A one-dimensional model of the electrical resistance sintering process

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ . This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s11661-014-2643-0 1 A one-dimensional model of the electrical resistance sintering process. J.M. Montesa, F.G. Cuevasb, J. Cintasa and P. Urbanª a Metallurgy and Materials Engineering Group, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla Camino de los Descubrimientos, s/n, 41092 Sevilla, Spain b Department of Chemistry and Materials Science Escuela Técnica Superior de Ingeniería, Universidad de Huelva Campus La Rábida, Carretera Palos s/n, 21819 Palos de la Frontera, Spain Corresponding author: J.M. Montes, Tel.: +34 954487305, Fax: +34 954460475, e-mail: [email protected] Abstract. In this work, a theoretical model for the processing technique usually known as electrical resistance sintering under pressure is proposed and validated. This technique consists of the consolidation of a metallic powder mass under compression by means of a high-intensity electrical current that passes through the aggregate. The electrical current heats the powder mass by the Joule effect. The model is numerically solved by the finite differential method, in which the added difficulty of the thermal–mechanical coupling of the process has been taken into account. To simplify the numerical resolution, a onedimensional scheme has been considered for both the powder densification mechanical problem and the heat generation and transmission problem. Furthermore, the theoretical predictions obtained after solving the model are compared with the data recorded by the electrical resistance sintering equipment sensors during electrical consolidation experiments with iron and titanium powders. The reasonable agreement between the theoretical and experimental curves suggests that the model, despite its simplifications, reproduces the main characteristics of the process. 2 Keywords: powder metallurgy; electrical resistance sintering; modelling; field-assisted sintering techniques 1. Introduction The conventional powder-metallurgy processing route fundamentally entails the cold compaction of powders to obtain a green compact, followed by a furnace sintering to obtain the final product. Although this method is the most commonly used technology in industrial processes, new alternatives are continuously investigated to improve, at least partially, the problems and deficiencies of the current technology. In this sense, the direct use of electricity as a means of sintering powders (both metals and ceramics) has been suggested and pursued numerous times, generally from an experimental viewpoint. A detailed review of the development of the different modalities of the electrical sintering has been carried out by Grasso et al. [1]. The great diversity of proposed modalities could be grouped under the generic name of electric-current-activated/assisted sintering (ECAS). Recently, the expression fieldassisted sintering techniques (FAST) has become popular. Indeed, the speed of the processes is the most remarkable characteristic and the common factor among these techniques. The high speed of the ECAS or FAST processes exhibits a significant advantage in that it renders the use of vacuum or inert atmospheres unnecessary. Moreover, similar to the effects of hot-press processes (where the FAST techniques are in some way included), the shape that parts acquired after their processing is definitive or near definitive (netshape or near-net-shape technique). However, one of the problems frequently mentioned by researchers that have pursued the problem of the FAST techniques is related to the dies, especially in cases where the die must be electrically insulating. The unresolved problem of choosing a material that is acceptably durable and economically worthwhile is of great interest and requires an exhaustive study. Although the different FAST variants may appear to have many common aspects, detailed studies lead to the conclusion that the physical phenomena in each of the modalities are very different. One of the FAST techniques, the so-called electrical resistance sintering (ERS), was already described in 1933 by Taylor [2], and later modified by Cremer [3] in 1944, 3 although its systematic study was not carried out until some years later by Lenel [4], around 1955. Later, the ERS technique has been revised by incorporating technical variations of different importance [5-7]. Essentially, the ERS entails the application a lowvoltage and high-intensity alternating current through a powder mass contained in an insulating die while pressure is simultaneously applied to the powder. Because the electrical requirements of low-voltage and high-intensity current are perfectly satisfied by a resistance welding equipment, this equipment can serve as the basis for the implementation of the ERS technique. The typical duration of the ERS process is approximately 1 s. During this time, the applied voltage and mechanical load induce powder densification due to softening. The softening occurs as a result of the temperature increase that, in turn, is caused by the current passing through the powders and the energy released by the Joule effect. In other FAST modalities, such as electrical discharge consolidation (EDC), the electrical current that passes through the powder is also of high intensity [8-11]. However, in comparison with the ERS process, the voltage can reach much higher values. This combination of high intensity and high or moderately high voltage can be achieved by the discharge of the energy stored in a bank of capacitors, which is a well-known technique in the electric-welding field. Because the electric current induces very quick micro-weldings at the particles’ contact points, the required consolidation time can be extraordinarily short, only a few microseconds. The increase in the applied voltage represents an important difference with respect to the ERS technique, where highly resistive initial powders are very difficult to sinter because the intensity of the current passing through the powder aggregate is insignificant (due to the significantly lower voltage applied to the powder mass). Therefore, the physics of the phenomena that take place during the EDC processes can be more complex than those of the ERS processes. Nevertheless, there is no doubt that, nowadays, the FAST modality most widely studied is the so called Spark Plasma Sintering (SPS), where a pulsed DC current directly passes through a graphite die, as well as through the powder compact, in case of conductive samples. This facilitates a very high heating or cooling rate (up to 1000 K/min), hence the sintering process generally is very fast (within a few minutes). Although slower than the aforementioned techniques (ERS and EDC), the typical speed of the SPS process ensures that, in certain cases, it has the potential of densifying powders with nanosize or nanostructure while avoiding coarsening which accompanies standard densification routes. 4 An extensive review about SPS works can be found in the recent work by R. Orrù et al. [12], with more than 1000 references. The FAST techniques, although known for many years, have almost always been approached from an experimental point of view as opposed to a theoretical one, which has severely limited the work conducted in this sense. The first theoretical studies include the work carried out by (mainly from the former USSR): Medvedenko et al. [13], Istomina et al. [14], Meshkov et al. [15], Raichenko [16], Raichenko and Chernikova [17], Burenkov et al. [18, 19], Yoneya and Ikeshoji [20] and Montes [21]. The later theoretical and numerical simulation advances have been clearly carried out in the context of the SPS technique by Matsugi et al. [22], Olevsky and Froyen [23], Anselmi-Tamburini et al. [24], Wang et al. [25] and Zhang [26]. A more detailed list of the theoretical, and also experimental works dealing with the FAST processes can be found in the papers by R. Orrù et al. [12], Olevsky et al. [27], Olevsky et al. [28] and Munir and Quach [29]. It is worth indicating that the differences between SPS and ERS are so significant (especially due to the thermal insulating character of the die in ERS and the conductive character in SPS, as well as the lower process duration of the ERS process) that, unfortunately, many theoretical advances developed for SPS cannot be translated to the description of the ERS process. Even the validation strategies of the theoretical predictions cannot be the same. For instance, due to its extraordinary shortness, it is very difficult to experimentally evaluate the evolution of the global temperature of a compact during an ERS process (the thermal inertia of sensor makes it impossible). This is not the case of the SPS, due to its larger duration. The problem is worsen because of the heterogeneity of the temperature distribution inside the compact in the ERS modality, and the common practice in SPS of placing temperature sensors in the external frontiers of the compacts is not sufficient. The fact that most of these modelling studies are uncompleted, ignoring determined relevant aspects, constitutes an additional problem. The reasons for this deficiency lie in the difficulty of modelling such a short process (sometimes on the order of only a few microseconds), as well as the inexistence of equations that adequately describe the thermal, electrical and densification behaviours of powder aggregates. Most proposed models for these constitutive equations are based on spherical particles with ordered arrangements. These assumptions are very different from the reality to produce results that are applicable to most actual powders, which are constituted by irregularly shaped particles with a random distribution. An additional point of special concern is the presence of oxide layers 5 that surround metallic particles, which, although very thin (on the order of a few nanometres), severely complicate the description of the global properties of the powder aggregate, primarily those of an electrical nature. The development of a model of the electrical sintering process could facilitate the monitoring of experimentally inaccessible variables inside the powder aggregate (i.e., temperature, porosity and stress fields, densification rate). A suitable model could also aid in the selection of suitable experimental conditions, such as current intensity, processing time, external pressure, and even the ideal materials for the die. Such a model could establish the basis for possible industrial-scale production. In this work, a theoretical model that constitutes an approach to describe the more relevant aspects of the ERS process will be developed. The model considers the modality of the insulating die and the regulated electrical current. The considered simplifying hypothesis, the equations modelling the behaviour of the different materials, the differential equations governing the heat generation, heat transmission and the powder densification, and the numerical strategies for tackling the numerical resolution of the model will be discussed. The model predictions and the results obtained in actual ERS experiments are also discussed. 2. Modelling 2.1. Preliminary considerations A system that consists of a metallic powder mass contained between two punches, which simultaneously act as electrodes, and enclosed in an electrically insulating die will be considered. The described system is schematically shown in Figure 1. FIGURE 1 A theoretical description of the process requires studying the transference of the thermal energy generated by the electrical current passing through the system. The system densification must also be addressed, because the loose powder between the electrodes can reduce its porosity by the effects of the applied pressure. This porosity reduction results in a strong thermomechanical coupling, with powders aggregate properties depending on temperature and porosity, and therefore exhibiting two time-based variations. 6 The resolution of the model should allow to know at any time: (i) the temperature distribution in the whole system (i.e., the temperature field evolution), (ii) the porosity distribution of the powder mass (i.e., the porosity field evolution) and (iii) the global properties of the compact (height, porosity and electrical resistance). The following simplifying hypothesis will be considered in the model: I. The lateral surfaces of the die are completely rigid and thermally and electrically insulating. II. Both punches are completely identical and are constituted of two parts: the electrode itself of length L1, and a thermal wafer of length L2, both with a crosssectional area equal to SN. The contact resistance between different components is neglected. III. The thermal expansions of the different materials are negligible. IV. The initial temperature of the whole system is T0. The first hypothesis sets the ideal conditions to use a one-dimensional model. This simplification will be justified if predictions maintain a good accuracy. Because of the symmetry of hypothesis II, the problem can be solved for one half of the system. Figure 2 shows a scheme of the final system under study. FIGURE 2 The thermal wafer is placed in direct contact with the powder to avoid thermal leaks. If a material with a high thermal conductivity, such as the one used in the electrode, was in contact with the powder, the heat would be evacuated so rapidly that the upper surface of the powder aggregate could not reach a temperature sufficient for its sintering. Its low electrical resistance ensures the passage of a high intensity current through the system and the powder heating. Another aspect to be considered in the model is that a typical ERS process, shown in Figure 3, starts with a cold-pressing period, during which no current passes through the powder. A heating period, during which pressure and electrical current are both applied, 7 subsequently takes place. The process finishes with a forging period, during which only pressure is again applied, although the system temperature is changing. FIGURE 3 Therefore, and taking into account that, because of a technical limitation in the ERS equipment, the regulation of the desired current intensity that passes through the system (I0) is only possible when this value is approached by an increasing ramp: V. The variation with respect to time of the effective current intensity that passes through the system results in: 0 0 00 0 0 00 () 0 S S SC CT if t t tt I if t t t It tt I if t t t if t t t             (1) where t0 represents the end of the cold-pressing period, tS is the constant intensity start, tC is the end of the heating period and tT is the end of the process. VI. The force (F) exerted by the punches starts at time t = 0 and is constant during the process. The nominal pressure results in NN P F S . 2.2. Materials properties Concerning the materials properties, we will assume that: VII. The properties (specific heat, cm ; thermal conductivity, km ; and electrical resistivity,  m ) of the bulk materials in the system (powder and punch) depend on the temperature: () mm c c T , () mm k k T and () mm T   (2) 8 (The exact dependence is modelled according to the appropriate tables or interpolation functions.) The case of the specific heat (cm ) deserves special consideration, because the eventual phase transformations that the powder could suffer should also be considered. Thus, ( ) ( ) ( ) m m D i i i c T ç T T T L       (3) where çm is the specific heat itself, Li are the different latent heats (expressed in J/kg) of the corresponding phase transformations at the temperatures Ti, and  D is the Dirac delta function. Regarding the oxide covering the metallic particles, only the thermal dependence of the electrical resistivity,  x, must be considered, which is modelled by the analytical function:   ( ) exp (2 ) x x x T A E RT   (4) where Ax is a pre-exponential factor expressed in ·m, Ex is the band-gap energy expressed in J/mol and R is the gas constant. In addition to the thermal dependence, the powder aggregate or compact properties also depend on the porosity level. Expressions that are clearly established include those for the apparent density and the specific heat of a porous material (see German [30]). The dependence of the electrical resistivity and the thermal conductivity on the porosity is a more complex matter, particularly during the first instants of the current passage, because of the oxides that cover the particles (see Montes et al. [31-32]). Thus: VIII. The apparent or effective properties (density,  ; specific heat, c; electrical resistivity,  ; and thermal conductivity, k) of any volume of the powder compact depend on its porosity level (  ) as follows: 15 where Tt is the temperature of the phase transition and  is a small quantity that represents the semi-width of the curve. 2.7. Predictable limitations of the model First tackling those aspects deliberately ignored in the model, Hypothesis I supposed that the die walls were perfectly rigid as well as electrically and thermally insulating. According to the working pressure of 80 MPa, the low compressibility modulus of the alumina and the supporting effect of the hoop surrounding the die, the non-deformability of the die is an acceptable hypothesis. On the other hand, the electrical insulating effect of the alumina is perfectly satisfied (electrical resistivity of ~1014 ·m at room temperature and ~1010 ·m at 800 K). However, the thermal insulating effect (thermal conductivity of 39 W/(m·K) at room temperature and 9 W/(m·K) at 800 K, as indicated by Kingery et al. [44]) is relatively poor, and a perfect insulating effect is not guaranteed, as the model supposes. Therefore, thermal leaks toward the die walls imply that the temperature in a section of the specimen is not constant, and, since the ERS model is one-dimensional, it cannot predict a non-laminar temperature distribution and the formation of an ovoid core of melted metal in the interior of some compacts, as observed in experiments carried out with iron. Hypothesis III ignores the materials’ thermal expansion. According to Kingery et al. [44], changing the temperature from 300 to 800 K in an alumina die without a hoop provokes a change of 0.4% of the die’s inner diameter (from 12 to 12.048 mm). The pressure variation resulting from such a change would be almost unappreciable. Thus, no important deviations because of the failure to comply with hypothesis III are expected. Because the ERS technique is based on the heat generated by the Joule effect, anything related to the current is crucial. There are two possible deviations with respect to the behaviour described in hypothesis V. The first one derives from the fact that the initial ramp is not exactly linear, as the model supposes. The second is caused by the regulation of the RMS value of the current, which is not reached in a reasonable time with respect to the total time of heating. Figure 5 shows a typical evolution of the efficient intensity for the entire ERS process. FIGURE 5 16 Although the intensity increase profile and the maximum intensity profile do not exactly coincide with the predicted profiles (for instance, it is relatively frequent to surpass the regulation value during the period of intensity increase), the deviations can be compensated in the model by adequately choosing the increasing time (  s). One possible criterion could be to choose a  s value such that the area under the intensity curves coincide, both experimentally and theoretically (as described by Eq. (1)). Mathematically, this implies that: 0 2 SC S I      (29) where C  is the duration of the heating period, S is the area under the experimental curve and I0 is the value of the regulated intensity. The constancy of the applied force during the process (as accepted in hypothesis VI) also deserves attention. Figure 5 shows the value of the force during a typical ERS experiment. As shown, the force constancy hypothesis is reasonably acceptable. The curve in Figure 5 only shows a slight decrease during the heating period because the punch suffers a bigger displacement due to the higher densification. Nevertheless, the regulation is satisfactory. Hypothesis IX accepts that after the application of pressure, the porosity takes a uniform value of 0 in the powder aggregate. It is well known that this is not true, and the influence of the initial porosity distribution in the cold compaction processes (when the powder is not previously vibrated in the die as usually happens industrially) is currently receiving much attention to theoretically model its effects. However, powders were always vibrated in the ERS experiments after pouring them into the die, and it is expected that the lack of uniformity in the initial porosity distribution is notably minimised. Although the initial distribution was uniform, the die walls cause this uniformity to disappear after the press starts. This situation worsens because of the uniaxial compression. Neither of these aspects has been taken into account in the model. The formation, evacuation and confinement of gasses that could appear during powder heating have neither been considered. These gasses result from the reaction of the powders with the atmosphere, or from the gasses already confined inside the powder particles. This is an aspect that falls outside the scope of this work’s first approach. 17 A final aspect that has not been considered is the evolution of the grain size of the sintered material, of interest in mechanical and magnetic properties. The far-reaching implications of this topic justify future experimental and theoretical studies. 3. Validation 3.1. The Electrical Resistance Sintering equipment There is not enough available commercial equipment to carry out electrical resistance sintering (ERS) experiments. However, the electrical requirements of the process –high intensity and low voltage– are adequately satisfied by a resistance welding machine, which is also capable of distributing the necessary mechanical strength to press the powders. A press-type resistance welding machine (Serra Soldadura S.A., Spain) has been adapted for this work. This equipment has a 100 kVA monophasic transformer, a pneumatic cylinder capable of providing a force of 1400 daN, and an electronic controller governing the process sequences and the current intensity. This machine has been fitted with sensors to measure the main parameters of the process, including the displacement of the mobile punch, applied force and efficient voltage and intensity. One of the main challenges in ERS experiments is obtaining adequate materials for the dies. These materials require good mechanical properties with electrical and thermal insulation. In this work, a similar arrangement to that of Lenel [4] was used, consisting of a ceramic tube externally reinforced by a two-part metallic hoop. The ceramic tube is made of alumina (class Alsint 99, W. Haldenwanger, Germany). The die is closed on its top and bottom with heat-resistant copper electrodes (98.9%Cu-1%Cr-0.1%Zr) and the powder is placed between them (Figure 6). Wafers of the alloy 75.3%W-24.6%Cu that are resistant to electro-erosion were placed in direct contact with the powders. They have lower thermal conductivity and dampen the heat flow generated in the powder mass towards the watercooled electrodes. The evolution of electrical resistance of the specimen during this process is known via the voltage measured between the electrodes and the circulating current intensity. The porosity variation of the powder mass was determined by calculating the punch position and the final height of the compact. The inner diameter of the ceramic die was 12 mm. The applied pressure was constant during the entire process and was 80 MPa. Lower pressures do not guarantee that the current passes in all experiments; however, higher pressures could deteriorate the die. The 18 pressure application starts 0.6 seconds before the application of current (50 Hz), being the time equivalent of 30 cycles. Pressure is maintained until the end of the process. Figure 7 shows a picture of the ERS equipment, including the punches and die used in the experiments. FIGURE 6 The follow-up sensors installed in the sintering equipment allow the monitoring of the evolution of: (i) the applied mechanical force, (ii) the displacement of the upper punch, (iii) the efficient voltage applied to the electrodes and (iv) the efficient intensity circulating through the system. By knowing the compact mass and absolute density of the powder and by measuring the final height of the compact, the height, global porosity and the electrical resistance of the compact can be known at any moment of the process by processing sensor data. The last two computed magnitudes will be compared with the theoretical predictions given by the model. The degree of agreement will allow estimates of confidence regarding other magnitudes experimentally inaccessible, such as temperature and porosity, which are important in selecting the optimal processing conditions. 3.2. Powder characterisation The ERS experiments have being carried out with two different powders: iron and titanium. The iron powder, Fe WPL200, is a water atomised (process describe elsewhere, for instance, ASM Handbook [45]) powder produced by QMP Metal Powders GMBH, with a purity of 99.4% Fe and main impurities of 0.2% Mn, 0.15% O, 0.01% C and 0.009% S. The absolute density of the powder is 7.87 g/cm3 with a tap density of 2.79 g/cm3. The granulometric analysis performed by laser diffraction gave a mean particle size of 84.4 m. Figure 7a shows a scanning electron microscopy (SEM) micrograph of the powder. FIGURE 7 The titanium powder was produced by Sejong Materials Co., with a reference Ti T325. It is a hydrated, milled and dehydrated powder obtained from a Kroll processed Ti sponge 19 (as described in ASM Handbook [45]). Purity is 99.5% Ti with 0.038% Fe, 0.035% Si, <0.045% O and <0.08% N. The absolute density is 4.51 g/cm3 with a tap density of 1.73 g/cm3.The granulometric analysis gave a mean particle size of 24.2 m. Figure 7b shows a SEM micrograph in which irregular- and angular-shaped particles are seen. The studies by Evans [46] indicate that the thickness of the oxide covering the Fe and Ti particles is 4.5 nm. The powder mass employed in the ERS experiments was 4 g for iron and 3 g for titanium. The final heights of the sintered compacts were nearly 5 mm for iron powders and nearly 6.5 mm for titanium powders. 3.3. Validation strategy The strategy used to validate the model considers two basic parameters, with values relatively unknown and difficultly measurable. The other parameters in the model have well known values. These parameters are A, which controls the creep densification rate (Eq. (19)), and Ex, which controls the decreasing rate of the electrical resistivity of the oxide layer by increasing the temperature (Eq. (4)). Fixed the Ex value, the value of the pre-exponential factor (Ax) will be automatically fixed because the value of the oxide resistivity is known at room temperature. Such fitting parameters are chosen such that they provide an initially good agreement between the theoretical predictions and the experimental data. Once A and Ex are chosen for a particular experiment they will be held constant for any other experiments carried out with the same powder under different processing conditions (different current dwelling time and/or current intensity). The model will be considered valid if the theoretical predictions for those new situations reasonably agree with the experimental data. (A study carried out altering the parameters A and Ex shows that small variations of their values do not significantly alter the global prediction of the model, therefore they are not very critical and the adopted strategy is reasonable.) Table 1 presents some of the more notable experiments. Table 1. Experiments analysed in this work (1 cycle = 0.02 seconds, 1 second = 50 cycles). Powder Experiment Working Conditions Fe WPL200 Fe_1 (6.5 kA, 1.2 s / 80 MPa) 20 Fe_2 (6.5 kA, 1.6 s / 80 MPa) Fe_3 (7.0 kA, 1.2 s / 80 MPa) Fe_4 (7.5 kA, 1.2 s / 80 MPa) Ti T325 Ti_1 (4.0 kA, 1.0 s / 80 MPa) Ti_2 (5.0 kA, 1.0 s / 80 MPa) Ti_3 (6.0 kA, 1.6 s / 80 MPa) For the experiments with Fe, the initial conditions were 6.5 kA and 1.2 s. Some variation was introduced in the other experiments. In Fe_2, the heating period was increased to 1.6 s maintaining the same intensity. In Fe_3, the heating period was maintained at 1.2 s, but the intensity was increased to 7.0 kA. The experiment Fe_4 was carried out with the same duration as Fe_1, but with a higher intensity of 7.5 kA to check whether the theoretical prediction correctly predicted the specimen core melting, something experimentally observed for this or more severe conditions. This is of special interest because this is the only possible way to check the validity of the temperature evolution predictions. Similarly, starting with Ti from 4.0 kA and 1.0 s, the intensity is increased in Ti_2 up to 5.0 kA. The intensity and the heating period in Ti_3 are 6.0 kA and 1.6 s. The conditions in the Fe and Ti experiments and other relevant details of the simulations are gathered in Table 2. Some parameters have been taken from the literature. Others have been experimentally measured for this work, including the mean particle radius (r) that was obtained by laser diffraction and the tap porosity (M) that was measured according to the MPIF standards [47]. Those parameters modelling the cold pressing period have been obtained by a least square fitting of the experimental compressibility curve to Eq. (11). The value of the activation energy for creep is 80% of the self-diffusion energy of the metal, according to Evans and Wilshire [48]. The value chosen for  S, according to Eq. (29), represents the mean value from various experiments with each type of material; it is 0.16 s (8 cycles) for the experiments with Fe and 0.12 s (6 cycles) for the experiments with Ti. The thermal dependence of different properties (specific heat, thermal conductivity, electrical resistivity) of the bulk metals and the punch material do not appear in Table 2, but can be found in handbooks as Smithels [49], ASM Handbook [50] and CRC Handbook [51]. 21 Table 2. Parameters considered in the simulations of the experiments with Fe and Ti. Parameter Description Fe_1 Fe_2 Fe_3 Fe_4 Ti_1 Ti_2 Ti_3 Working conditions T0 (K) Initial temperature 298 298 298 298 298 298 298 TRF (K) Cooling temperature 298 298 298 298 298 298 298 F (N) Applied force 9048 9048 9048 9048 9048 9048 9048 I0 (kA) Intensity maximum (regulated) value 6.5 6.5 7.0 7.5 4.0 5.0 6.0 t0 (s) Heating start 0.6 0.6 0.6 0.6 0.6 0.6 0.6  S (s) Intensity increase duration 0.16 0.16 0.16 0.16 0.12 0.12 0.12  C (s) Heating duration 1.2 1.6 1.2 1.2 1.0 1.0 1.6  T (s) Total duration (excluding cold pressing) 1.6 2.0 1.6 1.6 1.4 1.4 2.0 Geometrica l description L1 (mm)  Wafer length 5 5 5 5 5 5 5 L2 (mm)  Electrode length 40 40 40 40 40 40 40 SN (mm2)  Die inner section 113.1 113.1 113.1 113.1 113.1 113.1 113.1 Powder description  M (g/cm3) Bulk density of the metal, from [49] 7.870 7.870 7.870 7.870 4.507 4.507 4.507 M Tap porosity 0.645 0.645 0.645 0.645 0.616 0.616 0.616 r (m) Particle’s mean radius, Eq. (9) 42.2 42.2 42.2 42.2 12.1 12.1 12.1  (nm) Thickness of the oxide layer, Eq. (9), from [46] 4.5 4.5 4.5 4.5 4.5 4.5 4.5 md Used to model the descaling, Eq. (9), from [52] 13.6786 13.6786 13.6786 13.6786 11.0145 11.0145 11.0145 nd Used to model the descaling, Eq. (9), from [52] 0.2494 0.2494 0.2494 0.2494 0.1072 0.1072 0.1072  X (·m) Oxide electrical resistivity at 300K, Eq. (4), from [53, 54] 2100 2100 2100 2100 149.03 149.03 149.03 ∞ Used in the compaction law, Eq. (11) 0.016 0.016 0.016 0.016 0.019 0.019 0.019 a Used in the compaction law, Eq. (11) 304.93·106 304.93·106 304.93·106 304.93·106 614.54·106 614.54·106 614.54·106 b Used in the compaction law, Eq. (11) 0.797 0.797 0.797 0.797 0.666 0.666 0.666 nc Used in the creep behaviour, Eq. (19), from [55, 56] 4.5 4.5 4.5 4.5 4.2 4.2 4.2 Qc (J/mol) Used in the creep behaviour, Eq. (19), from [48, 55, 56] 192593 192593 192593 192593 1.53·105 1.53·105 1.53·105 EY (Pa) Used in the creep behaviour, Eq. (19), from [49] 211·109 211·109 211·109 211·109 103·109 103·109 103·109 M (g) Powder mass 4 4 4 4 3 3 3 Numeric parameters Tolerance Numerical tolerance 0.001 0.001 0.001 0.001 0.001 0.001 0.001  Used in numerical approach of Dirac , Eq. (28) 4 4 4 4 5 5 5 n Number of nodes in the powder 110 110 110 110 200 200 200 m Number of nodes in the system 150 150 150 150 300 300 300  Number of temporal nodes 800 1000 800 800 700 700 1000 Fitting parameters A Used in the creep behaviour, Eq. (19) 1·1029 1·1029 1·1029 1·1029 1·1023 1·1023 1·1023 Ex For the oxide electrical resistivity, Eq. (4) 2.939·104 2.939·104 2.939·104 2.939·104 4.818·104 4.818·104 4.818·104 Ax For the oxide electrical resistivity, Eq. (4) 5.788 5.788 5.788 5.788 9.484·10-3 9.484·10-3 9.484·10-3 22 3.4. Experiment Fe_1 (6.5 kA and 1.2 s) A. Global magnitudes evolution Figure 8a compares the experimentally obtained values (displacement sensor data) and the theoretical evolution (after empirical tuning of A and Ex) of the global porosity and electrical resistance of the compact. FIGURE 8 The theoretical global porosity curve in Figure 8 satisfactorily reproduces the experimentally obtained profile (the oscillations observed in the porosity experimental curve are due to noise in the acquisition sensor, an only the trend of the mean value should be considered). The main deviation appears during the first instants of the heating period when the experimental curve decreases more quickly. This could be due to the powder particles rearranging caused by local heating. Although this phenomenon has not been considered in the model, the deviation is too small to be considered without a large increase in the complexity and computation time. Similarly, the experimental and theoretical curves of electrical resistance are well fitted. Both the abrupt initial resistance fall (when the dielectric behaviour of the oxide layers is dominant) and the later smooth decrease (when the metallic behaviour dominates and the resistance increase by the higher temperature effect acts in an opposite way to the resistance decrease due to the porosity reduction) agree with the registered evolution. The acceptable concordance between the experimental and theoretical curves of the global porosity and electrical resistance permits confidence with respect to the other magnitudes not experimentally checked, such as the distribution and evolution of local magnitudes. These aspects will be tackled in the next sections. B. Distribution and evolution of the local porosity The final porosity distribution in the compact can be studied by optical microscopy. Figure 9 shows the result of the analysis carried out on a diametrical section of the cylindrical compact obtained with the experiment Fe_1. It is a more densified core than the periphery. FIGURE 9 23 As previously indicated for the temperature, the one-dimensional character of the model does not permit reproducing the ovoid distribution of porosity previously shown. Nevertheless, the model should account for the porosity increase from the centre to the compact bases in a laminar mode. Obviously, the information given by the numerical model is much more complete and is not restricted to the final porosity distribution. Figure 10a and b shows the porosity evolution at some representative distances of the compact as well as the distribution for some different moments of the process. FIGURE 10 Figure 10a shows a significant porosity gradient inside the compact. It is higher in the centre (plane 0 xx ), where the porosity varies from 0.46 to 0.10, than in the contact with the wafer (plane 1 xx ). Here, the porosity varies from 0.46 to 0.40. Figure 10b shows how this porosity gradient increases when the heating process advances. C. Local temperature distribution and evolution The theoretical predictions of the porosity can be compared to the final porosity distribution. However, there are no experimental data and possible comparison for the temperature distribution and its lack of uniformity, because it can only be known through the theoretical simulation of the process. Some limited information is available about the temperature distribution. The bases of the compact are in contact with cooled electrodes and in some experiments a melted area is found in the centre of the compact. Thus, the temperature is much higher in the centre than in the bases. Figure 11a shows the evolution and Figure 11b shows the distribution of the local temperature in some representative planes and instants. The evolution of the mean temperature (calculated according to Eq. (25)) of the compact along the process is also shown FIGURE 11 The central plane of the compact (x0) reaches 1004 K. During heating, the temperature difference between the centre and the bases (x1 vs. x0) reaches a maximum value at the end of 24 the heating period (330 K), which is translated into a thermal gradient of about 330 K / 2.65 mm or 124.5 K/mm. This difference slightly increases during the forging period. All the curves in Figure 11b show a similar trend. Their maximum value is placed at a plateau, which corresponds to the region occupied by the compact. The curves then abruptly decay in the region corresponding to the wafer and finally slowly decrease along the region of the electrode reaching a minimum value at the end of the electrode coinciding with the cooling temperature. The temperature inside the compact is less uniform when the time increases. For higher times, the aforementioned plateau is not observed. 3.5. Experiments Fe_2 (6.5 kA and 1.6 s) and Fe_3 (7.0 kA and 1.2 s) As explained previously, the simulation parameters (A and EX) have been arbitrarily chosen to give an optimal fitting with the experimental curves of the experiment Fe_1. Therefore, the model validation is not complete until checking that those parameters also give good predictions for experiments carried out with the same powder but different conditions of current intensity and dwelling time. The fitting between experimental and theoretical global porosity and electrical resistance curves in the iron powder experiments Fe_2 and Fe_3 is excellent in both cases, supporting the validity of the fitting parameter values resulting from the experiment Fe_1. The mentioned curves have a very similar shape to that shown in Figure 8, although the porosity reached at the end of the heating period for experiments Fe_2 and Fe_3 was about 0.14 and 0.15, respectively. The evolution of the global mean temperature and the local temperature at different planes of the compacts also result very similar to Figure 11a of experiment Fe_1. The higher temperatures in experiments Fe_2 and Fe_3 were 1027 K and 1021 K, respectively, both of which are lower than the melting temperature of pure iron (~1808 K). Checking the validity of the values of the chosen fitting parameters and the model itself are more critical for experiment Fe_4, in which the experimental conditions (intensity and duration) are more different. 3.6. Experiment Fe_4 (7.5 kA and 1.2 s) with melted core Experiment Fe_4 studies whether the model appropriately describes the experimentally observed phenomenon of the compact core melt. Figure 12 corresponds to the compact of experiment Fe_4 and illustrates this phenomenon. Not only is the porosity different between the periphery and the centre, but a near-equiaxial grain structure appears in the centre of the 31 [6] J.M. Montes, J.A. Rodríguez and E.J. Herrera: Rev. Metal. Madrid, 2003, vol. 39(2), pp. 99-106 (in Spanish). [7] J.M. Montes, J.A. Rodríguez, F.G. Cuevas, J. Cintas: J. Mater. Sci., 2011, vol. 46(15), pp. 5197-5207. [8] K. Okazaki: Rev. Partic. Mater., 1994, vol. 2, pp. 215-269. [9] D.K. Kim and K. Okazaki: 1991 P/M in Aerospace and Defense Technologies (Tampa, FL, 4–6 March 1991), Metal Powder Industries Federation/American Powder Metallurgy Institute, 1991, pp. 365-372 [10] D.K. Kim, H.-R. Pak, and K. Okazaki: Mater. Sci. Eng. A, 1988, vol. 104(C), pp. 191- 200. [11] T.J. Davies and S.T.S. Al-Hassani: Advances in Materials Technology in America, Vol 2, I. LeMay, Ed., American Society of Mechanical Engineers, 1980, p. 147 [12] R. Orrù, R. Licheri, A.M. Locci, A. Cincotti and G. Cao: Mat. Sci. Eng. R, 2009, vol. 63(4-6), pp. 127–287. [13] N.F. Medvedenko, M.Z. Kolchiskii and V.V. Skorokhod: Poroshkovaya Metallurgiya, 1977, vol. 6, pp. 562-565. (in Russian). [14] T.I. Istomina, A.A. Baidenko, A.I. Raichenko, M.A. Goldberg and A.V. Svechkov: Sov. Powder Metall. Met. Ceram., 1983, vol. 22(11), pp. 957-960. [15] V.V. Meshkov, N.K.Myshkin and A.I.Sviridenko: Sov. Powder Metall. Met. Ceram., 1984, vol. 23(3), pp. 200-203. [16] A.I. Raichenko: Sov. Powder Metall. Met. Ceram., 1985, vol. 24(1), pp. 26-30. [17] A.I. Raichenko and E.S. Chernikova: Sov. Powder Metall. Met. Ceram., 1989, vol. 28(5), pp. 365-371. [18] G.L. Burenkov, A.I. Raichenko and A.M. Suraeva: Sov. Powder Metall. Met. Ceram., 1987, vol. 26(9), pp. 709-712. 32 [19] G.L. Burenkov, A.I. Raichenko and A.M.Suraeva: Sov. Powder Metall. Met. Ceram., 1989. vol. 28(3), pp. 186-191. [20] M. Yoneya and T. Ikeshoji: Mater. Trans., 2001, vol. 42(11), pp. 2165-2171. [21] J.M. Montes: Modelado de la sinterización por resistencia eléctrica bajo presión de polvos metálicos, UNED thesis, Facultad de Ciencias. Thesis Directors: E.J. Herrera and J.A. Rodríguez, Escuela Técnica Superior de Ingenieros Industriales, Universidad de Sevilla, Spain, 2003. (in Spanish). [22] K. Matsugi, H. Kuramoto, O. Yanagisawa and M. Kiritani: J. Mater. Process. Technol., 2003, vol. 134(2), pp. 225-232. [23] E. Olevsky and L. Froyen: Scripta Mater., 2006, vol. 55(12), pp. 1175-1178. [24] U. Anselmi-Tamburini, S. Gennari, J.E. Garay, J.R. Groza and Z.A. Munir: Mater. Sci. Eng. A, 2005, vol. 394(1-2), pp. 139-148. [25] X. Wang, G. Casolco, G. Xu and J.E. Garay: Acta Mater., 2007, vol. 55(10), pp. 3611- 3622. [26] J. Zhang: Field Activated Sintering Technology: Multi-physics Phenomena Modeling, LAMBERT Academic Publishing AG & Co. KG, 2010. [27] E.A. Olevsky, W. L. Bradbury, C.D. Haines, D.G. Martin and D. Kapoor: J. Am. Ceram. Soc., 2012, pp. 1-8. [28] E.A. Olevsky, C. Garcia-Cardona, W.L. Bradbury, C.D. Haines, D.G. Martin and D. Kapoor: J. Am. Ceram. Soc., 2012, pp. 1-9. [29] Z.A. Munir and D.V. Quach: J. Am. Ceram. Soc., 2011, vol. 94(1), pp. 1–19. [30] R.M. German: Particle packing characteristics, MPIF (Metal Powders Industries Federation), Princeton, New Jersey, 1989. [31] J.M. Montes, F.G. Cuevas and J. Cintas: Granular Matter, 2007, vol. 9, pp. 401-406. 33 [32] J.M. Montes, F.G. Cuevas, J. Cintas and S. Muñoz: Met. Mat. Trans. A., 2012, vol. 43(12), pp. 4532-4538. [33] T. Nagae, M. Yokota, M. Nose, S. Tomida, T. Kamiya and S. Saji: Mater. Trans., 2002, vol. 43(6), pp. 1390-1397. [34] J. Groza, K.R. Anderson, M. Fendorf and C.J. Echer: Mater. Sci. Eng. A, 1999, vol. 270(2), pp. 278-282. [35] J. Secondi: Powder Metall., 2002, vol. 45(3), pp. 213–217. [36] A.N. Tijonov and A.A. Samarski: Ecuaciones de la Física Matemática, 2 nd ed., MIR, USSR, 1980. (in Spanish). [37] R. Haberman: Elementary applied partial differential equations, Prentice-Hall International, Inc, Englewood Cliffs, New Jersey, 1987. [38] F. P. Incropera, D.P. DeWitt, T.L. Bergman and A.S. Lavine: Fundamentals of Heat and Mass Transfer, 6 th ed., Wiley John & Sons, USA, 2006. [39] J.M. Montes, F.G. Cuevas, J. Cintas and P. Urban: Powder Metall., 2012, vol. 55(5), pp. 388-394. [40] O. Yanagisawa, T. Hatayama and K. Matsugi: Materia Japan , 1994, vol. 33(12), pp. 1489-1496. [41] M. Tokita: J. Japan Soc. Powder Powder Metall., 1993, vol. 30(11), pp. 790-804. [42] J. Groza and A. Zavaliangos: Mater. Sci. Eng. A, 2000, vol. 287(2), pp. 171-177. [43] S.V. Patankar: Numerical heat transfer and fluid flow, Hemisphere Publishing Corporation, USA, 1980. [44] W.D. Kingery, H.K. Bowen and D.R. Uhlmann: Introduction to ceramics, 2 nd ed., John Wiley & Sons, USA, 1976. [45] ASM Handbook, Vol 7, Powder metal technologies and applications, ASM International, USA, 2002. 34 [46] U.R. Evans: The corrosion and oxidation of metals: first supplementary volume, Edward Arnold Ltd., London, UK, 1968. [47] MPIF Standard 46, Determination of tap density of metal powders, in: Standard test methods for metal powders and powder metallurgy products, MPIF, Metal Powder Industries Federation, Princeton, NJ, USA, 2002. [48] R.W. Evans and B. Wilshire: Introduction to creep, The Institute of Materials, London, UK, 1993. [49] Smithells metals reference book, 8 th ed, Gale, W.F., Totemeier, T.C. (Eds.), Elsevier Butterworth-Heinemann Ltd, Oxford, UK, 2004. [50] ASM Handbook, Vol 2, Properties and selection: nonferrous alloys and special purpose materials, ASM International, USA, 1990, pp. 1099-1201. [51] CRC Handbook of Chemistry and Physics, 80th ed., CRC Press, New York, USA, 1999- 2000. [52] J.M. Montes, F.G. Cuevas, J. Cintas and P. Urban: Appl. Phys. A, 2011, vol. 105, pp. 935-947. [53] K.W. Vogt, P.A. Kohl, W.B. Carter, R.A. Bell and L.A. Bottomley: Surf. Sci., 1994, vol. 301, pp. 203-213. [54] A.A. Akl: Appl. Surf. Sci., 2004, vol. 221, pp. 319-329. [55] H.J. Frost and M.F. Ashby: Deformation-mechanism maps. The plasticity and creep of metals and ceramics, Pergamon Press, Oxford, UK, 1982. [56] C. Schuh and D.C. Dunand: Scripta Mater., 2001, vol. 45, pp. 1415-1421. 35 FIGURES Fig. 1. Electrical resistance sintering scheme. (The model assumes that the two electrodes are identical.) Fig. 2. The one-dimensional system under study. The powder is placed in the region from x = 0 to x = x1. A wafer of length 1 2 1 L x x is placed in direct contact with the powder to act as a thermal stopper, and then the electrode itself with a length of 22 L L x . 36 Fig. 3. Typical sequence in an ERS process and evolution of the effective current intensity that passes through the system during the ERS process. Fig. 4. Flow diagram of the numerical strategy adopted to solve the model of the ERS process. 37 Fig. 5. Intensity and applied force evolution during an ERS process. The vertical line marks the start of the applied current. Usually, time is measured in seconds, although it can also be measured in AC current cycles, with the equivalence 1 s = 50 cycles. Fig. 6. (a) Electrical resistance welding equipment adapted to act as the ERS equipment including (b) punches used in the experiments and (c) metallic hoop with the alumina die inside. (a) (b) (c) 38 Fig. 7. SEM micrographs of the powders used in the ERS experiments: (a) Fe WPL200, (b) Ti T325. Fig. 8. Experimental and simulated curves for the evolution of the global porosity and electrical resistance of the compact for Fe_1. The results of the simulation correspond to the heating and forging periods. Because of a technical limitation, however, the data acquisition is only possible during the heating period, not during the forging period (only the firsts are shown). (a) (b) 39 Fig. 9. Porosity distribution (general view and details) in the iron compact of Fe_1 experiment (after being etched with Beraha reagent). 40 Fig. 10. (a) Local porosity evolution in representative distances (planes) of the compact: centre (x0), middle point between the base and the centre (xm) and frontier powder/wafer (x1). (b) Porosity distribution inside the compact at some characteristic’s moments: heating period start (t0), end of the intensity increasing ramp (tS), middle of the heating period ( 02 mC tt   ), end of the heating period (tC) and end of the process (tT). The initial porosity value of 0.46 shown in the graphs corresponds to the porosity reached in the compact after the cold pressing period. The different curves in (b) finish at different values of the X axis because of the densification of the compact (only half compact is indeed represented because of the symmetry of the model). (a) (b)