect the end part of the solid-state sintering. The slower densification occurring later in the aired specimen can therefore be explained by an increase in the melting point of the material.

4.2. Magnetic Measurements

The coercive force and the magnetic saturation were measured after the dilatometer experiment and are presented in

Table 1. Magnetic properties of the specimen after the dilatometer experiment.

Specimen	Coercive Force (kA/m)	Magnetic Saturation (T·cbm/kg)	Density (g/cm3)
Reference	21.59	168.8	14.46
Separated debinding step	22.29	159.1	14.60
Separated debinding step—aired	22.23	137.9	14.63

. The magnetic saturation is linearly proportional to the carbon content of the cobalt-based binder phase [22], while the coercive force can indicate a variety of differences. For the same powder configuration, a higher coercivity indicates a smaller grain size [22]. As seen in Table 1, the coercive force is slightly smaller in the reference experiment, indicating that the WC particles are larger than those in the experiments with separated debinding. However, this difference is not deemed significant. The magnetic saturation, on the other hand, shows a larger difference in the results. Comparing the separated debinding experiments to the reference experiment, the magnetic saturation and thus also the carbon level are 6% lower for specimens kept in a vacuum and 18 % lower for the aired specimens.

It should be noted that the differences in magnetic saturation correspond to quite small, but important, differences in the carbon content. After sintering, the reference sample has a carbon content of approximately 5.5 wt-%; the other samples have carbon contents that are lower by some hundredths of a percent. The reduction in the carbon content is due to oxidation, which, quite naturally, becomes more pronounced in the aired sample.

4.3. Numerical Simulation

Simulations were performed in LS-Dyna [7]. Since densification is homogeneous, the geometry was irrelevant. To decrease the simulation time, one element was used in the implicit solver. The temperature and corresponding sintering time were set to replicate the dilatometer test.

To evaluate how well the constitutive model could replicate the experiments, the material parameters were optimised with LS-OPT [23]. The aim of the optimisation was to evaluate the possibility of describing the material with the given constitutive model. A previous sensitivity study found that the number of parameters used in the optimisation can be reduced to four ($d_{S1}$, $a_1$, ${\tau }_1$ and $m$) while still yielding accurate results [10], a method which was also utilised in this optimisation. In [10], the curves resulting from a Monte Carlo simulation were compared to the reference curve. The difference between the reference curve and Monte Carlo results was used to calculate the sensitivity of each parameter.

The four optimisation parameters were allowed to change by a factor of two, while the other parameters were kept constant. Remembering that quite similar hard metal materials (cemented carbides) are under consideration here, it was deemed sufficient to restrict the variable change to a factor of two. This was, however, also checked initially in the simulations and found sufficient in order to capture the variations in the material parameters. The numerical values of the parameters, being the same as the initial values used in the optimisation, are presented in Brandt et al. [1], and the optimised ones are presented in

Table 2. Parameters used in the optimisation and the resulting factor when optimised against the experimental measurements.

Parameter	Initial Value	Reference	Separated Debinding Step—Aired
$a_1$	12,700	$a_1·$1.675	$a_1·$1.761
$m$	1.18	$m·$0.500	$m·$0.501
${\tau }_1$	12.500	${\tau }_1·$0.500	${\tau }_1·$0.646
$d_{S1}$	1.22$·$105	$d_{S1}·$1.947	$d_{S1}·$0.929

. The objective of the optimisation was to minimise the root-mean-square value between the experimental values and simulation during solid-state sintering. Domain reduction with default settings [23] was applied in the optimisation loop. After 7 iterations, convergence was reached, but for all three optimisations, 10 iterations were used.

In previous work, the optimisation capability of the reference experiment was evaluated, resulting in the conclusion that the constitutive model was limited in its ability to capture the initial densification process seen in Figure 5. Figure 6 shows better agreement between the optimised curve and the experimental values, but there are still errors during the initial densification process. An optimisation for the separated debinding step experimental results is not presented because of its similarities to the reference curve.

5. Discussion

It was shown that the experimental setup affects the densification process and the carbon level in the powder. The reduction in carbon is assumed to correlate with the amount of tungsten and cobalt oxide in the powder before the sintering process starts, since the dilatometer chamber is closed with no addition of carbon. The air initially present in the closed dilatometer chamber will be removed during the debinding step, as seen in the experimental setup. In the case of separated debinding-aired, the vacuum effect can no longer be assumed, and air will be present during sintering. The wetting process and, thus, the densification, are initiated once all the oxygen has evaporated from the powder surface at around 800 °C [24] and has reacted with carbon. In the reference configuration, the amount of oxide is limited due to the presence of the pressing agent. In the separated debinding step, it is possible that a small amount of water is present during the cooling and resting period, allowing the oxygen in the water molecules to react with tungsten and cobalt during sintering. In the aired experiment, the specimen is taken out after the pressing agent is removed, allowing the oxygen to be present and react freely with the powder and significantly increasing the amount of tungsten oxide during sintering. The relationship between the amount of carbon in WC-Co powder and the wetting capability was previously studied by Konyashin et al. [12]. It was concluded that it increases with a lower amount of carbon, which corresponds to the results in Figure 3.

The constitutive model is better suited for simulations with a separated debinding step, which is a weakness when using this model for industrial applications. Therefore, the constitutive model could benefit from being modified to consider the amount of carbon or the wetting capability of the material. Since the industrial furnace used for sintering offers better control of the carbon level, the reference experimental method yields the lowest loss of carbon and is therefore a better experiment for replicating the manufacturing process. The experimental data for the reference strain and the two cases of separate debinding strain have the same final strain, showing that no significant densification occurs in the debinding step. It would be interesting to examine this situation for other powder materials, such as iron powders [25] and ceramics. The sintering of the latter material type was comprehensively reviewed by Bordia et al. [26].

In Figure 3, the reference curve and the curves for the two experiments with the separated debinding step diverge during the first 40 min in the graph, between 450 °C and 800 °C. One hypothesis for explaining this is that the additional heating time in the separated debinding step leads to early diffusion in the powder. This theory is not supported by the constitutive model, in which densification from the diffusion stage is not noticeable before higher temperatures, meaning that the effect of the additional time heating in the span of 20–450 °C is negligible. This theory was also experimentally tested in an experiment where the specimen was heated to 450 °C and held for at least double the holding time used in the reference experiment, after which the sintering process proceeded immediately. What we found was that the longer holding time had no effect on the densification during sintering, confirming the constitutive model and disproving the hypothesis.

6. Conclusions

Here, an experimental investigation of the solid-state sintering of cemented carbides was undertaken. An understanding of the importance of different features impacting on the mechanical behaviour is essential when building relevant constitutive models, and this was the aim of the present study. The results show that the experimental setup used affects the carbon level and the densification during sintering. In order to design the experiments to match an industrial situation, the amount of carbon lost in the process needs to be minimised. Furthermore, a comparison between the predictions from the investigated constitutive model and the experimental results indicate that the model does not properly describe the initial phase of sintering and is not able to capture, in an accurate manner, the effect on densification caused by changing carbon levels. This is a very important conclusion of the present work. Accordingly, improvements of the model must be implemented in order to better describe this feature from a mechanical point of view and, subsequently, account for it in the constitutive description.