Sintering and Densification (II)—New Sintering Technologies

Abstract

Besides the conventional sintering techniques discussed in the previous chapter, various new sintering methods, such as microwave (MW) sintering, spark plasma sintering (SPS), and flash sintering, have been developed for ceramic processing. These methods have also been used to fabricate transparent ceramics. This chapter is used to describe the principles, analysis and simulations of SPS and microwave sintering technologies, which could be useful to processing of transparent ceramics.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

6.1 Introduction

Besides the conventional sintering techniques discussed in the previous chapter, various new sintering methods, such as microwave (MW) sintering, spark plasma sintering (SPS), and flash sintering, have been developed for ceramic processing. These methods have also been used to fabricate transparent ceramics. In this chapter, two of them, i.e., SPS and MW, will be presented in detail. It is necessary to indicate that SPS has a number of other names, such as electrical current activated/assisted sintering (ECAS) and field-assisted sintering (FAST). When these names are used in this chapter, they are meant to be the same, unless otherwise mentioned [1]. Additionally, because there are no theoretical studies on sintering phenomena specifically for transparent ceramics by using these new sintering technologies, simulations or modelings are all discussed for all types of materials in general.

6.2 Electric Current Activated/Assisted Sintering (ECAS)

6.2.1 Brief Description

Comprehensive description and latest development of the SPS (or ECAS) have been presented in several recent review papers [1–4]. During experiments, loose powders or cold-formed compacts to be consolidated are loaded into a container, which is heated to a targeted temperature and then held at the temperature for a given period of time, while a pressure is applied and maintained at the same time. Thermal energy is supplied by applying an electrical current that flows through the powders and/or their container, thus facilitating the consequent Joule heating effect. SPS is a newly developed method for obtaining fully dense and fine-grained transparent ceramics at low temperatures within short time durations [5].

As compared with the conventional sintering methods, this technology possesses various technological and economical advantages, such as high heating rate, low sintering temperature, and short sintering time duration. Moreover, it can be used for consolidation of powders that are difficult to be sintered by the conventional sintering methods. In addition, it is not necessary to use sintering aids. In some case, the step of cold compaction can be skipped if necessary. Also, it is less sensitivity to the properties of the initial powders.

Due to the relatively low sintering temperature and short time durations offered by this technique, it is possible to fabricate ceramics with ultrafine or even nanosized grains, because grain growth can be avoided or suppressed. Also, the rapid processing makes it possible to consolidate metastable powders without variation in physical and chemical properties. Furthermore, due to the fast consolidation rate, the processing can be conducted in air atmosphere, without the need to control environments. Additionally, it is also possible to prevent undesirable phase transformations or reactions in the initial powders of the processed materials, owing to the relatively short processing time. More importantly, short processing time means short production cycle and thus high productivity.

6.2.2 Working Principles

Figure 6.1 shows a schematic diagram of a representative ECAS experimental setup [1]. An electrical current, together with a mechanical pressure, is applied during the consolidation of the sample. The applied electrical current and mechanical pressure either can be constant throughout the sintering process or can be varied at selected stages of the densification, thus offering a high flexibility in varying sintering conditions or processing parameters.

Fig. 6.1

Schematic diagram of a representative electric current assisted sintering (ECAS) experimental setup

The powders to be consolidated can be either electrically conducting or insulating. They are placed in a container, consisting of die, tube, and so on, which are then heated by applying an electrical current. If the powers are conductive, either conductive or insulative container can be used. Otherwise, the container must be conductive, so that the electrical circuit is guaranteed to be closure. Similarly, all electrodes, blocks, spacers, and plungers used in the circuit should be made of electrically conducting materials, such as copper (Cu), graphite, and stainless steel. Conducting powders are heated due to the Joule heating effect and the heat transferred from the container and electrodes, whereas nonconductive powders are heated only due the heat from the container and electrodes, without the presence of Joule heating. Conductive containers can be graphite, conductive ceramics, or steel. In ECAS processes reported in the open literature, graphite containers are the most widely used. The use of graphite limits the mechanical pressure levels that can be applied, which generally is not higher than 100 MPa. However, the presence of graphite makes the sintering environment to be reductive.

In terms of the application of a mechanical pressure, ECAS process shares a similarity with hot pressing (HP). However, the application of the electrical current makes ECAS process to be much rapid and efficient. The heating rate during the ECAS process is dependent on the electric power supplied, as well as the geometry of the container/sample ensemble and the thermal and electrical properties of all the materials. The heating rates can be as high as 1000 °C min⁻¹. As a consequence, the processing time ranges from fraction of seconds to minutes depending on the characteristics of the material, size of the sample, configuration of the setup, and the capacity of the whole equipment.

In comparison, in the conventional HP process, the sample container is heated due to the radiation from the enclosing furnace through external heating elements and convection of inert gases if used. In other words, the sample is heated through the heat transfer occurring by conduction from the external surface of the container to the powders. Therefore, the heating rate is much slower, and thus, the process can last hours or even longer. In addition, its total efficiency is much lower than that of ECAS process, because a lot of heat is wasted when the whole volume of space is heated while the sample compact indirectly absorbs heat from the hot environment. Due to the high thermal efficiency, ECAS processes can consolidate samples within a very short time duration, especially when electrically insulative containers are used.

However, ECAS process currently has its own problems, such as inhomogenous temperature distribution when the electrical conductance of the powders is not adequate. Actually, the electrical current and the consequent temperature distribution within the samples are very sensitive to the homogeneity of density distribution of the final products. Furthermore, significant density spatial variation, especially at the beginning of the current flow, may result in high local overheating or even melting of the samples. This is the reason why almost all the samples studied until now have either cylindrical or rectangular shape. This problem should be addressed so that the ECAS process will find more practical applications.

Depending on the characteristics of the power supplies, the electrical current delivered the samples during ECAS processes can be different in intensity and waveform. However, regardless of the waveform of the applied current during the ECAS processes, it is necessary to emphasize that the Joule heating, which in turn determines the level of temperature experienced by the samples, is related to the root-mean-squared instantaneous current intensity, which is defined by the following equation:

$$I_{\text{RMS}} = \sqrt {\frac{1}{\tau }\int\limits_{t}^{t + \tau } {I^{2} (t){\text{d}}t} } ,$$

(6.1)

where I is the instantaneous current and t the sampling time.

The mechanical load during ECAS processes is typically applied in the uniaxial direction in most of the currently available facilities. However, specific experimental setups have been designed to apply isostatic or quasi-isostatic [6] pressures to the samples. Ultrahigh isostatic pressure of up to 8 GPa has been reported [7, 8]. According to the characteristics of the applied currents, ECAS processes can be classified in several ways, with two essential types, i.e., resistance sintering (RS) and electric discharge sintering (EDS) [7, 8].

RS is usually observed when the electric power supply has a low voltage, with the order of few tens of volts, but a large current, with the order of thousands of amps, with current waveforms that can be direct current (DC), alternate current (AC), rectified current (RC), pulsed current, and so on. For the occurrence of EDS, the power supply is electrical energy that is stored in a capacitor bank, which is released to the powder compacts as a pulse. Due to this pulsed release mode, EDS process offers higher voltages and higher currents to the samples, as compared with RS. Additionally, the change of current versus time, dI/dt, during the EDS process may trigger electromagnetic phenomena in the compaction. Another significant difference between EDS and RS processes is in characteristic processing time. Generally, the processing time of EDS is in the range 10⁻⁵–10⁻² s, whereas that of RS is of the order of 1–1000 s. In practice, RS is far more frequently observed than EDS. The two types of process are described as follows.

6.2.2.1 Electric Discharge Sintering (EDS)

Electric discharge sintering (EDS) is also known as electric discharge compaction (EDC) or environmental electrodischarge sintering (EEDS), during which electrical energy is discharged from a capacitor bank so that it passes through a column of powders contained in an electrically nonconductive tube container [9]. The high transient current flowing through the column causes heating and sintering of individual particles of the powdery sample. At the same time, this current generates an intense magnetic field in the azimuthal direction, which makes the body of the powder to shrink radially. Therefore, after the discharge, the sample is compacted and the sample can be easily taken out after the sintering. Because of this, the containers can be used for serval times, which is one of the most important advantages of EDS [10].

The first electrical discharge unit is a magnetic forming machine. The capacitor bank generally consists of several capacitors with a total capacitance up to 25 mF, while a charging voltage up to 30 kV can be achieved [11]. The capacitor bank can be charged by using a variable transformer, together with a rectification and smoothing unit. The column of the metal powder acts as a short-circuit resistance across the capacitor bank. The applied discharge current density and intensity can be as high as 2800 MA m⁻² and 90 kA, respectively. During experiments, the current decreases monotonically with increasing length of the sample, whereas it increases with increasing diameter of the sample.

The typical waveform of the current that flows through the powder column can be in critical, overdamped, and underdamped states. Sometimes, only two stages are observed [12]. During the first stage, the powders are compacted due to the application of the high-voltage pulse. At the second stage, sintering of the sample takes place, at a current density with the order of 10²–10³ A cm⁻². The reason for the presence of the multistage (two-stage) sintering is electrical resistance of the samples could decrease by nearly six orders of magnitude during the EDS processes. Such process can be conducted with or without the application of mechanical pressure. If mechanical pressures are applied, they can be either static or dynamic. Static mechanical pressures of up to 710 MPa have been reported. When dynamic axial pressure and electrical discharge are applied at the same time, the time duration between the onset of the discharge and the onset of the maximum axial force can be optimized to maximize the densification of the samples.

It has been found that, in some cases, in order to achieve high density, e.g., >95 % of theoretical density, the green density should be higher than 80 %, thus requiring cold pressing step [13]. Also, a minimum level of pressure is necessary for effective discharge to occur. The application of the pressure is actually to compact the powdery sample to a desired density. This is because when discharge is passed through a loose powdery compact, densification does not take place, but instead, intense sparking occurs between adjacent particles, which could lead to rupture of the container. EDS has been extensively used to process metallic materials, especially ferrous materials [13]. Various parameters, including electrical circuit parameters, properties of the powders and dimensions, and geometries of the samples, have influence on the effectiveness of compaction at a given energy input. Generally, if no mechanical pressure is applied, it is difficult to achieve sufficiently high density of the final products. In this case, a swaging and a sintering step are necessary to obtain dense samples.

Porous products processed by using EDS on the other hand have also various potential applications, such as debris filtration, fluid flow control, capacitors, catalyst supporter, and pressure surge protection units. It has been observed that tensile strengths of the EDS-processed porous structures are much greater than those of the samples prepared by using isostatic press of the same density. In fact, isostatically formed samples cannot be used to measure their mechanical strengths. The high strengths of EDS-processed samples are attributed to the formation of interparticle welds due to the heating effect caused by the electrical discharge that passes through the metallic powders, so that strong metal–metal bonds between the base metallic particles are developed. If metallic powders have relatively large particles, e.g., up to 150 mm, the very short sintering time of EDS cannot trigger sufficient diffusion to accomplish homogenization or reactions, even at a temperature that is slightly below the point at which a liquid phase is formed. In other words, it is necessary to provide a liquid phase in order to achieve a considerable degree of alloying or reaction between the metal powders in the mixture. In some cases, continuous axial wires or fibrous structures that are aligned in the direction of current flow can be obtained, due to the interparticle welding.

To form bars through EDS process, the container is first filled with the powders to be consolidated, which are then plugged between two copper electrodes that are pushed into contact with the powder column. To maintain an effective discharge, the powders must be confined in electrically insulating containers. Therefore, Pyrex glass tubes are usually used as the containers, because they can be used repeatedly without pitting in the inner surface and cracks or deformation. Electric energy is charged into capacitors, which is then discharged directly to the loose powder column through a high-voltage switch. The energy injected into the powder column due to the discharge is determined by the current. The density, distribution, and period of the current can be controlled by the capacitance, resistance, and inductance of the circuit.

Usually, the change in inductance during the process is very insignificant, but the variation in resistance of the circuit, due to the heating and progressive welding of particles, can be very pronounced. Also, the higher the current that flows in the circuit, the larger the magnetic body forces will be, which is beneficial to increase green density and strength of the samples. In this respect, the density of the instantaneous current passing through the specimen is the most significant parameter that determines the effectiveness of the compaction process. Therefore, it is expected that the degree of compaction is increased with increasing current density. However, it is necessary to mention that welding and compaction occur only within a certain range of discharge energy, which is depending on the dimension of the powder column and the type of the materials processed.

There is also a minimum voltage that is required to maintain the sample to be sufficiently strong. If the energy discharge is too low, the induced magnetic field will not be strong enough to reduce the diameter of the powder body. As a result, it will be difficult to remove the sample after the processing. This can be understood in terms of electrical resistance of the compact. If the resistance is too high, the current will be too low to facilitate the sintering. In contrast, if the resistance is too low, sintering will not be effective, because the current through the cross section of the compact is not uniform. In this case, a very low current will first pass through the area of least resistance in the compact, which will be overheated to be broken down and more current will then flow through.

EDS is also known as high-energy high-rate (HEHR) consolidation, which converts stored rotational kinetic energy into electrical energy by using the Faraday effect, with a homopolar generator (HPG) [14]. The HPG is a low-voltage (5–25 V) high-current device that is operated in a pulsed mode. In this case, pulse-resistive heating is produced at the interparticle interfaces, so that bonding takes place to achieve the consolidation. The applied current pulse can be as high as 250 kA with a few hundreds of milliseconds, with current densities ranging over 100–500 MA m⁻². The process happens within 3 s, with most of the energy to be delivered in the first 0.5 s. At the onset of the pulse, sufficient pressure is applied and maintained for 3–5 min. The specific energy input varies in the range of 0.4–14.25 kJ g⁻¹.

6.2.2.2 Resistance Sintering (RS)

RS process has been extensively studied in terms of different electrical current waveforms that have been applied, and various names and acronyms have been employed to designate the RS process. It is worth mentioning that although the word ‘‘plasma’’ has often been used to designate one of the RS processes, the samples are actually not in an external plasma environment, as in the real plasma sintering or microwave sintering. In addition, sometimes, different names and acronyms have been used to describe the processes, in which, however, the same electric current waveform is applied or the same facility is used. Also, in some cases, RS processes with different electric current waveforms or different equipment have identical names. Therefore, it is suggested to present all parameters or conditions clearly when describing whatever RS process, in order to avoid any unnecessary confusion. It is well known that the application of an electric current is the key feature of RS processes, which can be used for classification. Figure 6.2 shows the types of current waveforms that have been most widely used in the open literature [1].

Fig. 6.2

Typical electric current waveforms applied in the RS processes: a constant DC, b AC, c pulsed DC, and d pulsed DC + DC. Reproduced with permission from [1]. Copyright © 2009, Elsevier

The simplest current waveform is the constant DC, as shown in Fig. 6.2a, which is characterized by one parameter, i.e., the current intensity, I. The AC current waveform shown in Fig. 6.2b is determined by a maximum current intensity, I _max, and its frequency, ω. The third waveform in Fig. 6.2c is pulsed DC, which has four independent parameters, including maximum current, I _max, pulse duration τ _pulse, and the on- and off-time, τ _ON and τ _OFF. In the RS process of Fig. 6.2d, the electrical current is applied at two stages. A pulsed electrical current is imposed in the first stage, while a constant DC is followed in the second stage. The first stage is defined with parameters of maximum current, I _max, the on-time, τ _ON, and the off-time, τ _OFF, while the second stage has only one parameter, i.e., current intensity, I. In addition, the relative durations, τ _I and τ _II, of the two stages must be clearly indicated. The squared pulses shown in Fig. 6.2d have been used in experiments, as illustrated in Fig. 6.3 [15]. Therefore, a precise description of the level of the current applied, together with a complete set of all operating parameters, is important to characterize a RS process [1].

Fig. 6.3

Experimental measurements of the electrical currents with the type of the waveform shown in Fig. 6.2d. Reproduced with permission from [15]. Copyright © 1996, Elsevier

In practice, there are more possibilities to synthesize current waveforms, through the combination of the basic ones shown in Fig. 6.2. Also, the waveforms or pulses can be further modified or changed by having different shapes. In this respect, the room for operation for RS process is unlimited. Different types of commercially available equipment, which are capable of applying different types of current waveforms, have been studied and can be found in the open literature [16, 17].

6.2.3 Brief History of ECAS Processes

The first example could be a US patent in early 1920s could be the first example of ECAS, regarding the production of dense articles starting from oxide powders to be sintered through the application of an electrical current [1]. The idea of simultaneously applying a uniaxial mechanical pressure and an electrical current to sinter metal powders reported in later 1920s. The mechanical pressure is applied through the electrodes to the confined WC/Co powder mixture and a direct alternate electrical current is simultaneously supplied through the electrodes connected to an external electric circuit, thus heating the powdery material to a temperature that is sufficiently high to sinter the sample thoroughly. The processing is finished within minutes. This method was used to consolidate elemental powder mixture of W, C and Co. There are the earliest examples of reactive sintering assisted by electrical current. Complete densification of the WC/Co cemented carbide in about one second was reported in early 1930s. The reason why such a shorter processing time is possible is because the apparatus is capable of discharging a pulse of direct current to the samples followed by imposition of an alternate current. The electric discharge was realized by using a condenser connected to a 2.5 kV source of direct current, which is in parallel with the sample to be sintered. The development of the technique was continuously progressed.

Various applications of ECAS process were patented later on. Progress in commercialization of ECAS process was achieved in the 1960s. The proposed ECAS method combines pressing and sintering of powdery metal parts in one shot due to the impulsive force of an unspecified ‘‘spark energy,’’ and therefore, it is called spark sintering. This process had been claimed to be capable of sintering materials that cannot be processed by using the conventional sintering methods, thus having versatility and almost unlimited capabilities.

In 1966, a patent of ECAS process was issued, which is able to apply simultaneously an electrical current and a mechanical pressure of lower than 10 MPa. The electrical current is applied through the superimposition of a periodic current on a direct current. The light contact between the particles due to the relatively low pressure, along with the imposition of the periodic/alternate current, leads to the formation of spark discharges within the initial voids within the sample to be sintered. These electric sparks have a power of hundreds to even thousands of joules, which allow the particles to bond one another with a pressure that is even higher than the applied mechanical pressure. In earlier ECAS techniques, high pressure is required to reduce the contact resistance of the powdery samples. As the contact resistance is substantially equal to the internal resistance of the particles, a high current could be passed through the sample to develop the necessary bonding heat. In contrast, in the new design, the contact resistance must be greater than the internal resistance, preferably several times as great, so that at least during the initial stages of the sintering, most of the applied energy is in the form of spark discharge, instead of being dissipated in resistive heating of the particles. Since the sintering takes place immediately upon the spark discharges, densification can be achieved within seconds, as compared with earlier methods requiring tens of minutes and even hours to complete the sintering.

The process, named as spark sintering instead of electric discharge sintering, was presented as the latest method for producing powder metal parts, due to its capability of combining pressing and sintering in a single operation. In addition, it was already advanced to industrial scale since a semi-automatic system was developed. It has been reported that spark sintering could be a very effective process for consolidating alloy powders that have high creep resistance at elevated temperature because the processing parameters can be readily adjusted to produce a temperature that lies within the very narrow range between the onset of plasticity and melting. Considerable progress had also been achieved in forming and simultaneously bonding powders into solids by this technique. Complex configurations with minimized machining can be produced [18].

Although spark sintering has been realized as a practical and cost-effective process, a fundamental understanding on the real role of the electric current/voltage is lacking. There are also some problems related to the ECAS processes [19]. For example, the application of this process may be limited by the typical use of graphite dies, which are characterized by low wear resistance with a consequent increase in the processing costs. Therefore, the use of wear resistance alumina dies has been proposed to replace the graphite ones. Also, the use of short-time high-current pulses cannot produce dense parts of conducting powders in nonconducting dies, due to the inhomogeneous current distribution within the powder compact. To address this problem, a strategy has been developed so that alternating and direct current can be applied simultaneously. It is believed that continuous development of ECAS process will be advanced in the future.

6.2.4 Modeling and Simulation

Because RS represents the dominant majority, modeling considerations only related to this process will be presented and discussed in this subsection. RS process can be taken as a thermo-electromechanical system, so that the corresponding modeling is meant to deal with the complex coupling of thermal, electrical and mechanical phenomena involved in the powders inside the experimental setup, in which there are several elements characterized by different geometries and material properties. Generally, properties of the materials exhibit a nonlinear dependence on temperature and pressure. Therefore, modeling of such a system is particularly challenging, because nonlinearities are accompanied by model uncertainties. At the same time, the difficulties to obtain reliable experimental data may cause masking effects that can be macroscopically misinterpretable.

Usually, the description of the mechanical behavior of RS process has been completely neglected. Thus, the analysis is limited to solid (fully dense) or presintered compact specimen instead of sintering powders. In this respect, the results obtained are generally not applicable to RS-processed powder compacts, because the mechanics of sintering and densification is strongly coupled to the corresponding thermal and electrical phenomena. Furthermore, even the analysis is limited to a thermoelectric perspective suitable for compact samples, too many assumptions are necessary when developing the models, thus making it difficult to provide quantitative results. In almost all cases, simulation results more or less have deviations from actual operating conditions experimentally used, which thus are used as a qualitative reference of the behavior of the real system.

In practice, modeled results are not used to directly compare with experimental data. Even though comparison is necessary, it is only limited to temperature profiles. In contrast, electrical variables, such as measured voltage and current, are not used for comparisons. The purpose of modeling is to provide a quantitative track for these variables. If temperature profiles can be well modeled, it means that the modeling is reliable. This is because temperature distribution is the main factor having effect on heterogeneity of the sintered product, so that all other variables and their effects can be neglected.

Heat transfer and generation, e.g., Joule effect, in an electrode, such as punch–specimen–electrode (punch) system without a die, have been well modeled [20–23]. In this case, 1D axial thermal balances, i.e., for steel (50S2G or 20KhN3A) electrode/(punch) and for electrically conductive powder (VK16 or VK8VK hardalloys) samples, coupled through continuity boundary conditions at the interface contacts, as shown in Fig. 6.4, are considered [20]:

$$\rho_{\text{i}} C_{{{\text{p}},{\text{i}}}} \frac{{{\partial }T}}{{{\partial }t}} = k_{i} \frac{{{\partial }^{2} T}}{{{\partial }z^{2} }} + \rho_{\text{el,i}} j^{2} ,\quad {\text{i}} = \left\{ {\begin{array}{*{20}c} {{\text{specimen}}\,{\text{powder}},} \\ {{\text{electrode}}\,{\text{steel}} .} \\ \end{array} } \right.$$

(6.2)

Fig. 6.4

Integration domain and boundary conditions of the model. Reproduced with permission from [20]. Copyright @ 1989, Springer

The heat in the model is generated uniformly in such a way that it is proportional to the corresponding electrical resistivity, ρ _el,i, and the square of same current density, j, that flows in series through the system of electrode (punch)–specimen–electrode (punch). During the modeling, axial symmetry is assumed, so that it is necessary to consider only half of the real domain of integration. A constant temperature that is equal to the initial value, T ₀, can be assigned, because the extreme ends of the electrode (punch) are cooled, due to the contact with a high thermal capacity of the rest of the RS system. In order to obtain analytical solutions of the model, it is assumed that the current density and all thermophysical parameters, including density, ρ _i, thermal capacity, C _p,i, thermal conductivity, k _i, and electric resistivity ρ _el,i, are constant.

The effect of actual changes in thermophysical parameters with temperature has been evaluated when solving the model by starting with different initial temperatures, with which a representative mean value is derived and used. The sintering behavior of the powder sample can be modeled by using effective thermophysical parameters that are corrected with the corresponding values of its fully dense solid counterpart, for a given porosity (θ), which are expressed by the following equations:

$$\rho_{\text{powder}} = \rho_{\text{solid}} (1 - \theta ),$$

(6.3)

$$C_{\text{p,powder}} = C_{\text{p,solid}} (1 - \theta ),$$

(6.4)

$$k_{\text{powder}} = k_{\text{solid}} (1 - \tfrac{3}{2}\theta ),$$

(6.5)

$$\rho_{\text{el,powder}} = \rho_{\text{el,solid}} (1 - \tfrac{3}{2}\theta )^{ - 1} .$$

(6.6)

Modeling results and experimental data have been compared, in terms of temperature temporal profile, for the system of VK8VK powder sample and 20KhN3A steel electrode, at two constant current densities of 429 and 784 A cm⁻² [24]. The modeling results are derived with the thermophysical parameters evaluated at 20 and 500 °C, respectively. The temperature profile along the axis of the system inside the electrically conductive powdery specimen predicted by the model is relatively flat, whereas the temperature gradient through the electrode (punch) according to the simulation is higher than experimental observation, which can be attributed to the effect of cooled ends and a lower thermal conductivity of the materials.

A new model has been developed, dealing with the current density and the consequent Joule heat distribution between the specimen and the die [25]. Thermal balances, as given in Eq. (6.2), where Joule heat is expressed in terms of voltage gradient, are coupled to the current density balances, i.e., Kirchhoff law with distributed parameters in a 2D cylindrical coordinate system:

$$\begin{aligned} \frac{1}{r}\frac{{{\partial }\left[ {\left( {1/\rho_{\text{el,i}} } \right)\left( {r{\partial }\varphi /{\partial }r} \right)} \right]T}}{\partial r} & + \frac{{\partial \left[ {\left( {1/\rho_{\text{el,i}} } \right)\left( {\partial \varphi /\partial z} \right)} \right]T}}{\partial z} = 0, \\ & {\text{i}} = \left\{ {\begin{array}{*{20}l} {\text{specimen,}} \\ {{\text{graphite}}\,\,{\text{die,}}} \\ \end{array} } \right. \\ \end{aligned}$$

(6.7)

where φ is the electric potential, i.e., current density vector is proportional to voltage gradient according to Ohm’s law, $\overline{j} = - (1/\rho_{{{\text{e}} . {\text{l}}}} ){\overline{\nabla }}\varphi$. Radiation heat losses from outer surface of the die are included in the boundary conditions, while the effective heat conduction dissipation in axial direction from the plunger to the electrodes is not modeled. Because the PDS apparatus chamber is operated in vacuum, the convection heat losses from outer surface of the die can be neglected. It has been assumed both temperature and voltage are continuous, i.e., the voltage and current density between the contacting sides are the same. Thermophysical parameters of solid graphite or copper powder specimen are used in the model, which is solved by using a numerical technique based on the method of fundamental solutions, so that the voltage and temperature spatial gradients and their discontinuity at the junctions of different materials can be captured easily.

Modeling results have been available about the axial profiles of the graphite solid and copper powder sample, in terms of voltage and current density under a uniform temperature distribution at 303 K, with a voltage drop of 1 V. For isotropic thermophysical properties, the copper powder sample is assumed to have a constant porosity. It can be concluded that the electrical current flows through the sample instead of the surrounding graphitic die when copper powders are used instead of solid graphite, because copper has higher electrical conductivity than graphite. Therefore, at same voltage drop, current density of the copper sample is higher than that of graphite one. As a result, axial voltage drop is concentrated through graphite plungers when a copper powder sample is inserted into the graphitic die, whereas an almost uniform distribution over the whole system is observed when the sample is graphitic. It is also found that heat generation is concentrated on the plungers, due to their smallest cross-sectional area, as demonstrated by theoretical potential, Joule heat, and temperature distribution contour for graphite solid and copper powder when a steady state is arrived at voltage drops of 0.8 and 0.6 V to the system. Therefore, temperature is almost uniform in the sample.

A different result has been reported in simulating TiB₂/BN composite solid that is inserted in a graphitic die [26, 27]. The difference in the simulation results between this study and the previous one could be attributed to the difference in properties between the two materials. Different from that of copper powder, electric resistivity of TiB₂/BN is almost equal to that of graphite, but with a lower thermal conductivity. The system simulated in this case has relatively larger dimensions. Thermal balances for the different materials are taken into account only in 2D cylindrical coordinate system. Thermophysical parameters are assumed to be constant in order to obtain steady-state analytical solution through an original derivation which is based on radial and axial symmetry.

The solution of the original 2D problem is represented by the product of the solutions of the two 1D problems, as shown in Fig. 6.5 [28]. The radial profile is derived by coupling thermal conduction and Joule heat generation inside the two infinitely long coaxial cylinders with radiuses equal to those of the real system, whereas axial profile is obtained by using sandwiched slabs of infinite surface area with heights equal to those of real system. Continuous boundary conditions at contact surfaces are assumed, with radiation heat losses from lateral outer surface of the external cylinder to mimic lateral outer surface of the die. The temperature of the exposed surfaces of slabs is assumed to be constant, i.e., ambient temperature, which simulates axial cooling from plungers to rams.

Fig. 6.5

Integration domain and boundary conditions of the model for TiB₂/BN composite solid in a graphite die: a 2D configuration, b r-direction, and c z-direction. Reproduced with permission from [28]. Copyright @ 2002, Elsevier

Steady-state temperature difference from the center of the sample along radial direction is predicted by using the model, while experimental temporal profiles of temperature measured by using two thermocouples positioned at center of the sample, which are radially apart, at plunger/die contact interfaces [28]. In this study, conditions for the experimental data and the parameters used for the simulations, such as electric current densities and corresponding heat generation, are not mentioned. Thermophysical data of the materials and height of sample are also not available. Nevertheless, the model predicts large differences in radial temperature, 350 °C inside the sample and 100 °C inside the die, i.e., the TiB₂/BN sample is hotter than the graphite. The temperature difference inside the sample measured experimentally is 450 °C. Therefore, the modeling results are in a qualitative agreement with the experimental data. Steady state is not reached during the experiments.

Similar simulation results have been reported for insulating powder compaction by using SPS, for example, Al₂O₃ powder [27]. Temperature distribution is predicted by considering 1D radial thermal balance with conduction and Joule heat generation for a sample of 2 cm diameter with finite element method (FEM). The necessary boundary condition of thermal balance inside the sample is determined by taking surrounding graphitic die into account. The temperature at the sample/die interface is evaluated according to the Joule heat developed by the current flowing inside the die as a function of its electrical conductivity and cross-sectional area, which are compared with those of the sample. The sample and the die are considered as resistors connected in parallel. In this study, thermophysical parameters and system geometry are not described and boundary conditions are not clearly defined. Based on the predicted temperature distribution, grain growth of the powder is mathematically described by using Monte Carlo simulations through moving grain boundaries and peak points of a fine cell structure. Radial temperature profiles inside the Al₂O₃ sample at two steady states have been simulated, with boundary temperature at the sample/die contact be 1450 and 1650 °C, respectively. The temperature at center of the sample is always lower than that at its outer surface, with a maximum difference of 77 °C. This is because most of the electrical current flows through the graphite die, due its lower resistivity as compared with that of the sample. In this case, Al₂O₃ powder is heated by thermal conduction from the die, which is reflected by the distribution of grain growth.

A modeling study to simulate direct synthesis of MoSi₂–SiC composites from element mixture of Mo, Si, and C by using field activation has been reported [29]. The experimental process of spark plasma synthesis (SPS) is simulated, with simultaneous synthesis and densification of the materials, including also nanophases. A parallel combination is used to determine the distribution of flowing current density, between the electric resistances of the inner sample and the surrounding die. The reaction is highly exothermic. A schematic of the modeled sample is shown in Fig. 6.6, in which the cylindrical coordinate thermal balances for the sintering/synthesizing powder sample and graphitic die are coupled with the corresponding current density balances for properly evaluating Joule heat distribution [29]. The diameters of the sample are 2.0–8.0 cm, while the thickness of the sidewall of the graphite die is about 1.0 cm.

Fig. 6.6

Schematic of the cylindrical sample and die. Top and bottom graphite plates are modeled as boundary conditions for the sample and die. Reproduced with permission from [29]. Copyright © 2002, Elsevier

The reaction times as function of voltage for four sample sizes of 1–4 cm have been studied [29]. The formation of SiC is always slower than that of MoSi₂. This difference becomes less pronounced at high voltages or for large samples. There is a threshold voltage, below which ignition cannot be triggered within 200 s. This voltage is also different between MoSi₂ and SiC within one sample. Smaller samples correspond to faster reactions. As the voltage is increased, the 50 % reaction time decreases. The reaction time decreases with decreasing sample size. Above certain values of voltage, the effect of voltage becomes less pronounced and the difference in reaction time between the samples of different sizes diminishes.

The completion of the reaction is examined at the same point in space, i.e., halfway between the center and the surface of the sample and halfway between the top and the bottom [29]. The time, at which the completion of the reaction is measured, is either 200 s or the time at which less than 5 % of the original amount of is silicon remained at the outer edge of the sample. It is clearly demonstrated that the completion of MoSi₂ is always earlier than that of SiC. At higher voltages, for SiC, the larger the samples, the higher the degree of completion of reaction will be.

For the sample with 80 % MoSi₂, the 50 % reaction times for both MoSi₂ and SiC are much faster with the dies with higher conductivity [29]. According to total current in a cross section of the die and the sample, as a function of time, the conversion data of MoSi₂ and SiC are at the usual point, i.e., halfway between the center and the surface and halfway up. In the beginning, the current is carried nearly completely by the sample. The amount of current carried by the sample increases as the amount of MoSi₂ is increased, reaching a maximum at a point where the reaction of MoSi₂ is almost completed.

At that point, the die starts to carry the current, which increases gradually, and at the same time, the reaction to SiC also increases. Similar results are obtained for a die with lower conductivity, in which the current is carried by the sample throughout the whole reaction. For the 20 % MoSi₂ samples, the 50 % reaction times are also much faster with the dies of higher conductivities. According to plots of current distribution of the samples with this composition, it is observed that the difference between these two plots is pronounced. With a factor of 5.0, the sample carries almost no current throughout the reaction, while for the factor of 0.5, the sample carries the current transiently, with a peak at about 30 s, following which the current is carried primarily by the die.

However, in this study, plungers are not simulated. The thermal balance inside the powder compact is characterized by an additional source term, which is related to the enthalpy released during the highly exothermic reaction between starting powders. The rate of the corresponding heat generation is taken into account by coupling material balances of the reacting components with pseudo-homogeneous second-order kinetics and the kinetic constants of Arrhenius-type relation. Thermophysical parameters, such as specific heat, thermal, and electric conductivities, as a function of temperature, are determined by using the mixing rules, with the thermophysical properties of fully dense reactants, according to the composition that is varied as the combustion reaction is progressed. However, porosity variations during the sintering under mechanical loads are not evaluated.

Radiation and convective losses from all exposed surfaces of the dies are considered when defining the boundary conditions for thermal balances. Different effective heat transfer coefficients and emissivities are used to express heat losses from top to bottom or from lateral surfaces of the dies. This choice is made in order to simulate the presence of contact resistances at top and bottom surfaces of the dies. Boundary condition at contact surfaces between the sample and die is not clearly defined, instead they are assumed to be continuous. In boundary conditions for current density balances, electrical current is confined into the conducting materials and a voltage drop between the top and bottom equipotential surfaces of the die is applied. By the way, comparison with experimental data is not available and the model clearly predicts volume or wavelike combustion reaction, depending on sample size, initial powder composition, and electric conductivity of graphite die. The simulation results are obtained for the application of 15–40 V voltages along graphitic dies, which are relatively high values in SPS.

All the relevant qualitative results reached by modeling activities described so far, including hot spots in plungers, electrical current, and related Joule effect distribution field as a function of electric resistances of sample and die in parallel combination, have been confirmed [30]. Spark sintering behavior of presintered Ti powders (electrical conductive material) and alumina powders (electrical insulator material) are compared with model results obtained by considering the SPS system, as schematically shown in Fig. 6.7 [31]. Compacts of Ti or Al₂O₃ powders, with average particle sizes of 35 and 0.4 μm, are tested. The upper punch, die, and presintered compacts are drilled with holes of 3 mm diameter, and temperatures are measured by using thermocouples at fixed positions, as shown in Fig. 6.8, for 3.6 ks in the continuous current discharging conditions, at pressure of 37.5 MPa and current densities of 290 and 700 A for the punch–die–Ti and the punch–die–Al₂O₃ compact systems, respectively.

Fig. 6.7

Schematic diagram of the punch–die–compact system for modeling, with all units in mm. Reproduced with permission from [31]. Copyright © 2003, Elsevier

Fig. 6.8

Positions of measuring temperature for the punch–die–titanium and punch–die–alumina compacts in the system, with the values within the parenthesis for those in the punch–die–alumina compact and unit in mm. Reproduced with permission from [31]. Copyright © 2003, Elsevier

This is a simulation for the first time in which pulses of electrical current, with rectangular shape and duration of 100 ms, and mechanical load applied during the SPS experiments are mentioned and considered [32]. Besides the usual thermal balance, the model also considers heat conduction and Joule effect, coupled to current density conservation equation for the presintered powder compact samples, graphitic die, and plungers. The change in relative density of the presintered powder sample is taken into account when evaluating the effective thermophysical properties throughout the processing. The corresponding thermophysical properties of the full-dense materials, together with empirical functions, i.e., third-order polynomials of relative density (D), have been used for the simulation. The density is treated as a function of temperature (T) and applied nominal pressure (P) according to the following expression [1]:

$$D(T,P) = D_{\text{m}} + aP + b(T - T_{0} )c,$$

(6.8)

where D _m is the initial relative density of the presintered powder sample and T ₀ is the initial temperature of the system, whereas a, b, and c are the parameters that can be adjusted accordingly. Their values are fitted through the comparison of Eq. (6.8) and the relative density values determined experimentally. The density is estimated by measuring longitudinal displacements between the two electrodes, with the assumption that temperature and pressure distributions inside the sample are uniform.

Thermal and electric contact resistances are taken into account as a function of local temperature and are referred to a specific contact surface (horizontal and vertical) between the different elements of equipment, i.e., die/compact, punch/compact, punch/die, and graphite-spacer/punch, as shown in Fig. 6.7 [31]. As a result, the obtained contact conductances and thermal conductivities are in the range of 5 × 10³–7 × 10⁶ S m⁻² and 3 × 10⁻²–2 × 10⁵ W m⁻² K⁻¹, respectively.

The comparisons of the experimental data and modeling results, for the two samples, are shown in Figs. 5.9 and 5.10, respectively [31]. In these figures, temperature contour plot for Ti and Al₂O₃ compacts at steady state at a pressure of 37.5 MPa and constant currents of 290 and 700 A are compared with the temperatures experimentally measured by using thermocouples at six different positions, with two in plungers, three in die, and one at center of the sample. According to Fig. 6.9, the matching for the Ti sample, with maximum deviation of 90 °C that corresponds to an error of about 10 %, is better than that for the Al₂O₃ sample. For the Al₂O₃ sample that is an insulator, current flows through the die. As a result, temperature is lower inside the sample than inside the die, with a difference of about 10 °C. In other words, the Al₂O₃ sample is heated due to the thermal conduction from the surrounding die. In contrast, Ti is a conductor, so that an opposite direction of heat flux is observed, whose temperature is higher than that of die, with a difference of about 30 °C. It means that the temperature gradient for conductive sample is larger for insulative one. In addition, show hot spots in plungers as a result of their smallest cross-sectional area in both cases (Fig. 6.10).

Fig. 6.9

Comparison between experimental data and model results in terms of temperature distribution (Kelvin scale) of Ti compact at t = 3600 s subjected to 37.5 MPa and 390 A. The measurements by six thermocouples at positions P ₁, P ₂, D ₁, D ₂, D _3, and C ₁ are indicated in brackets. Reproduced with permission from [31]. Copyright © 2003, Elsevier

Fig. 6.10

Comparison between experimental data and model results in terms of temperature distribution (Kelvin scale) of Al₂O₃ compact at t = 3600 s subjected to 37.5 MPa and 700 A. The measurements by six thermocouples at positions P ₁, P ₂, D ₁, D ₂, D _3, and C ₁ are indicated in brackets. Reproduced with permission from [31]. Copyright © 2003, Elsevier

A punch–die–two step system has been developed to further increase the homogeneity of the sintered product with complex shapes [33]. The strategy proposed is the reduction in heterogeneity of temperature field distribution inside the samples, as shown in Fig. 6.11. The material is pure Ti powder with average particle of 35 μm. The compact has a length of 10 cm, with two steps of 15 and 30 mm in diameter, at a relative density of 100 %. Ti powder is filled into the graphite die, which is plugged at both ends with graphite punches. Spark sintering is conducted with a combination of electric pulse discharging and continuous electric current discharging, at upper and lower punch pressures of 10 and 38 MPa, respectively. The current density on top surface of the upper graphite punch is 0.14 A mm⁻², while that on bottom surface of the lower punch is 0.56 A mm⁻². The temperatures are measured with thermocouples at eight positions, including two in plungers, three in die, and three in sample axis, as shown in Fig. 6.12, for 3.6 ks under continuous current discharging conditions of 900 A and constant upper and lower punch pressure of 10 and 38 MPa, respectively [33]. The current densities, on top and bottom surfaces of the upper and lower punches, are 1.3 A mm⁻² and 5.1 A mm⁻², respectively. Three types of relative density are used to characterize the compact, including (i) mean density throughout the compact according to Archimedes’ principle, (ii) density measured by using an image analyzer on polished section, and (iii) apparent density according to the longitudinal displacement measured directly between the electrodes during the continuous current discharge.

Fig. 6.11

Schematic of the punch-die-two step cylindrical titanium compact system, with units in mm. Reproduced with permission from [33]. Copyright © 2003, Elsevier

Fig. 6.12

Temperature measurement locations, with units in mm. Reproduced with permission from [33]. Copyright © 2003, Elsevier

The first step is called presintering stage, which is carried out for 15 min with pulsed current, i.e., rectangular pulses of 100 A and duration 100 ms, 1 ON and 1 OFF operations, and constant mechanical load equal to 7 kN, which is about 10 MPa on top plunger and 38 MPa on bottom plungers, respectively. The second step then conducted for 1 h, with constant continuous current of 900 A, while mechanical load is kept unchanged. The whole process is modeled and compared with experimental results. In Figs. 5.13 and 5.14, the experimental data and modeling results in terms of temperature contour plot at 3 and 10 min, correspondingly, are compared [33]. The temperatures experimentally measured with thermocouples at the eight positions are indicated in brackets. A very agreement can be observed between experimental and simulation results, with maximum difference to be about 3 %. It is found that temperature reaches maximum value inside the lower plunger, i.e., at the smallest cross-sectional area, and the Ti sample is always hotter than the surrounding die in the radial direction (Figs. 6.13 and 6.14).

Fig. 6.13

Temperatures in the y-z plane at locations, P1, P2, D1, D2, D3, C1, C2 and C3 at the holding time of 0.18 ks, and calculated isothermal contours under constant current discharge conditions, with nits in K. The values within parentheses were the temperatures measured by the thermocouples. Reproduced with permission from [33]. Copyright © 2003, Elsevier

Fig. 6.14

Temperatures in the y-z plane at positions, P1, P2, D1, D2, D3, C1, C2 and C3 at the holding time of 0.6 ks, and calculated isothermal contours under constant current discharge conditions, with units in K. The values within parentheses were the temperatures measured by the thermocouples. Reproduced with permission from [33]. Copyright © 2003, Elsevier

The presence of thermal conduction heating from the surrounding die to an insulator-like material sample has been further confirmed when analyzing SPS behavior of BN powder [34]. Densification behavior of the powder is well modeled, with the assumption that the SPS process consists of three stages. At the first stage, no obvious shrinkage can be observed although temperature increases continuously. When the process enters the second stage, there is a sharp increase in density due to the abrupt shrinkage, while the temperature is kept almost constant. At final stage, both temperature and density of sample are a constant. Therefore, only the first stage is modeled. The usual 2D cylindrical coordinate thermal and current density balances for the graphite die, plungers, and the powder compact are solved by FEM method. Boundary conditions for thermal balances, i.e., radiation losses from all exposed surfaces, assigned temperature at the plungers’ ends mimicking a perfect cooling system in the SPS system and continuity condition at every contact surface are specified. Natural convection is neglected, because the processing is performed in vacuum conditions.

Model results are obtained at 3 V voltage drop between the plungers’ ends, as temperature distribution at four different times, for a quarter of domain that is due to the cylindrical symmetry of the system. The intensive heat generation in the plungers regulates the heat flux direction inside the sample and die, especially in the beginning of the heating process. Because the BN powder has higher resistivity than the graphite, the temperature of the sample is lower than that of the die at earlier stages. This is because electrical current preferentially flows through the less resistive die. As a result, thus Joule heat effect is more pronounced in the die than inside powder sample. The radial gradient of temperature is reversed near the end of the process, due to the radiation losses from outer surfaces of the die, which increases with increasing temperature. In this case, temperature radial profile inside the powder sample possesses a wavy shape.

Generally, temperature difference in radial direction inside the sample is lower than inside die, which could be about 10 and 40 °C, respectively. The modeling results have been qualitatively validated by the experimentally measured temperatures at different positions inside the sample. However, the experiments are carried out with operative conditions that are different from those ones used for the simulations. In experiments, the temperature is controlled by adjusting electric power input, whereas a constant voltage drop is used for the modeling. Nevertheless, it is suggested that temperature gradient inside the sample should be as low as possible, so that the sample will have a relative density distribution and minimized heterogeneity. This can be realized by lowering the heating rate [34].

The model has also been used to simulate the densification behavior of a conductor-like material, which is TiB₂/BN powder [35]. In this study, electrical current flows through the sample, so that the sample experiences a temperature higher than that of the die. In the axial direction, the temperatures experienced by the plungers continuously increase to the highest values in the whole system, due to the intensive Joule heat generation. Therefore, the sample is heated axially through the thermal conduction from the plungers, while heat is also losed to the die in the radial direction. A high radial difference in temperature inside the sample has been predicted, which is about 150 °C. However, the voltage drop between the rams that is applied as an input to the model is not mentioned and thus has not been compared with experimental data. Similarly, the transient spatial temperature differences inside the sample can be decreased, by lowering the heating rates, which is beneficial to increasing homogeneity of the final sintered products.

Electroconsolidation® is a process that can be used to densify preformed materials with complex-shaped to near-net-shaped by using electrically conductive particulate solids as a pressure transmitting medium [36]. The densification of preformed materials is realized by passing an electrical current through a bed of particulate graphite that surrounds the preform, while transmitting pressure is applied simultaneously by using a double-acting hydraulic press. The surrounding graphite powder, heated up by Joule heating effect, facilitates the preform to densify, due to the increase in temperature through thermal conduction and the pressure applied.

A 2D cylindrical coordinate model, which is developed with respect to the system geometry, consisting of three modules, is used for simulation. The first module describes the density and electrical resistivity of the graphite particulate medium, as a function of the magnitude of mechanical pressure applied. It consists of two empirical algebraic equations, which are determined by fitting the experimental data. A die with sufficiently large diameter is used to obtain the data, so that the effect of powder–die wall friction can be minimized. With this module, together with the temperature-dependent electric resistivity of fully dense graphite, the value of the graphitic powder, as a function of pressure and temperature, is obtained. With the results obtained in the first module, the resistive heating, voltage, and current distribution field inside the whole system can be derived in the second module. After that, the distribution field of resistive heating can be provided to the third module.

The innovative contribution of the model is the consideration of friction between the powder and the internal die, when determining the axial pressure distribution inside the particulate graphite medium [1, 36]. As a result, model results can be used to predict the temporal profiles, which are in a good agreement with the measured values by using thermocouples at two different positions inside the particulate graphite medium.

The relevance of contact resistances in SPS process has been simulated and confirmed, with the simulation shown in Fig. 6.15 as an example [4]. The system simulated is a Model 1050-Sumitomo SPS, where a solid graphitic cylinder is inserted into the die. The 2D cylindrical coordinate system of coupled thermal and electrical problems is numerically solved by using Abaqus (FEM). The heat losses due to radiation from all exposed surfaces, except those on the ends of the rams, have been considered, where a constant temperature of 25 °C is used for the simulation. Thermophysical parameters of all materials are available in that study. A proportional feedback controller based on the outer surface temperature of the die is modeled, in order to determine the voltage drop applied at two ends of the rams. This controller is used to imitate the actual proportional integral derivative (PID), which is observed in real SPS facilities. It is used to apply electric power input to the system when experiments are conducted in terms of temperature controlling.

Fig. 6.15

Geometry of the SPS machine, the corresponding finite element mesh, (right) and schematic boundary conditions (left) for the simulations. Curved arrows represent potential radiations. Contact resistances are also marked. θ _surface is the controlling temperature, and $\varphi$ is the controlled potential applied to the system. Actual calculations are performed in one half of the geometry shown due to axial symmetry. The components referred by the letters A–G can be found in the original publication. Reproduced with permission from [4] by Zavaliangos et al. Copyright © 2004, Elsevier

Boundary conditions of both the horizontal and vertical contacting surfaces, as shown in Fig. 6.15, have been mathematically described with the assumption that all contact conductances are kept to be constant [4]. It is well known that the temperature and the electrical potentials are continuous functions across a perfect interface. Practically, it is difficult to have perfect contact when two real surfaces are considered, due to the geometrical irregularities and surface deposits. Therefore, in a real system, both temperature and electric potentials are discontinuous instead. In other words, a finite temperature (Δθ) or voltage difference ($\Delta \varphi$) should be considered at the interface, as shown in Fig. 6.16, where the thermal flux and electric current density across it can be expressed as:

$$\dot{q}_{\text{c}} = h_{\text{g}} (\theta_{1} - \theta_{2} ),$$

(6.9)

$$j = \sigma_{\text{g}} (\varphi_{1} - \varphi_{2} ),$$

(6.10)

where h _g and σ _g are the thermal and electrical conductances of a gap, respectively. Besides the localized drop of temperature, the heat flux is also discontinuous, because Joule heat is generated due to the electrical contact resistance at the interface, given by:

$$\dot{q}_{\text{ec}} = j(\varphi_{1} - \varphi_{2} ) = \sigma_{\text{g}} (\varphi_{1} - \varphi_{2} )^{2} .$$

(6.11)

Therefore, the heat flux at the contacting surfaces of the parts (1) and (2), as shown in Fig. 3.22, can be expressed as:

$$\dot{q}_{1} = \dot{q}_{\text{c}} - \tfrac{1}{2}\dot{q}_{\text{ec}} = g_{\text{h}} (\theta_{1} - \theta_{2} ) - \tfrac{1}{2}\sigma_{\text{g}} (\varphi_{1} - \varphi_{2} )^{2} ,$$

(6.12)

$$\dot{q}_{1} = \dot{q}_{\text{c}} - \tfrac{1}{2}\dot{q}_{\text{ec}} = g_{\text{h}} (\theta_{1} - \theta_{2} ) + \tfrac{1}{2}\sigma_{\text{g}} (\varphi_{1} - \varphi_{2} )^{2} ,$$

(6.13)

where the Joule heat due to the electric contact resistance is assumed to be evenly divided between the two contacting surfaces. The contact conductances (h _g and σ _g) can be fitted by comparing the model results and the experimental data, which are obtained in the system shown in Fig. 6.21, at a pressure of 15 MPa and a heating rate of 15 °C min⁻¹. Electrical conductances are 1.25 × 10⁷ and 7.5 × 10⁶ S m⁻², while thermal conductances are 2.4 × 10³ and 1.32 × 10³ W m⁻², for horizontal and vertical interfaces, respectively. Therefore, the vertical contact resistances are larger than horizontal ones, which have been attributed to the fact that the mechanical load is applied in the horizontal direction, thus leading to relatively “perfecter” contacts.

Fig. 6.16

Drops of temperature or voltage at an imperfect interface with apparent contact area (SA), due to the thermal contact resistance or electrical contact resistance as heat ($\dot{q}_{c}$), or electric current flux (j), which flows across the interface, respectively. Joule heat ($\dot{q}_{\text{ec}}$) is also generated and evenly distributed into two parts. $\dot{q}_{1}$ and $\dot{q}_{2}$ are heat flux in parts 1 and 2, respectively. Reproduced with permission from [4] by Zavaliangos et al. Copyright © 2004, Elsevier

One of the significances of this model is its ability to differentiate between the horizontal and vertical contact resistances, which are not considered in the previous simulations. Figure 6.17 shows experimental data and simulated results for the differences in temperature at the center of the sample and surface of the die [4]. The model results, with and without considering the thermal contact resistances, have also been compared. Without considering the thermal contact resistance, the simulated results are much lower than the experimental observations. Experimental results indicate that the temperature difference can be more than 200 °C, with the center of the sample being much hotter than outer die, even though the conducting materials used for the two SPS elements are the same.

Fig. 6.17

Comparison between experimental data and model results in terms of temperature difference at the center of the sample and surface of the die. Reproduced with permission from [4] by Zavaliangos et al. Copyright © 2004, Elsevier

Fig. 6.18 shows relative contributions through all the possible heat loss mechanisms, as a function of temperature at surface of the die [4]. The temperature evolution in the plungers/sample/die assembly is a reflection of the synergistic effects of Joule heat generation and heat transfer. The heat is lost from the assembly to the loading train and eventually the water-cooled electrodes through thermal conduction and to the wall of the SPS chamber through radiation. It has been demonstrated that, at low temperatures, most of the heat is lost due to the thermal conduction ($\dot{q}_{\text{c}}$) to the loading train, whereas the loss due to the thermal radiation ($\dot{q}_{\text{r}}^{\text{ps}} + \dot{q}_{\text{r}}^{{{\text{d}}t}} + \dot{q}_{\text{r}}^{{{\text{d}}s}}$) becomes more and more significant, at higher temperatures. Among all mechanisms, the loss due to thermal conduction to the loading train always counts for a significant portion, which is in a range from 40 to 80 %. These conclusions have been supported by the simulation results on the cooling water system, with the assumption that the temperature at ends of the rams is the ambient temperature and constant. Therefore, the heat loss by conduction to the loading train should be included and considered, whenever heat loss is simulated.

Fig. 6.18

Relative contributions of various mechanisms to the heat losses as a function of the controlling temperature. Reproduced with permission from [4] by Zavaliangos et al. Copyright © 2004, Elsevier

Contact resistances have also been well studied in a simulation of the system, as shown in Fig. 6.19 (Model HP D 25/1-FCT, Germany), where a solid cylinder specimen is inserted into the die [20]. The system has two boreholes, where the upper one is used to measure the temperature near the sample with a pyrometer and the lower one is used to fix and position the whole assembly and maintain symmetry. In the balance equations, radiation heat losses are considered from the lateral surfaces. At the same time, an effective convective heat transfer to simulate the axial cooling directly from the ends of the spacers is also included. Experiments are carried out by controlling the temperatures measured at bottom of the upper borehole as close as possible to the center of the sample. The model is solved by using Abaqus (FEM), assuming a total current flowing through the assembly. In order to properly evaluate the Joule heating effect when a pulsed DC is applied, the root-mean-squared (RMS) values of the current and voltage are given by:

$$I_{\text{RMS}} = \sqrt {2/3} I_{\text{peak}} ,$$

(6.14)

$$V_{\text{RMS}} = \sqrt {2/3} V_{\text{peak}} ,$$

(6.15)

where I _peak and V _peak are the peak values of the corresponding pulsed signals experimentally measured as independent variables which are demonstrated in Fig. 6.20, where a pulse sequence of 10 ms ON and 5 ms OFF is used. It means that 10 ms is the duration of direct pulses, which is followed by a period of 5 ms without pulses [20].

Fig. 6.19

Relative contributions of various mechanisms to the heat losses as a function of the controlling temperature. Reproduced with permission from [20] by Vanmeensel et al. Copyright © 2005, Elsevier

Fig. 6.20

Typical “on”-part of a 10/5 ms on/off pulsed current signal provided by the FCT system. The pulsed voltages over the transformer and over the tool are shown. The calculated resistance over the tool is obtained by dividing the pulsed voltage over the pulsed current. The constant DC current signal used during the FEM calculations generating the same power as the pulsed DC signal is also indicated. Reproduced with permission from [20] by Vanmeensel et al. Copyright © 2005, Elsevier

Thermal and electric resistances at contacts between the adjacent elements of the die–plunger–sample setup have been evaluated experimentally by comparing the thermal and electrical responses of three different graphite configurations, with increasing complexity, so as to allow for demonstrating the influence of horizontal vertical resistances of the contacts. The dummy geometries used are schematically shown in Fig. 6.21, where graphite sheets with anisotropic thermophysical properties are inserted at the horizontal and vertical contacts [20]. In the case of the full dummy geometry (F), there is no contact, and the electrical resistivity of the solid graphite (ρ _{el,solid graphite}) has been experimentally verified, when comparing the modeled results with the experimental data, while fitting the effective heat transfer coefficient at ends of the spacers.

Fig. 6.21

Dummy geometries used to establish the influence of horizontal and vertical contact resistances. Thick horizontal lines indicate the presence of horizontal graphite papers, while thick vertical lines indicate the presence of the vertical graphite papers. Reproduced with permission from [20] by Vanmeensel et al. Copyright © 2005, Elsevier

The electrical resistances of the horizontal contacts between the plungers and the spacers as a function of temperature can be evaluated, by subtracting the experimentally measured electrical resistances of the F geometry from those of the SPP geometry (Fig. 6.21) at various steady states. These contact resistances are used to determine the electrical resistivity of the graphite paper ($\rho_{{{\text{el,graphite}}\,{\text{paper}}}}^{{{\text{horizo}}\,\tan }}$) sandwiched at horizontal contacts between the spacers and the plungers, which are modeled by using the thin-film approximation. The corresponding thermal conductivity of the sandwiched graphite paper ($\kappa_{{{\text{el,graphite}}\,{\text{paper}}}}^{{{\text{horizo}}\,\tan }}$) is empirically given by:

$$\kappa_{{{\text{graphite}}\,{\text{paper}}}}^{\text{horizontal}} (T) = \mu \kappa_{{{\text{solid}}\,{\text{graphite}}}} (T)\frac{{\rho_{{{\text{el,solid}}\,{\text{graphite}}}} }}{{\rho_{{{\text{el,graphite}}\,{\text{paper}}}}^{\text{horizontal}} (T)}},$$

(6.16)

where the value of the proportional constant μ is 2.85, which is derived from the fitting results through the comparison of modeled results and the experimental data of the configuration with the SPP dummy geometry. This empirical expression means that the ratio of the thermal and electrical conductivities of the graphite paper sandwiched at the horizontal contacts is proportional to that of solid graphite at any temperature.

Thermal and electrical resistivities of the graphite paper at the invertical contacts between the plungers and the die are given by:

$$\rho_{{{\text{el,graphite}}_{{}} {\text{paper}}}}^{\text{vertical}} (T) = \alpha_{\text{el}} \rho_{{{\text{el,graphite}}_{{}} {\text{paper}}}}^{\text{horizontal}} (T),$$

(6.17)

$$k_{{{\text{el,graphite}}\,{\text{paper}}}}^{\text{vertical}} (T) = \alpha_{\text{therm}} \frac{1}{{\kappa_{{{\text{graphite}}\,{\text{paper}}}}^{\text{horizontal}} (T)}},$$

(6.18)

where the values of the proportional constants, α _el and α _therm, are equal to 7, which is obtained through the fitting procedure, in which the modeled results are compared with the experimental data of the configuration with the GRA dummy geometry (Fig. 6.21) [20]. Therefore, the vertical, thermal, and electrical contact resistances are higher than horizontal ones. However, in this paper, the temperature dependence of contact resistances is determined. In this study, the effect of the applied mechanical load is not simulated, although preset mechanical load temporal changes are used during the experiments, which can be a subject of further study.

The effect of contact resistances can be readily observed, when comparing the radial temperature differences between the center of the sample and external surface of the die as a function of time, for both simulated results and the experimentally measured data. The model results are in a good agreement with the experimental data. The effect and contribution of vertical contact resistances has been clearly observed. The temperature difference increases from 100 to 200 °C, when the configuration is changed from the F and SPP geometries to the GRA one. The higher temperature difference of the GRA geometry is attributed to the combined effect of the vertical thermal contact resistances and vertical electrical resistances. In this case, the electrical current can be confined to flow into the plungers and the sample in series combination. Therefore, Joule heating effect is enhanced by reducing the cross-sectional area [20].

After evaluation with the solid graphitic samples, the model is applied to a fully dense conducting TiN and an insulating ZrO₂. Without experimental data, the modeling results indicate that the radial temperature difference inside the conducting sample (80 °C) is larger than that (25 °C) inside the insulating one. This has been ascribed to the difference in distribution of the current density and thus the Joule heating, due to the parallel combination of the sample and the surrounding die. Also, the conductive heat transfer rates between the two elements (the sample and the die) and the heat loss due to the radiation from outer surface of the die have played an important role. The overall effect is also closely related to the relative sizes of the different elements. Therefore, the measured temperature at the outer surface can be used to represent the sintering temperature of insulating samples, at least for the specific system geometry used in the study. When the samples are conductor-like materials, the real sintering temperature of the samples is significantly different from the temperature measured at outer surface of the die. In this case, if the surface of the die is covered with an insulation layer of porous graphite, the temperature measurement would be more reliable. This is because thus, the use of the porous insulation graphite layer can minimize the heat losses due to the radiation.

The model has also been applied to the FAST of composite powder, ZrO₂–TiN, with compositions from 65/35 to 10/90 vol% [37]. Mixture rules are adopted for the evaluation of thermophysical properties of the sintering powder. According to the electrical (σ) and thermal (κ) conductivities of fully dense ZrO₂ and TiN as a function of temperature, the Polder–Van–Santen (PVS) mixture rule is used to obtain the thermophysical parameters of fully dense ZrO₂–TiN composite, given by:

$$\begin{aligned} \kappa_{\text{mix}} & = \tfrac{1}{4}\left\{ {2\kappa_{\text{m}} - \kappa_{\text{p}} + 3V_{\text{p}} \left(\kappa_{\text{p}} - \kappa_{\text{m}} \right)} \right. \\ & \quad \quad + \left. {\sqrt {\left[ {2\kappa_{\text{m}} - \kappa_{\text{p}} + 3V_{\text{p}} \left(\kappa_{\text{p}} - \kappa_{\text{m}} \right)} \right]^{2} +\, 8\kappa_{\text{m}} \kappa_{\text{p}} } } \right\}, \\ \end{aligned}$$

(6.19)

$$\begin{aligned} \sigma_{\text{mix}} & = \tfrac{1}{4}\left\{ {2\sigma_{\text{m}} - \sigma_{\text{p}} + 3V_{\text{p}} \left( {\sigma_{\text{p}} - \sigma_{\text{m}} } \right)} \right. \\ & \quad \quad + \left. {\sqrt {\left[ {\sigma_{\text{p}} - 2\sigma_{\text{m}} + 3V_{\text{p}} \left( {\sigma_{\text{p}} - \sigma_{\text{m}} } \right)} \right]^{2} + \, 8\sigma_{\text{m}} \sigma_{\text{p}} } } \right\}. \\ \end{aligned}$$

(6.20)

The samples are treated as mixtures consisting a continuous matrix phase (m) and a secondary phase as solute or filler (p). The solute phase has particles with spherical shape which are homogeneously dispersed in the matrix. V _p is volumetric concentration of the solute phase in the mixture, which is either ZrO₂ or TiN, depending on the composition of ZrO₂–TiN mixtures. Thermophysical parameters of the fully dense ZrO₂–TiN composites with different compositions, as a function of temperature, have been studied [37].

The PVS mixture rule is also applied to study the effect of residual porosity on sintering behaviors of the ZrO₂–TiN composite powders, by incorporating porosity as a third phase with zero conductivity. With this assumption, Eqs. (6.19) and (6.20) are still applicable, by using a reduced volume concentration of the secondary phase, $V_{\text{p}}^{*}$, due to the presence of the pores. The reduced volume concentration can be related to relative density (D) of the sintering powder compacts, given by $V_{\text{p}}^{*} = D - V_{\text{m}}^{*}$, where $V_{\text{m}}^{*}$ is the volume fraction of the matrix phase in partially sintered compacts.

Heat capacity of the sintering powder is determined by directly multiplying the temperature and the composition-dependent heat capacity of the fully dense material with the relative density. The relative density of the sintered powder compact (D) varies with time during the FAST process depending on composition of the samples. It can be measure by using the Archimede’s method. Figure 6.30 shows experimental results of representative samples [37]. The corresponding experimental conditions, such as heating rate and pressure cycle, are demonstrated in the inset of Fig. 3.30. It has been assumed that the distribution of temperature inside the sample is uniform.

Thermal and electrical conductivities of the ZrO₂–TiN powder composites with different compositions can be well described as a function of temperature or relative density of the powder compact [37]. It is found that the samples with TiN contents of <70 vol% all experience a sharp increase in electrical conductivity, which is similar to percolation. In other words, during the FAST process, the samples change from an insulator to a conductor, due to the increase in relative density during the sintering. In this case, electrical current and Joule heat generation take place inside the sample rather than in the surrounding die.

It is found that percolation occurs for the sample at temperatures between 1200 and 1300 °C. Figures 5.22 and 5.23 show current density distribution, temperature distribution inside the sample and that inside the system of the sample processed under two sets of processing conditions [38]. The model predicts that the current density is concentrated inside the die when the sample approaches a temperature of 1050 °C, as shown in Fig. 6.22a. When the sample reaches a higher temperature of 1500 °C, the current flows through the sample (Fig. 6.23a). By inspecting Figs. 5.23b, c and 5.33b, c, it is readily observed, at the earlier stage of sintering, that ZrO₂–TiN powder composite behaves as an electrical insulator, and the temperature at center of the sample is lower than that of the surrounding die. In this case, the sample is heated due to thermal conduction from the die. As the sintering is proceeded, the sample center becomes hottest, with a temperature much higher than that side of the die. This is because, at this stage of sintering, the sample experiences a transition from insulator to conductor, so that the current flows through it and thus the Joule heating becomes to be dominant. At the same time, the heat loss due to radiation from the exposed surface of the die is more and more pronounced. As a result, high temperature differences in radial direction are observed inside both the sample and the die, which are about 150 °C and about 400 °C, respectively.

Fig. 6.22

Current density distribution (a), temperature distribution inside the sample (b), and inside the system (c), for the SPS of the ZrO₂–TiN composites (60/40), at a controlling temperature of 1050 °C at a pressure of 28 MPa. Reproduced with permission from [38]. Copyright © 2007, Elsevier

Fig. 6.23

Current density distribution (a), temperature distribution inside the sample (b), and inside the system (c), for the SPS of the ZrO₂–TiN (60/40) composite, at a controlling temperature of 1500 °C at a pressure of 56 MPa. Reproduced with permission from [38]. Copyright © 2007, Elsevier

Densification behaviors of electrical conducting Cu and insulator Al₂O₃ with SPS have been compared and modeled, with the portion of the system schematically shown Fig. 6.24 [39]. In this modeling study, stainless steel electrodes (rams) are included in the model. The boundary conditions of axial cooling and the consequent temperature distribution change dramatically. The mathematical equations used in the modeling, i.e., the enthalpy and current density conservation equations in cylindrical coordinates applied to the elements, are solved in full 3D version through CFD-Ace (FEM). Temperature dependences of thermophysical properties for all materials have been considered.

Fig. 6.24

Schematic of the portion of the spark plasma sintering apparatus used for modeling. Reproduced with permission from [23]. Copyright © 2005, Elsevier

During the simulation, radiation heat losses from the exposed surfaces are considered, with the assumption that the ambient temperature, the voltage drop, or total flowing current at end surfaces of rams are constant, while contact resistances at the interfaces between different adjacent elements are neglected. The simulation results on the distribution of flowing current density are similar to those that can be found in the open literature for RS. The behavior of the Sumitomo SPS DC pulse generator is analyzed and monitored by using an oscilloscope [23]. The complex temporal profiles of the voltage between the ends of the rams are measured under the 8/2 ON/OFF operation. The current pulse pattern consists of eight direct voltage pulses of fixed time duration of 3 ms (ON), followed by two pulses of time duration of 3 ms with zero voltage (OFF). This operation leads to a low-frequency signal, which is less than 100 Hz, as described by the Fourier analysis of the pulse pattern.

In this case, the relevant skin effect would occur at higher frequencies due to the electrical conductivities and dimensions of the different elements used in the SPS system, because skin depth is given by $\delta = 1/\sqrt {f\mu \sigma }$, where f is frequency, μ is magnetic permeability, and σ is electrical conductivity. Therefore, under the operation with the 8/2 current pulse pattern, the electrical current would not be confined within a thin layer of the external surfaces due to the electromagnetic induction, but instead, it is distributed along the cross-sectional areas, according to the relative electrical conductances in series or parallel combination. This allows the model to consider just ohmic behavior of the materials, so that all the assumptions for the modeling are justified and that the current density simulated with the conservation equations would have well represented the real scenario of the experiments.

Figure 6.25 shows current distribution profiles of the Al₂O₃ and Cu samples at room temperature, which are calculated with a constant voltage of 4 V that is applied across the entire assembly [23]. The samples have a thickness of 3 mm. Contact resistances are assumed to be zero during the simulation. A distinct difference in the current distribution can be observed, especially the currents flowing through the samples. Because Al₂O₃ is an insulator, no Joule heating is expected at this temperature. In other words, the initial heating of the sample is not due to the current. In contrast, Cu is a conductor, so that Joule heating starts immediately. In both cases, the highest current densities are observed along the exposed portions of the plungers [23].

Fig. 6.25

Schematic of the portion of the spark plasma sintering apparatus used for modeling. Reproduced with permission from [23]. Copyright © 2005, Elsevier

The simulated results on dependences of the fraction of the total current that flows through the sample, as a function of electrical conductivity at room temperature, over a range of eighteen orders of magnitude, have been available [39]. The lower end corresponds to the electrical conductivity of Al₂O_3, while the upper end the conductivity of Cu. Two samples have thicknesses of 3 and 10 mm, while the rest parameters of the system geometry are kept constant. It is found that no current flows through the samples whose conductivity is less than about 10³ S m⁻¹. For 10 mm thick sample, almost all current is carried by the conducting sample (such as Cu). This can be simply attributed to the relative resistances of the two current paths, either through the sample (for conducting samples) or the surrounding die (for insulating samples). Because Cu has higher conductivity than graphite of the die, when the thickness of the sample is increased, the increase in resistance of the die is more significant than that of the sample (Cu), and the portion of current flows through the sample is increased with increasing thickness.

Electrical resistances of samples with a thickness of 3 mm measured by using a four-point method at room temperature across the die with the plungers in place for Al₂O₃ and Cu have been examined, as a function of applied mechanical pressure. There is a critical point of pressure that is about 50 MPa, below which the resistance of the case with Cu in side the tie is lower, whereas above which the resistances are the same for the two types of samples. In addition, the measured resistances decrease with increasing pressure for both Al₂O₃ and Cu samples. Theoretically, the overall resistance for the assembly of die and plungers, evaluated based on the room temperature resistivity of graphite, without including the contact resistance, is about 1.5 × 10⁻³ Ω. As the applied pressure is increased, the resistance approaches this theoretical value for both samples. This observation implies that contact resistance has a stronger effect than the resistivity of the samples and the contact resistances can be safely excluded, if all experiments are carried out at sufficiently high pressure levels, e.g., ≥100 MPa.

Experimental data and modeling results of the Al₂O₃ sample have been compared, in terms of the temporal profiles of the temperatures at the center of the sample and surface of the die, when the system is subjected to a 400 s constant current step at 1000 A [23]. It is found that modeling results underestimate the real experimental results by about 200 °C. The modeled temperature at the center of the sample is lower than that at the surface of the die, which is opposite to the experimental observation. Therefore, further study is necessary to improve the occurence and capability of the modeling.

A constitutive model has been developed to model SPS behaviors of compacts of conducting powdery materials, such as Al powder [40]. The model is able to take care of the effect related to the flowing of electrical current, which is different from Joule heating. In this model, a constitutive law is proposed for sintering and densification of the poser, which is stress–strain mechanics. Grain-boundary diffusion, power-law creep driven by externally applied loading, sintering stress due to surface tension, and steady-state electromigration are all taken into account when developing the model. Schematic diagram of the system for modeling is shown in Fig. 6.26 [40]. The sample is a simply packed powder compact, with rectangular grains of semi-axes a and c and elliptical pores located at the grain quadra-junctions. Here, the electromigration is to represent the contribution of electric field to the diffusive mass transfer during the sintering process of the powder compacts.

Fig. 6.26

Representative unit cell including rectangular grains and ellipsoidal pores in quadra-junctions. The pore–grain structure is subjected to a simultaneous action of external load, surface tension, and electromigration. The electric field is macroscopically unidirectional and can branch locally. Reproduced with permission from [40]. Copyright © 2006, Elsevier

According to Nernst–Einstein equation, matter flux $\overline{J}$ caused due to the grain-boundary diffusion is driven by stress and electromigration, which can be described as:

$$\bar{J} = C_{\text{E}} \bar{E} + C_{\sigma } {\bar{\nabla }}\sigma ,$$

(6.21)

where $\bar{E}$ is electric field and ${\bar{\nabla }}\sigma$ is gradient of stresses normal to grain boundary, whereas C _E and C _σ are temperature-dependent parameters related to mass diffusivity of the grain boundary.

The electric field is used an input of the model, which is evaluated as E = V/l, where V is the voltage drop along a characteristic length of the grain, l. ${\bar{\nabla }}\sigma$ can be obtained by solving the equilibrium mechanical problem, which allows to simulate the stress field distribution due to the surface tension (free-sintering) and the sintering under external loading to the powder compact. After that, linking the strain rate $\dot{\varepsilon }$ to the matter flux $\bar{J}$ in orthogonal direction leads to the shrinkage kinetics due to the diffusion of grain boundary, $\dot{\varepsilon }_{\text{gb}}$, which is given by:

$$\dot{\varepsilon }_{\text{gb}} = \dot{\varepsilon }_{\text{gb}}^{\text{E}} + \dot{\varepsilon }_{\text{gb}}^{\text{st}} + \dot{\varepsilon }_{\text{gb}}^{\text{load}} ,$$

(6.22)

which includes the three contributions of electromigration, surface tension, and external load. The total shrinkage rate $\dot{\varepsilon }$ is the summation of the shrinkage rate due to the grain-boundary diffusion and the power-law creep mechanism, thus given by:

$$\dot{\varepsilon } = \dot{\varepsilon }_{\text{gb}} + \dot{\varepsilon }_{\text{creep}} .$$

(6.23)

The shrinkage rate due to power-law creep, $\dot{\varepsilon }_{\text{creep}}$, can be evaluated based on the continuum theory of sintering, which deals with the dependences of normalized shear and bulk viscosity modules on porosity, θ, along with the effective sintering.

It has been found that the contribution of external loading to grain-boundary diffusion shrinkage kinetics, $\dot{\varepsilon }_{\text{gb}}^{\text{load}}$, can be neglected, so that it only has influence on the power-law creep phenomenon. A densification map for Al powder has been established, in which electromigration, $\dot{\varepsilon }_{\text{gb}}^{\text{E}}$, has a significant contribution to the total shrinkage, when the modeled powders have relatively low porosity and small grain size [40]. For micron-sized powders, power-law creep is the dominant mechanism for the shrinkage kinetics.

In a 2D axial coordinate system, x and y, porosity rate is related to strain rate, which can be expressed as:

$$\dot{\theta } = (1 - \theta )(\dot{\varepsilon }_{x} + \dot{\varepsilon }_{y} ).$$

(6.24)

It is assumed that, during the SPS process, powder compact is pressed in a rigid die, i.e., $\dot{\varepsilon }_{y} = 0$, the densification kinetics can be given by:

$$\dot{\theta } = (1 - \theta )(\dot{\varepsilon }_{{{\text{gb}},x}}^{\text{E}} + \dot{\varepsilon }_{{{\text{gb}},x}}^{\text{st}} + \dot{\varepsilon }_{{{\text{gb}},x}}^{\text{load}} + \dot{\varepsilon }_{{{\text{creep}},x}} ).$$

(6.25)

With this equation, the temporal profile of porosity during the SPS of Al powder can be derived numerically, with the available experimentally measured temporal profile of temperature, together with the applied voltage gradient and constant pressure. It is found that the model satisfactorily predicts the shrinkage kinetics of the powder compacts.

ECAS behaviors of electrical conducting Cu and electrical insulating Al₂O₃ by using the system shown in Fig. 6.27 have been modeled [41]. In this system, copper electrodes with a height of 2 cm are considered. In this ECAS study, a DC generator is used to provide the electrical current, while a mechanical loading is applied on the top ram, whereas the bottom ram is blocked by its end at a fixed vertical position. The model is for the first time to couple the usual enthalpy and current density conservation equations to the stress–strain mechanical equilibrium equations. The 2D cylindrical coordinates, with FEM minimization of the total mechanical potential energy related to elastic and thermal expansion, are solved to determine the stress distribution field inside the system, due to the mechanical and thermal loads. For boundary conditions of the thermal and electrical equations, radiation heat losses from all the exposed surfaces have been considered, while the ambient temperature and the voltage drop or total current flowing the cross section of the rams are assumed to be constant, whereas all contact resistances at the interfaces between different elements of the system are neglected. The later assumption could be valid, because during the experiments, a constant pressure of 140 MPa is applied, as discussed previously. This model is similar to that in the previous studies. A pretty good agreement can be observed between the experimental and the modeled results during the heating and cooling process [41].

Fig. 6.27

Schematic of the current-activated sintering apparatus, showing dimensions and coordinate system, with the origin to be at the center of the sample. The inset in the top right-hand corner shows the relationship of the three-dimensional stresses in the radial, angular, and vertical directions. Reproduced with permission from [41]. Copyright © 2007, Elsevier

However, for the conducting Cu sample, simulation results indicate that the temperature is usually lower than that of Al₂O₃ case, whereas the temperature difference inside the sample at steady state is higher for Cu than for Al₂O₃. Figure 6.28 shows the radial mechanical displacement distribution at steady state, as the system is applied with a current of 1000 A and a pressure of 140 MPa for the case of alumina and copper sample. The corresponding radial and axial components of the stress distribution field, due to thermal and Poisson-type expansion, are illustrated in Fig. 6.29, for the Al₂O₃ (a and c) and Cu (b and d) samples [41].

Fig. 6.28

Steady-state contour plots at 1000 A current input: a radial displacements of sample–die area of the Al₂O₃ sample (≈705 °C at center of sample) and b radial displacements of sample–die area of the Cu sample (≈692 °C at center of sample). Reproduced with permission from [41]. Copyright © 2007, Elsevier

Fig. 6.29

Steady-state contour plots at 1000 A current input: a radial stress of sample–die area of the Al₂O₃ sample (≈705 °C at center of sample), b radial stress of sample–die area of the Cu sample (≈692 °C at center of sample), c vertical stress of sample–die area of the Al₂O₃ sample (≈705 °C at center of sample), and d vertical stress of sample–die area of the Cu sample (≈692 °C at center of sample). Reproduced with permission from [41]. Copyright @ 2007, Elsevier

In the plunger–die–sample region, the radial displacement increases relatively uniformly with increasing radius, and thus, the highest displacement is observed near the surface of the die. These radial displacements are due to the synergistic effect of the thermal expansion and a Poisson-type expansion due to the pressure applied in the vertical direction. In contrast, the radial distribution, in the case of the Cu sample (Fig. 6.29b), does not increase uniformly with the radial position. Instead, the portion where the copper is in contact with the die wall has larger displacement than the portions where the plunger is in contact with the die wall, i.e., graphite on graphite. The top portions have higher displacement due to the uniform pressure applied to the top electrode and the boundary condition that the vertical displacements of the lower end of the apparatus are set to be zero [41].

Under steady-state conditions, the differences in temperature inside the sample in radial direction are modest, with a maximum difference of 2 °C, corresponding to a variation of 1 %, whereas those in stress are relatively higher, with a maximum difference of 50 MPa, corresponding to a variation of 33 %. Therefore, the heterogeneity of the final sintered product is more due to the uneven distribution of stress than the distribution of temperature. As a result, the nominal sintering pressure, i.e., that obtained by the mechanical load applied over the cross-sectional area of plungers, and the sintering temperature recorded by using the pyrometer on the outer surface of the die may not be proper indications of the operative conditions of the experiments.

It is obvious that quantitative comparisons with experimental data are important to demonstrate the reliability modeled results. The agreement between the real experimental parameters and conditions, such geometry of system and properties of materials involved, and corresponding representations used for the simulation should be maintained. An actual modeling could be a complicated process, because various adjustable parameters for the mathematical description of the control system have to be used during the modeling. Therefore, it is more convenient to provide quantitative comparisons with experimental data in temperature than in voltage/current and mechanical displacement. It is therefore suggested to model the ECAS processes by separately analyzing a gradually increasing complexity according to the system behavior, so that the modeling could be conducted in the framework of a step by step procedure. Various physicochemical phenomena can be gradually included, along with their unknown modeling parameters. As a result, it is possible to independently fit the complete set of unknown parameters of the comprehensive ECAS model, thus avoiding the masking effect given by the various phenomena involved in the whole process [1].

These considerations have been accomplished in an example to model the SPS behavior of graphitic elements that are inserted between the two stainless steel rams used in the model 515S system (Sumitomo), which is schematically shown in Fig. 6.30 [42]. The geometry of the graphitic elements has been designed in such a way that it not necessary to consider the effect of vertical interfaces between them. As a result, the horizontal contact resistances can be excluded in the mathematical models. A 2D model in cylindrical coordinates based on the usual enthalpy conservation equation that takes into account the Joule heat generation is developed, which is coupled to density current balances expressed in terms of the RMS portion of the electric potential and the mechanic equilibrium equations due to elastic behavior and thermal expansion of the materials. Thermophysical properties of graphite and stainless steel are used to model different graphitic elements inserted between the two rams. Monolithic graphitic elements are used to mimic the plungers positioned in place within the die.

Fig. 6.30

Schematic of the SPS experimental setup (not to scale). Reproduced with permission from [42]. Copyright © 2007, John Wiley & Sons

In the real geometry of the system shown in Fig. 6.30, stainless steel rams and the corresponding water cooling system with different lengths have been modeled [42]. Because of this, the consideration of only one quarter of the integration domain is not applicable, because axial direction is not symmetric. To solve the model with FEM method, boundary conditions related to the thermal problem, including the radiation losses from all exposed surfaces, a constant ambient temperature at ends of the rams, and the forced convective losses from the rams along the contact interface with the cooling water, are adopted. To maintain the radial direction to be symmetric for simplicity of simulation, an hypothetical geometry of water cooling circuit according to that in the real system is calculated, with the assumption that the heat flow rate loss and the Joule heat generation inside the rams are equal, whereas the real heat transfer coefficient is determined by using the typical correlations of forced convections in closed ducts.

Only the horizontal thermal contact resistances between the stainless steel rams and the graphitic spacers are considered. Experimental results indicate that the horizontal thermal contact resistances between the graphitic elements can be neglected. Therefore, the continuity condition in terms of temperature and conductive heat flux is valid.

Regarding boundary conditions for the electrical problem, since current-controlled experimental runs are performed, the RMS portion of voltage drop between the two ends of the rams is determined by assigning the RMS value of the total electrical current flowing through the system, i.e., an integral equation of the gradient of RMS portion of electric potential is applied. Similarly, horizontal electrical contact resistances are considered to be present only between the stainless steel rams and the graphitic spacers, because also horizontal electrical contact resistances between graphitic elements are experimentally found to be negligible. Accordingly, the continuity condition for the resistive portion of the RMS electric potential and current density is applied. In particular, by simultaneously fitting experimental data in terms of temporal profiles of RMS portion of electric potential and temperature, thermal and electrical contact conductances are determined as a function of temperature and applied mechanical pressure through the following relationships [42]:

$$C_{\text{T}} (T,P) = \alpha_{\text{T}} \kappa_{\text{Harm}} \left( {\frac{P}{{H_{\text{Harm}} }}} \right)^{{\beta_{\text{T}} }} ,$$

(6.26)

$$C_{\text{E}} (T,P) = \alpha_{\text{E}} \sigma_{\text{el,Harm}} \left( {\frac{P}{{H_{\text{Harm}} }}} \right)^{{\beta_{\text{E}} }} ,$$

(6.27)

where k _Harm and σ _el,Harm are the parameters that have included temperature dependence of the contact conductances, which are expressed by using the harmonic mean of the individual thermal and electrical conductivities of graphite (G) and stainless steel (SS), expressed as:

$$\kappa_{\text{Harm}} = \frac{{2\kappa_{\text{G}} (T)\kappa_{\text{SS}} (T)}}{{\kappa_{\text{G}} (T) + \kappa_{\text{SS}} (T)}},$$

(6.28)

$$\sigma_{\text{el,Harm}} = \frac{{2\sigma_{\text{el,G}} (T)\sigma_{\text{el,SS}} (T)}}{{\sigma_{\text{el,G}} (T) + \sigma_{\text{el,SS}} (T)}}.$$

(6.29)

The use of the harmonic mean is the requirement of the series combination between the two interfacing materials, in order to characterize the properties of the real contacts. The parameter H _Harm is the harmonic mean between the hardness of graphite and stainless steel, given by:

$$H_{\text{Harm}} = \frac{{2H_{\text{G}} H_{\text{SS}} }}{{H_{\text{G}} + H_{\text{SS}} }}.$$

(6.30)

α _T, β _T, α _E, and β _E are adjustable parameters, which are dependent on surface roughness and plastic behavior at contact interface between the graphite and stainless steel elements. They can be derived from fitting procedures. The corresponding electrical and thermal conductances in the range of temperature and mechanical load in that study are 0.25–1.7 × 10⁶ Ω⁻¹ m⁻² and 0.007–2 × 10³ W m⁻² K⁻¹, respectively, which are comparable with the literature values [43]. The experimental data used for the fitting procedure are obtained from the system, with only one graphitic element of 8 cm in diameter between the two rams, which is subject to a RMS current of 1200 A with varying mechanical loadings of 0.6–10 MPa [42].

Figure 6.30 also shows that a Hall effect transducer and a voltage isolation amplifier are used in the system, which provides independent measurement of the pulsed current and voltage drop between the two rams, respectively. The RMS values of the electrical variables are important for the correct evaluation of the Joule heat distribution, which is demonstrated by the comprehensive analysis of the system behavior from the electromagnetic perspective. With these instantaneous measurement results, the corresponding average values of various parameters can be calculated as [42]:

$$\bar{I} = \frac{1}{\tau }\int\limits_{t}^{t + \tau } {I(t){\text{d}}t,}$$

(6.31)

$$\bar{V} = \frac{1}{\tau }\int\limits_{t}^{t + \tau } {V(t){\text{d}}t,}$$

(6.32)

$$I_{\text{RMS}} = \sqrt {\frac{1}{\tau }\int\limits_{t}^{t + \tau } {I^{2} (t){\text{d}}t} } ,$$

(6.33)

$$V_{\text{RMS}} = \sqrt {\frac{1}{\tau }\int\limits_{t}^{t + \tau } {V^{2} (t){\text{d}}t} } ,$$

(6.34)

A magnetic self-inductive response due to the pulsed electrical loading could be observed, depending on current level, pulse cycle, and system geometry. As stated earlier, a low-frequency (<100 MHz) pulse is not able to induce skin effect on electrical current path through the system [23]. The 515S SPS machine by Sumitomo has a built-in measurement system, providing the average values of the current flowing through the system and the average values of voltage drops between the rams. Depending on the pulse cycle adopted and the level of current flowing through the system, the RMS values of the current can be more than two times higher than the corresponding average values. According to circuit theory, this behavior is mainly caused by nonideal inductor, which is a series combination of resistance (R)–inductance (L), whose relationship is given by [42]:

$$V = V_{\text{R}} + V_{\text{L}} = {\text{R}}I + {\text{L}}\frac{{{\text{d}}I}}{{{\text{d}}t}},$$

(6.35)

where V and I are the instantaneous voltage and electric current, respectively.

This observation can be readily attributed to the fact that SPS systems involve time varying electromagnetic fields. However, in this case, the assumption of current density conservation is not valid. As a result, the Maxwell’s equations should be developed in such a way that the magnetic induction and the corresponding Joule heat distribution are included. Moreover, the whole model should be solved in the timescale of the generated electric pulses (fractions of millisecond), while temperature and displacement change in a much larger timescale of at least serval seconds. Obviously, such a mathematical description could have a very high degree of stiffness and its solution could be very time-consuming. To address this problem, a macroscopic nonideal inductor has been used, through a distributed parameter modeling approach, in which two models are superpositioned. One is a steady-state magnetic model outside the conductor, i.e., outside the element assembly of the SPS system, by considering the inductive character of the system. The other is a steady-state conduction model inside the conductor, which takes care of the resistive character of the system.

The conduction model is not different from the usual current density balance expression in mathematical form in terms of electric potential. The only difference is that this model is able to determine only the resistive portion of the electric potential. Also, the measurable voltage drop between the two rams represents both the resistive and the inductive contributions. This problem is solved by using a Fourier series development which evidences that the experimental counterpart of the modeled resistive portion of the RMS voltage can be obtained with φ = RI _RMS, where the resistance (R) is determined from the measured average voltage and electrical current by ${\text{R}} = \bar{V}/\bar{I}$. As for the boundary conditions of the mechanical problem, the bottom surface of the lower ram is assumed to be subject to a uniformly distributed load, whereas a blocked displacement is applied at the top surface of upper ram in both the radial and the vertical directions. It is also assumed that all the exposed surfaces are free, which means that there is no radial and shear component of stress. Friction forces in horizontal contacts are neglected, so that continuity condition can be applied on stress axial component and vertical displacement between the two contacting sides of the interfaces.

Mathematically, the model has been coupled with both temperature and electrical current distributions. Therefore, as the thermal and electrical current balances are derived, the stress–strain distribution and the displacement temporal profile can be determined, because the temperature distribution represents the input for the mechanical problem. The assumption that the spatial domain of integration does not change substantially due to thermal expansion and mechanical load is reasonable for the SPS system without powder sample involved. The magnitude of the negative displacement due to the thermal expansion is about 1 mm, as compared with the height of the system which is 1 m. It is also reasonable to assume that the contributions of thermal expansion and mechanical loading are equally distributed among the different elements, i.e., the stainless steel electrodes. However, if powder samples are involved in the system, a positive displacement with a magnitude of about 3–5 mm will be observed in the shrinking powder sample. In this case, these assumptions may no longer be valid for the modeling. Instead, moving boundaries should be considered in modeling, especially when simulating the spatial–temporal profiles of density of the final products.

It is found that a pretty good agreement is observed, during both the heating and cooling processes. The problem is that the model prediction without including the contact resistances is not satisfactory. This conclusion is supported by independent simulation results [42], where axial temperature and RMS portion of electric potential profiles at steady state are assumed for the system assembly that is subject to a RMS current of 1200 A and a mechanical load of 10 MPa. Because the graphite elements have lower electrical resistance, the voltage drop is mainly at the contact interface with the stainless steel rams. Thermal contact resistances are less significant, because the conductive heat flux at the contact interface due to the Joule heat contribution is discontinuous. Even if, for the main part of their length, almost the whole length of the stainless steel rams is at ambient temperature, while only a very small portion at the end contacting with the graphite element is at high temperature, which however contributes significantly to the displacement measured experimentally. This is simply because the coefficient of thermal expansion of stainless steel is three times higher than that of graphite. Therefore, the water cooling system inside the full length of the rams must be modeled when simulating the displacement that is measured at bottom end of the lower ram [42].

More recently, there has been a report on systematic study of the ECAS behaviors, with Al₂O₃ (insulator) and Mo (conductor) as targeted materials [44]. The effects, including die geometry (thickness and height), heating rate, heat dissipation, material properties, and sintering temperature, on microstructural nonuniformities have been analyzed. Figure 6.31 shows reference geometry of the system for simulation. The parameters used in the analysis are divided into three groups. The first group defines the geometry of the plunger–die–sample assembly, including dimension of the sample, wall thickness and height of the die, and the exposed section of the plunger [44]. The second group includes the parameters controlling the thermal cycle, such as target temperature and heating rate. The final group includes the amount of radiative heat losses and the presence/absence of electrical and thermal contact resistances. Electrical and thermophysical properties of graphite used for the die are assumed to be constant.

Fig. 6.31

a Reference geometry of the spark plasma sintering apparatus, with dimensions in millimeters. b Control temperature profile of the reference experiment. Reproduced with permission from [44]. Copyright © 2013, Springer

Figure 6.32 shows simulation results on electrical current density and temperature distribution in radial direction for the two samples [44]. For insulative Al₂O₃ (Fig. 6.32a), the current cannot flow through the sample, but through the die which is around it instead. For different wall thicknesses, the current profiles are similar, which are characterized by the presence of a peak at near the edge of the sample and a marked drop toward the external surface of the die. The temperature profile within the sample (Fig. 6.32c) is determined by the temperature of the wall of the die. Therefore, a decrease in the wall thickness means an increase in the local current density, which thus produces higher localized Joule heating and higher temperatures at the edge of the sample. As a result, the temperature profile within the wall of the die decreases in the radial direction, due to both the current distribution and the heat losses by radiation. A discontinuity across the interface between the sample and the die is always observed, which is attributed to the presence of a thermal resistance at the interface. It is worth mentioning that the presence of both positive and negative radial temperature gradients for the insulative samples is of particular interest, because it offers a possibility to minimize the temperature gradients through the proper design of the die.

Fig. 6.32

The effect of the die thickness. Electrical current density distribution of Al₂O₃ (a) and Mo (b) samples. Temperature distribution of Al₂O₃ (c) and Mo (d) samples. All results are obtained with the reference geometry. Reproduced with permission from [44]. Copyright © 2013, Springer

For the conductive sample (Mo), the electrical current flows through both the sample and the die (Fig. 6.32b). In this case, the current density distribution is only affected by the wall thickness of the die, resulting in the temperature distribution profiles as shown in Fig. 6.32d. Because a large fraction of the electrical current flows through the sample and consequently through the plungers that are adjacent to top and bottom surfaces of the sample, higher temperatures are always observed at the central region of the sample (Fig. 6.32d) in this case. As the thickness of the die wall is decreased, a more homogeneous temperature distribution within the sample is observed, but the effect is very marginal [44].

The effects of other factors, such as the free length of the plungers and contact resistances, have also been systematically studied [44]. From this systematic study, several conclusions have been arrived. First of all, there is possibility to optimize the experimental conditions so as to reduce the temperature inhomogeneities within the samples. As for the geometrical parameters, the temperature gradient in the sample is highly sensitive to the wall thickness and height of the graphite dies, while the exposed length has a negligible effect in this aspect. It is further found that the effects of various parameters on temperature distribution profiles are more significant for insulative samples than for conductive samples, due the difference in the flowing electrical current in the two cases.

The temperature distribution has no significant dependence on the heating rate if the analysis is carried out after the temperature is stabilized. The transient effects on the sintering of the materials can be safely neglected. The positive effect of insulation of the external surfaces of the die on homogeneous temperature distribution of the samples is only pronounced for conductive samples. For insulative samples, the actual result even could be opposite, depending on the size of the samples. The effect of contact resistances is also related to the type of the samples. The presence of contact resistances means an increase in the temperature inhomogeneities for the insulative samples, while for the conductive samples, the effect is negligible.

Due to the intrinsic complexity of the problem of the sintering technique and the strongly correlated relationship among the various variable parameters, it is still a challenge to predict the effect on the temperature distribution of the samples, by just adjusting one of the experimental parameters. In most cases, a thorough optimization of all the experimental conditions is necessary, especially for samples with large dimensions and for scale-up production for industrial applications [44].

Another example, which is necessarily mentioned, is the study on the scalability of SPS [45, 46], because the previous common sense is that the technique is not suitable for large-scale production [47]. Although efforts have been made in trying to scale-up the production, the progress is far away the requirement for real applications [48, 49].

Experimental studies are carried out first, with four dies fabricated to produce final products with diameters of 15, 40, 48, and 56 mm, as shown representatively in Fig. 6.33 [45]. The assembly components include external spacer, transition spacer, punch, insert, and sample. The die is used to encase the sample, as well as the two inserts and portions of the upper and lower punches. The lateral surfaces of the die and end faces of the punches are covered with carbon paper to prevent material adhesion to the graphite components. The inner and outer diameters and height of the die are maintained in an approximate aspect ratio of 1:2:2. A commercial Al₂O₃ powder is used for this study.

Fig. 6.33

Tooling assembly for the spark plasma sintering (SPS) experiments: a 15 mm and b 56 mm. Reproduced with permission from [45]. Copyright © 2012, John Wiley & Sons

According to the experimental results, in terms of relative density and grain-pore structure, it is found that, despite certain deviations in some cases, the samples with different tooling sizes and temperature regimes are quite similar, when they are processing with the same SPS facility (FCT). The samples also have relatively uniform density and grain size spatial distributions, which means that SPS is potential to be scaled up. At the same time, a significant size impact in case of high heating rates and large specimen sizes has been observed. In addition, when different SPS systems are used, the outcomes could be possibly different.

By using a thermo-electromechanical finite element framework, shrinkage and grain growth kinetics, as well as the temperature, porosity, and grain size gradients, of the four samples with different sizes have been analyzed. The thermal and electrical conduction problems are strongly coupled, which has been taken into account by expressing the electrical properties of the materials as functions of temperature. The thermophysical parameters, such as heat capacity and thermal conductivities, are also expressed as functions of temperature. The thermal, electrical, and porosity components can be linked because the electrical and heat transfer properties can be expressed as functions of porosity. The consolidated system of partial differential equations has been solved with all the boundary conditions, including electrical, thermal, and mechanical, which are assigned simultaneously at the same time step.

As boundary and initial conditions, for the electrical component of the modeling framework, the end of the lower external spacer is considered to be grounded and thus its potential is set to be zero. A potential is applied at the end of the upper external spacer, which is specified to generate a preset time–temperature profile on the outer die surface. The free surfaces are regarded to be electrically isolated.

For the temperature component of the modeling framework, the initial temperature is set to be the ambient temperature (300 K). Thermal boundary conditions on the tooling surfaces are considered in the forms of a specified boundary temperature or heat flux. The temperatures of the two extreme upper and lower surfaces of the external spacers are assumed to be constant of 300 K. Heat loss due to radiation is considered at the lateral surfaces to be: $f = \nu \varepsilon \left( {T_{\text{W}}^{4} - W_{0}^{4} } \right)$, where f is the heat flux per unit area, m is the emissivity, which is assumed to be 0.8, ɛ = 5.6704 × 10⁻⁸ W m² K⁴ called the Stefan–Boltzmann constant, T _W is the temperature of at surface of the die, and T ₀ is the ambient temperature [23]. Contacts are modeled as perfect interfaces without losses. All simulation results are compared with experimental results [46].

Figure 6.34 shows numerical results of the finite element modeling code in terms of longitudinal and transversal slices for the final temperature distributions of the 56 mm–100 °C min⁻¹ simulation (a) and the 15 mm–100 °C min⁻¹ simulation (b). The transversal slices are taken at the middle of the die, so as to expose the temperature distribution inside the specimen.

Fig. 6.34

a 56 mm configuration–100 °C min⁻¹ regime: temperature distribution. Left longitudinal view. Right transversal view, with unit of °C. b 15 mm configuration-100 °C min⁻¹ regime: temperature distribution. Left longitudinal view. Right transversal view, with unit of °C. Reproduced with permission from [46]. Copyright © 2012, John Wiley & Sons

Temperature evolution profiles at the center of the samples for the three heating rates for the 56 mm–100 °C min⁻¹ simulation and for the 15 mm–100 °C min⁻¹ simulation are shown in Fig. 6.35a, b, as a comparison. It is observed that the temperature at the center of the samples reaches the holding temperature earlier than for the 300 °C min⁻¹ case, but with similar stabilized value to the other two cases. The 100 °C min⁻¹ heating regime is slower for the 56 mm sample, but reaches a slightly higher temperature. This trend is opposite for 15 mm sample. Therefore, the two samples of different size should experience different evolution in material structures, such as relative density and average grain size [46].

Fig. 6.35

Modeling results for 56 mm (left) versus 15 mm (right) specimens: a and b temperature evolution in the center of the specimen; c and d relative density; e and f grain size. The calculation results are given for different heating rates of 100, 225 °C min⁻¹ (200 °C min⁻¹ for the 15 mm specimen), and 300 °C min⁻¹. Reproduced with permission from [46]. Copyright © 2012, John Wiley & Sons

Figure 6.35c, d demonstrates the evolution of relative density of the two samples, both of which can achieve a high level of densification of about 95 % [46]. The 56 mm sample exhibits a much slower densification progress, delayed for about 300 s, for the 225 °C min⁻¹ simulation, as compared with the 300 °C min⁻¹ simulation. The densification is slower than for the 100 °C min⁻¹ simulation, which is readily attributed to the lower holding temperature that it can achieve and the shorter holding time. For the 15 mm sample, the higher temperatures developed at 100 and 200 °C min⁻¹, as compared with at 300 °C min⁻¹ regime, are ascribed to their higher shrinkage rates.

Figure 6.35e, f shows evolutions of average grain size of the two samples for the three regimes. For both the 56 and 15 mm samples, the average grain size is larger for the 300 °C min⁻¹, probably due to the earlier significant levels of densification, on the one hand, and larger temperature gradients, on the other hand. However, for the two lower heating rates, the increase of the heating rate shows an opposite impact in 56 and 15 mm samples. This implies that there has been a complex interplay of the time of processing maximum temperature and temperature/relative density gradients [46].

Gradients of temperature, relative density, and grain size, as functions of sample size, are shown in Fig. 6.36 for two heating rates: 100 and 300 °C min⁻¹ [46]. The results are obtained for two time durations: an intermediate and the final time. The intermediate time corresponds approximately to the point at which the holding period starts for each combination of the sample size—heating rate. Similarly, the final time corresponds to the end of the holding time, which also represents the end of the simulation. The simulated gradients depending on the sample size are a reflection of the heating patterns of the samples with different sizes. Therefore, establishment of reliable modeling approaches is an important requirement for large-scale industrial production of ceramics by using the SPS techniques, which could be a subject of future studies in this area.

Fig. 6.36

Computed dependence on the spark plasma sintering (SPS) tooling size for 100 °C min⁻¹ (left) and 300 °C min⁻¹ (right) of: a and b temperature gradient; c and d relative density gradient; e and f grain size gradient. The calculation results are given for an intermediate and final time of SPS. Reproduced with permission from [46]. Copyright © 2012, John Wiley & Sons

Distributions of temperature and stress in a functionally graded material (FGM) based on Ti and TiB during a SPS process have been simulated, with the development of a coupled electrical–thermal–mechanical finite element model [32]. It is found a stable axial temperature gradient in the sample is observed if a die with an area-changing cross section is used in the SPS process, which can be used as a strategy to ensure homogeneity of FGM during the sintering with SPS. High stress and the stress gradient are observed at bottom of the specimen, which could result in microstructural heterogeneity and cracks between adjacent layers. The simulation results also indicate that both the temperature and stress gradients increase with increasing heating rate, suggesting a principle to use heating rate as low as possible.

A schematic diagram of the SPS equipment is shown in Fig. 6.37a, which consists of a graphite die, a specimen, two plungers, and six protection spacers [32]. The die, plungers, and spacers are all made of dense graphite. Figure 6.37b shows schematic of the die used in the simulation. The sintered FGM sample has five layers of equal thickness, with a diameter of 30 mm and a total thickness (height) of 20 mm. Weight percentages of Ti in the Ti–TiB layers are 45, 55, 65, 75, and 85 %, respectively. The sample is considered to be isotropic in the simulation in order to establish a simple model. The properties of the different layers in the sample are calculated by using the mixture rule, as discussed earlier.

Fig. 6.37

Configuration of the SPS system: a schematic diagram of the SPS equipment and b die dimensions used in the simulation. Reproduced with permission from [32]. Copyright © 2012, Elsevier

During SPS process, heat is generated due to the application of an electrical current; therefore, the temperature distribution in the die is determined by the distribution of the electrical current [32]. Also, the density of the electrical current is inversely proportional to the cross-sectional area of the die. As a result, the current density decreases gradually from top to bottom of the die, so that Joule heating is more significant at top of the die, leading to a temperature gradient in the axial direction. The highest temperature at Point A is 1481 K, while the lowest temperature at Point L is 1385 K. The maximum temperature difference in this direction is 88 K. This axial temperature gradient is crucial to fabricate FGMs with desired homogeneity.

Because all the three stresses, radial, axial, and circumferential, are compression stress, it is less likely to form cracks and fractures in the sample during the stage of heating [32]. It is also found that both the radial stress and the circumferential stress are maximized at the bottom of the sample, whereas the maximum axial stress is observed in the center of the sample. The hydrostatic stress along Lines CD, EF, GH, and IJ decrease at an increasing rate, while the hydrostatic stress at top and bottom of the sample increases in the radial direction. In addition, the hydrostatic stress at the bottom of the sample is significantly higher than that at all the other horizontal planes. The hydrostatic stress for Line AK is in the range of 74–100 MPa, whereas that for Line BL is from 45 to 109 MPa, which means that the difference in stress for Line BL is about 2.5 times larger than that for Line AK. The difference in hydrostatic stress between adjacent layers in the inner part of the sample is generally less than 10 MPa, while that in the rest part is higher. The maximum difference in stress at the bottom portion is 35 MPa. Highest stress and largest stress gradient are located at the edges of the sample.

Figure 6.38 shows SEM images of two samples, A and B, which are processed by using dies with area-changing cross section and hollow cylinder, respectively [32]. It is observed that the sample A has a denser microstructure than the sample B. Also, the sample B has macro-crack at the interface between the second layer and the third layer. There are also many pores at that interface, which is responsible for the low bonding strength of the interface. Therefore, the use of the die with area-changing cross section is a solution to ensure the homogeneity of the FGM during the processing with SPS.

Fig. 6.38

SEM images of the FGM samples sintered by using the SPS process with dies of different structures: a the die with an area-changing cross section and b a hollow cylinder die. The inset shows a magnified image of the squared area. Reproduced with permission from [32]. Copyright © 2012, Elsevier

According to this study, serval conclusions can be arrived as follows [32]. (i) A stable temperature gradient can be achieved in the axial direction of the sample by using a die with an area-changing cross section. (ii) The axial temperature gradient increases with increasing heating rate, due to the increase in the rate of Joule heat generation and the decrease in heat diffusion at top of the die. (iii) Both the stress and the stress gradient at the bottom portion of the specimen are significantly higher than those at the samples’ center. (iv) The maximum hydrostatic stress is observed at the final stage of heating, and both the hydrostatic stress and the stress gradient increase with increasing heating rate. (v) The sample processed by using the die with an area-changing cross section has much better microstructure than the sample processed with the normal die, which would ensure desired mechanical properties of the final products.

Although the studies on simulations of densification behaviors of transparent ceramics by using ECAS (SPS) process have not been widely reported, the results that have been discussed above can be used as a reference. This is especially true for those of oxide materials, because they share similar thermophysical properties with transparent ceramics.

6.3 Microwave Sintering

6.3.1 Brief Introduction

Microwave sintering is a special sintering process that is fundamentally different from the conventional sintering techniques. Microwave energy belongs to electromagnetic energy, with frequencies ranging from 300 MHz to 300 GHz, corresponding to wavelengths between 1 mm and 1 m, as shown in Fig. 6.39 [50]. The fundamental difference between microwave sinterings is essentially different from the conventional sintering in the heating mechanism. The temperature profile for both methods is shown in Fig. 6.40 [50]. In microwave process, the processed materials themselves absorb microwave power and then microwave is converted into heat within the samples rapidly [34, 35, 50]. Because the power produced by the microwave source can be used only to heat the materials, while the furnace elements are not heated, much higher heating rates can be achieved when using microwave sintering. In addition, a high sintering rate is benefit to bring out products with finer and less defective microstructures and thus enhanced functional properties [36]. Therefore, microwave sintering to produce ceramics has various advantages, such as higher energy efficiency, enhanced reaction and sintering rate, time-consuming and cost-effectiveness. Specifically, transparent ceramics can be fabricated at lower sintering temperatures for much shorter sintering times by using microwave sintering [43].

Fig. 6.39

Electromagnetic spectrum and frequencies used for microwave processing of materials. Reproduced with permission from [50]. Copyright © 2010, Elsevier

Fig. 6.40

Temperature profiles within the materials to be sintered by using different techniques: a conventional heating, b microwave heating, and c microwave hybrid heating. Reproduced with permission from [50]. Copyright © 2010, Elsevier

As compared with the traditional furnaces, MW furnaces have relatively more complicated configurations, due to the involvement of MWs. Two examples are demonstrated here. The first configuration is able to sinter samples batch by batch [51], while the second type can support continuous experiments and thus is suitable for large-scale production [52], as shown in Figs. 6.41 and 6.42, respectively.

Fig. 6.41

Configuration of heating chamber of a microwave sintering setup: 1 samples, 2 bubble alumina thermal insulation in sintered alumina container, 3 sintered alumina disk, 4 porous alumina parts providing thermal insulation, 5 Pt-sheathed S-type thermocouple, 6 MW insertion points, 7 applicator wall, and 8 metal stand. Reproduced with permission from [51]. Copyright © 1999, Elsevier

Fig. 6.42

Schematic drawing of apparatus for continuous microwave sintering. Reproduced with permission from [52]. Copyright © 1998, Elsevier

6.3.2 Theoretical Aspects

Theoretical understanding is important for the development and analysis of microwave sintering technique. For example, optimization of processing parameters is tedious and very time-consuming. By using modeling and numerical simulations, this work could be significantly simplified. Microwave technique is an interdisciplinary subject that involves several areas, including electromagnetics, physics, and materials science. A model of microwave sintering process is to obtain the temporal evolution and spatial distribution of (i) electromagnetic field in the materials to be processed, (ii) temperature, and (iii) other physical properties of the materials, such as stresses, mass fluxes, porosity, and grain size. These properties are dependent on the evolution and distribution of temperature within the body of the materials, while they could also be influenced by the nonthermal effect of electromagnetic field.

The distribution of electromagnetic field within the materials to be processed is determined by various factors, such as effective dielectric and magnetic parameters of the materials that are dependent on temperature and microstructure, dimensions and properties of the applicator, and the degree of matching between the microwave source, transmission line, and applicator. Consequently, the distribution of the electromagnetic field will affect the distribution of temperature within the sintered object. Of course, other factors, such as effective absorption properties, heat capacity, and thermal conductivity of the materials, as well as heat transportation characteristics of the materials and the furnaces, will also have their specific effects. Ultimately, all these will be reflected by the properties of the final sintered products. While detailed theoretical elaboration of microwave sintering can be found in Ref. [39], a brief description will be presented in this section.

6.3.2.1 Theoretical Modeling

For simplicity, normal incidence of a plane electromagnetic wave on a slab of material, with a thickness of a, from vacuum, is considered, as shown in Fig. 6.43 [39]. In the regions (1) and (2), there are waves traveling in both forward and backward directions, because of the reflections at the front and back surfaces of the material slab. However, in the region (3), there is only wave traveling in the forward direction. Complex amplitudes, E _x and H _y , of the electric and magnetic fields, of the electromagnetic wave, are given by the following equations [53–55]:

$$E(z,t) - x_{0} \text{Re} \left[ {E_{x} (z){\text{e}}^{i\omega t} } \right],$$

(6.36)

$$H(z,t) - y_{0} \text{Re} \left[ {H_{y} (z){\text{e}}^{i\omega t} } \right].$$

(6.37)

Then, the electric field, in the three regions, can be written in the form:

$$E_{x} = E_{0} \left( {{\text{e}}^{{ik_{0} z}} + R \cdot {\text{e}}^{{ - ik_{0} z}} } \right),\,\,{\text{region }}\left( {\mathbf{1}} \right),$$

(6.38)

$$E_{x} = E_{0} \left( {A \cdot {\text{e}}^{ikz} + B \cdot {\text{e}}^{ - ikz} } \right),\,\,{\text{region }}\left( {\mathbf{2}} \right),$$

(6.39)

$$E_{x} = E_{0} T \cdot {\text{e}}^{{ik_{0} z}} ,\,\,{\text{region }}\left( {\mathbf{3}} \right),$$

(6.40)

where E ₀ is the amplitude of electric field in the incident wave, ω is frequency, k ₀ = ω/c is the wave number of vacuum, c is the velocity of light, $k = k_{0} \sqrt {\varepsilon \mu }$ is the wave number in the material, ε is relative dielectric permittivity, and μ is relative magnetic permeability of the material.

Fig. 6.43

Normal incidence of a plane electromagnetic wave on a slab of material from vacuum. Reproduced with permission from [39]. Copyright © 2013, John Wiley & Sons

Similarly, expressions for the magnetic field in the three regions are given by:

$$H_{y} = H_{0} \left( {{\text{e}}^{{ik_{0} z}} - R \cdot {\text{e}}^{{ - ik_{0} z}} } \right),\,\,{\text{region }}\left( {\mathbf{1}} \right),$$

(6.41)

$$E_{y} = \frac{{H_{0} }}{\zeta }\left( {A \cdot {\text{e}}^{ikz} - B \cdot {\text{e}}^{ - ikz} } \right),\,\,{\text{region }}\left( {\mathbf{2}} \right),$$

(6.42)

$$H_{y} = H_{0} T \cdot {\text{e}}^{{ik_{0} z}} ,\,\,{\text{region }}\left( {\mathbf{3}} \right),$$

(6.43)

where H ₀ is the amplitude of magnetic field in the incident wave and $\zeta = \sqrt {\mu /\varepsilon }$ is the impedance of the material. For a plane wave in vacuum, the amplitudes of the electric and magnetic field, E ₀ and H ₀, can be related by the following equation:

$$E_{0} = \zeta_{0} H_{0} ,$$

(6.44)

where $\zeta_{0} = \sqrt {\mu_{0} /\varepsilon_{0} }$ is the impedance of free space, ε ₀ is the relative dielectric constant, and μ ₀ is the relative magnetic permeability of vacuum, with $c = 1/\sqrt {\mu_{0} \varepsilon_{0} }$.

In Eqs. (6.38–6.40) and (6.41–6.43), A, B, R, and T are unknown coefficients, which are determined the boundary conditions that require continuity of the electric and magnetic field components, E _x and H _y , at the front and back surfaces, i.e., z = 0 and z = a. By imposing four boundary conditions and solving four algebraic equations, the coefficients can be obtained as follows:

$$A = \frac{{2\left( {\frac{1}{\zeta } + 1} \right)\left( {\cos \,ka - i\sin \,ka} \right)}}{{\frac{2}{\zeta }\cos \,ka - i\left( {\frac{1}{{\zeta^{2} }} + 1} \right)\sin \,ka}},$$

(6.45)

$$B = \frac{{2\left( {\frac{1}{\zeta } - 1} \right)\left( {\cos \,ka + i\,\sin \,ka} \right)}}{{\frac{2}{\zeta }\cos \,ka - i\left( {\frac{1}{{\zeta^{2} }} + 1} \right)\sin \,ka}},$$

(6.46)

$$R = \frac{{i\left( {\frac{1}{{\zeta^{2} }} - 1} \right)\sin \,ka}}{{\frac{2}{\zeta }\cos \,ka - i\left( {\frac{1}{{\zeta^{2} }} + 1} \right)\sin \,ka}},$$

(6.47)

$$T = \frac{{\frac{2}{\zeta }\left( {\cos \,ka - i\sin \,ka} \right)}}{{\frac{2}{\zeta }\cos \,ka - i\left( {\frac{1}{{\zeta^{2} }} + 1} \right)\sin \,ka}},$$

(6.48)

The energy flux density in the plane electromagnetic wave is equal to the average value of the Poynting vector, given by:

$${\bar{\varXi }} = \left| {\bar{E} \times \bar{H}} \right| = \frac{1}{2}{Re} \left( {E_{x} H_{y}^{*} } \right),$$

(6.49)

where the overline denotes averaging over the field period, while the asterisk denotes complex conjugation. Therefore, the energy flux densities, in the incident wave, in the wave reflected from the slab and that transmitted through the slab, are (E ₀ H ₀)/2, (E ₀ H ₀|R|²)/2 and (E ₀ H ₀|T|²)/2, respectively.

If the materials have dielectrically and magnetically lossless, both ε and μ are their real parts. In this case, the wave number k also has only real part and there is:

$$\left| R \right|^{2} \,+\, \left| T \right|^{2} = 1,$$

(6.50)

which means that the total energy flux density, carried away by the reflected and transmitted waves, is equal to the energy flux density in the incident wave.

However, if the material has dielectric and magnetic losses, both ε and/or μ will have imaginary parts, ε″ and μ″. As a result, the wave number k will be complex. The power density, w, with a unit of per unit square, which is the energy absorbed by the slab of the material, can be calculated by using following equation:

$$w = \frac{\omega }{2}\int\limits_{0}^{a} {\left( {\varepsilon_{0} \varepsilon^{\prime\prime}\left| {E_{x} } \right|^{2} \,+\, \mu_{0} \mu^{\prime\prime}\left| {H_{y} } \right|^{2} } \right)} {\text{d}}z,$$

(6.51)

where the components, E _x and H _y , within the material slab, are given by the Eqs. (6.38–6.40) and (6.41–6.42) for the region (2), with coefficients A and B being derived from Eqs. (6.45) and (6.46). Therefore, the energy absorbed in the material slab is the difference between the energy flux density in the original incident wave and the total energy flux density due to the reflected and the transmitted waves, i.e.,

$$w = \frac{1}{2}E_{0} H_{0} (1 - \left| R \right|^{2} - \left| T \right|^{2} ).$$

(6.52)

For materials with nonuniform properties, the most general method is to divide the material layers, with each layer having a fixed value of complex dielectric permittivity and magnetic permeability. In this case, the incident wave can be calculated by using the boundary conditions at all boundaries between every adjacent layers and solving the linear algebraic equations of the field amplitudes. Therefore, the power absorbed in each layer can be calculated accordingly [39, 56]. Besides planar slabs, materials with other shapes, such as cylinders and spheres, have also been simulated, although the electromagnetic field distribution and structure in these configurations would be much more complicated [57–60].

In addition to the interaction between the electromagnetic wave and the materials, the heating effect of applicators is also an important issue in microwave sintering. This is simply because applicators are necessary to build the cavity resonators. Various methods, such as finite-difference time domain (FDTD) [61–63], finite element method (FEM) [64], transmission line matrix (TLM) [65, 66], method of moments (MOM), have been used for such purposes.

Most transparent ceramics are nonmagnetic materials; therefore, their dielectric properties, i.e., dielectric constant and loss tangent, will play very important roles in determining the effectiveness and efficiency of microwave sintering. Complex dielectric permittivity is expressed as ε = ε′ − jε″, where ε′ and ε″ are real and imaginary parts, respectively. In this case, dielectric loss tangent is defined as tanδ = ε″/ε′. While dielectric permittivity of a given material is somehow a constant, its loss tangent could be varied over a wide range, depending on various factors, including purity, density, grain size, size distribution, and defects [67–69].

6.3.3 Heat Transfer and Sintering

As mentioned earlier, microwave heating is much faster than the conventional methods, because the microwave power can be directly delivered into the material to be sintered. In this respect, thermal conduction to dissipate and transport heat is not the only determining factor of microwave heating process. The temperature within the material during microwave heating also follows the heat conduction equation:

$$c_{\text{p}} \rho \frac{\partial T}{\partial t} - {\nabla } \cdot (\kappa {\nabla }T) = w,$$

(6.53)

where c _p is specific heat capacity, ρ is density, κ is thermal conductivity of the material, and w is local density of energy released in the material. For microwave heating, w is usually given by:

$$w = \frac{1}{2}\omega \left( {\varepsilon_{0} \varepsilon^{\prime\prime}{\rm E}^{2} + \mu_{0} \mu^{\prime\prime}{\rm H}^{2} } \right).$$

(6.54)

It was found that the distribution of temperature in the materials processed by using microwave heating is generally non-uniform. The distribution is parabolic with a maximum at the center of the material, which is usually referred to as an inverse temperature profile and is a specific characteristic of microwave sintering. At short timescales, the temperature distribution follows the power distribution. Otherwise, it is determined by thermal conduction of the entire dimensions of the materials [70].

If the distribution of the microwave power is uniform or follows certain trend, e.g., exponentially decaying from the surface into the bulk of the materials, heat conduction equation alone is sufficient for the modeling of the process. This is especially true for highloss materials. Simulation methods, including finite elements and finite differences, have been used to analyze the microwave sintering process based on heat conduction equation [71, 72].

In practice, however, the distribution of electromagnetic field of a material is strongly affected by the variation in dielectric and magnetic properties with temperature and time. Therefore, it is necessary to simulate the microwave sintering process by considering the coupling of electromagnetic and thermal aspects. It is knows as self-consistent model, in which there is an electrodynamic part to describe the electric and magnetic fields and a thermal part to determine the temperature. At the same time, the temperature in turn has influence on all the material parameters, including ε, μ, c _p, and κ. If the temperature dependencies of these parameters can be expressed with certain describe forms, the modeling would have analytical solutions [73–76]. In addition, it is also very important to consider the electromagnetic field in the whole microwave cavity [77–79]. This is because the variation in material parameters in turn would alter the matching characteristics of the cavity [80]. For such purposes, various models have been developed and applied to microwave sintering [81–83].

The ultimate goal is to simulate the densification of the ceramics during microwave sintering. Microwave sintering is different from the conventional sintering in certain aspects, e.g., inhomogeneity of temperature distribution in the materials [84]. There have been only limited attempts to simulate the temperature-dependent densification by coupling the electromagnetic and thermal effects. A simple way is to fit the relation of density and temperature [85]. However, in this case, the evolution of dimension of the samples and the properties of the materials are not described. An alternative simple approach is the use of master sintering curves [86, 87]. More sophisticated simulations include continuum sintering models and finite-difference electromagnetic simulation, which also have their disadvantages [88].

6.3.4 Nonthermal Effects

Another important issue in microwave sintering is the nonthermal effects of electromagnetic field, which should be considered in the modelings and simulations [39]. As stated previously, there is an essential difference in principle between microwave and conventional sintering processes. Microwave sintering is to use coherent electromagnetic field to drive oscillatory motions of charged particles in the materials, while conventional sintering is realized through energy exchange between the furnace and materials, due to the quasi-equilibrium thermal electromagnetic radiation with a continuous spectrum predominantly in the infrared range. The thermal radiation is absorbed by the materials through the excitation of oscillations with an equilibrium spectrum that can be described by the Maxwell–Boltzmann statistics. This absorption is usually at the surface of the materials, which is then transferred to the whole materials through thermal conduction. In summary, accurate simulation and modeling of microwave sintering either for all materials in general or for transparent ceramics in particular are still a great challenge.

6.4 Summary

Various transparent ceramics have been fabricated by using SPS (ECAS) and MW sintering. Both sintering techniques have similar advantages, such as rapid heating rate, low sintering temperature, and short sintering time duration. However, comparatively, their application in fabrication of transparent ceramics is still much less popular, as compared with the conventional sintering techniques. Possible reasons include the requirement of sophisticated facility in general, as well as expensive setup for SPS and relative inaccuracy in measurement of temperature for microwave sintering in particular. In addition, theoretical understanding on mechanisms of the densification and grain growth of transparent ceramics is still lacking, as compared with that of the nontransparent materials. It is expected that more and more works will be reported to process transparent ceramics with these two new sintering technologies.