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DAEM kinetics analysis and finite element simulation of thermal debinding process for a gelcast SiAlON green body
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DAEM kinetics analysis and finite element simulation of thermal debinding process for a gelcast SiAlON green body ⁎
Jing Lia, , Chuanfu Zhanga, Ruiming Yinb, Wenhai Zhanga,c a
School of Metallurgy and Environment, Central South University, Changsha, Hunan 410083, China College of Metallurgical Engineering, Hunan University of Technology, Zhuzhou, Hunan 412008, China c China Nerin Engineering Co., Ltd., Nanchang, Jiangxi 330002, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Gelcast SiAlON Thermal debinding Distributed activation energy model Finite element method
Thermal debinding is an important step in a gelcasting process, and the proper selection of the debinding technique is crucial for the quality of a green body. In this work, the pyrolysis characteristics of an N,N-dimethylacrylamide/N,N′-methylenebisacrylamide gel system in a thermal debinding process of a gelcast SiAlON green body were investigated through nonisothermal thermogravimetric analysis and a thermogravimetric analyzer coupled with Fourier transform infrared spectroscopy analysis. In addition, the pyrolysis kinetics of the thermal debinding process of the gelcast SiAlON green body were described by using a three-parallel-distributed activation energy model. The results showed that the conversion (α) and reaction rate (dα/dT) curves predicted by the kinetic model agreed well with the experimental data. The kinetic parameters (E0,i, k0,i, and σi) of the global thermal debinding process were 116.0–158.0 kJ/mol, 9.31 × 108 s−1 and 2.19–20.52 kJ/mol, respectively. Finally, a solid–fluid–thermal–mechanical coupling numerical model of the thermal debinding process was established through finite element methods on the basis of theories of the porous medium seepage and thermodynamics. The distributions of the residual content of the gel polymer, the temperature, the pressure, and the stress in the green body throughout the entire thermal debinding process were evaluated using this model. It was shown that a high heating rate leads to a high temperature gradient and von Mises stress inside the green body. Moreover, a reasonable debinding technique was proposed to effectively prevent and eliminate defects caused by excessive stress or a temperature gradient inside the green body. In a given DMAA/MBAM gel system, the strongest stress peak can be effectively eliminated by holding at 310 °C for no less than 2 h at a heating rate of 1 °C/min. This work aims to provide a theoretical basis for studying thermal debinding kinetics and optimizing debinding techniques for the gelcasting of various materials.
1. Introduction Gelcasting is a novel in situ solidification molding technology for near-net-shaped ceramic bodies that have been invented in recent years [1,2]. Compared to other molding techniques, gelcasting provides ceramic bodies characterized by favorable uniformity, excellent sintering properties, high strength, and easy deep processing and has been gradually applied to the preparation of various ceramic materials [3–8]. In recent years, numerous research reports have focused mainly on the application of gelcasting in various materials [9,10] and the development of new gel systems, such as selecting low-toxicity organic monomers [11–15] and applying environmentally friendly macromolecular gel systems [16–18]. However, there has been a lack of studies on the evolution and control of defects in gelcasting ceramic bodies during drying, debinding, and sintering. These basic theoretical problems are ⁎
essential for controlling and eliminating harmful defects in ceramic materials and improving their reliability. Thermal debinding is an important step in the gelcasting technique. Moreover, the proper selection of a debinding technique is crucial for the quality of a green body. Experimental results indicate that cracks may occur on the surface and inside gelcast green bodies in the debinding stage. This phenomenon is particularly prominent for large green bodies [19]. In fact, the thermal debinding process of the green body is a complex physical and chemical procedure. This procedure includes a heterogeneous pyrolysis reaction of the gel polymer, a gas precipitation reaction, and seepage mass and heat transfers of pyrolysis gases in the green body. The distribution of the residual gel content and the temperature, pressure, and stress fields in the green body constantly change with the continuous pyrolysis of the gel polymers. Excessive pressure or stress inside the body can easily cause cracking and
Correspondence to: School of Metallurgy and Environment, Central South University, Changsha, Hunan 410083, China. E-mail address: [email protected] (J. Li).
https://doi.org/10.1016/j.ceramint.2019.01.118 Received 27 October 2018; Received in revised form 21 December 2018; Accepted 16 January 2019 0272-8842/ © 2019 Elsevier Ltd and Techna Group S.r.l. All rights reserved.
Please cite this article as: Li, J., Ceramics International, https://doi.org/10.1016/j.ceramint.2019.01.118
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deformation. Furthermore, different gel systems possess various molecular structures and elemental compositions, thereby leading to different pyrolysis behavior [10,11]. Moreover, factors such as debinding temperature and time, heating rate, and body size considerably affect the quality of the green body [20]. Therefore, the pyrolysis kinetics characteristics of the thermal debinding process and the seepage mass and heat transfer mechanism of pyrolysis gases in the green body must be explored. This work aims to provide an important theoretical guide for effectively eliminating the internal defects in gelcast bodies caused by the debinding stage. In recent years, thermogravimetric analysis (TGA) has become a common method for studying the thermal stability of organic matter. Pyrolysis kinetics research is an important aspect of thermal analysis [21]. The distributed activation energy model (DAEM) has been proven to be able to describe the pyrolysis behavior of complex organic polymers, which contains infinite parallel first-order reactions. This model can be established by combining the weight loss data at different heating rates, thereby decreasing the influence of heating rate on the solution of kinetic parameters; furthermore, the model has been extensively used in pyrolysis kinetics studies [22]. Wang et al. [23] obtained the kinetic parameters of coal pyrolysis by adopting nonisothermal TGA and DAEM. Chen et al. [24] divided the pyrolysis process of lignocellulose biomass into three stages using the second derivative and determined the main pyrolysis kinetic parameters (i.e., activation energy E and pre-exponential factor k0) by using a threeparallel-distributed activation energy model (T-DAEM). However, few reports on the thermal debinding kinetics of gelcast ceramic bodies are currently available. The finite element method has been reported as an effective technique for studying the thermal debinding process of green bodies by coupling the pyrolysis kinetic equation with thermal stress and strain. Belgacem et al. [25] estimated the kinetic parameters of 316 L stainless steel feedstocks prepared by powder injection molding in the thermal debinding process through the Kissinger and Ozawa method. The authors conducted a numerical simulation of the thermal debinding process using finite element methods. The simulation results agreed well with the experimental data. At present, however, numerical simulation of the thermal debinding process of the gelcast green body that incorporates gel pyrolysis kinetics remains unexplored. In the present work, gelcast SiAlON green bodies were prepared with a low-toxicity N,N-dimethylacrylamide/N,N′-methylenebisacrylamide (DMAA/MBAM) gel system, and the thermal stability of the SiAlON green bodies was investigated through TGA and a thermogravimetric analyzer coupled with Fourier transform infrared spectroscopy (TG-FTIR) analysis. Then, a T-DAEM was adopted to investigate the thermal debinding kinetics and obtain the kinetic parameters. Finally, a 3D solid–fluid–thermal–mechanical coupling numerical model of the thermal debinding process was established through finite element methods on the basis of the theories of porous medium seepage and thermodynamics. The distribution characteristics of the residual gel content, the temperature, the pressure, and the stress in the body throughout the entire thermal debinding process were studied using this model. The proposed numerical model allowed us to obtain a reasonable debinding technique.
Fig. 1. Experimental flowchart of the gelcast SiAlON green body. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
N,N,N′,N′-tetramethylethylene-diamine were used as the dispersant, initiator, and catalyst, respectively. The pH of the slurry was adjusted with ammonia water. 2.2. Preparation of gelcast SiAlON green bodies Fig. 1 illustrates a schematic flowchart of the preparation process of the SiAlON green body by gelcasting. First, deionized water was used as a solvent to prepare a premixed solution (the monomer and crosslinking agent content is 12.4 wt% with a 16:1 ratio of DMAA to MBAM). Ammonia water was utilized to adjust the pH value of the premixed solution to approximately 11. A thermal treatment at 850 °C for 2 h in air was applied for the surface modification of AlN such that a dense alumina film was formed on the surface of AlN powders, thereby resulting in hydrolytic resistance [26]. Then, 22.5 g of Si3N4, 2 g of Al2O3, 2.8 g of AlN, 0.9 g of Y2O3, and 1.8 g of Ce2O3 were added to the premixed solution, with a solid loading of 42 vol%. After 4 h of ball milling with a planetary ball grinder, 1.0 wt% catalyst and initiator were added to the slurry and degassed in a vacuum device for 20 min. Subsequently, the degassed slurry was poured into a designed polyethylene mold and placed at room temperature for 30 min so that the monomers sufficiently polymerized and cured; then, the slurry was demolded after being dried at 80 °C for 24 h. Finally, the green body was transferred to a drying oven at 90 °C for further drying for 24 h to obtain the gelcast SiAlON green bodies required for the subsequent thermogravimetric experiment. 2.3. TGA experiment A thermogravimetric analyzer (STA-449 F3 Jupiter, NETZSCH, Germany) was used for TGA to study the thermal debinding kinetic characteristics of the green body. Nonisothermal runs in a temperature range of 35–900 °C were conducted at heating rates of 10, 15, and 20 °C/min. The loaded gas was highly pure argon (99.999%) with a flow of 20 mL/min. TG-FTIR analysis (TGA 8000-Frontier, PerkinElmer, USA) was utilized to analyze the pyrolysis reactions of the gel polymer in the green body. The loaded gas was highly pure helium (99.999% purity), and the purge-gas-flow and shielding gas rates were 40 and 20 mL/min, respectively. The thermal analysis temperature was determined in the range of 35–900 °C at a heating rate of 10 °C/min. The infrared spectrum range was 500–4000 cm−1, and the detection resolution was 4 cm−1. The scans were conducted 16 times.
2. Experiments and methodologies 2.1. Raw materials Si3N4 (SN-E10, UBE Industries, Ube, Japan), Al2O3 (99.9% purity, 0.5 µm, AKP-50, Sumitomo Chemical, Japan), and AlN (99.5% purity, 2.0 µm, Aladdin Industrial Co., Ltd., China) were used as raw materials. Y2O3 (99.99% purity, grade fine, H.C. Stark, Germany) and Ce2O3 (99.99% purity, 5 µm, Aladdin Industrial Co., Ltd., China) were used as sintering additives. DMAA and MBAM (Aladdin Industrial Co., Ltd., China) were used as the monomer and crosslinker, respectively. Ammonium polyacrylate, ammonium persulfate (self-prepared), and
3. Kinetic model and numerical simulation by finite element methods 3.1. Theory of DAEM and T-DAEM The DAEM is an effective method for studying the complex kinetics of pyrolysis processes [27,28]. In the DAEM, the reaction system is 2
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of the thermal debinding of the SiAlON green body are reasonable and reliable. The objective function is as follows [23]:
assumed to be composed of numerous independent and irreversible chemical reactions. These chemical reactions possess different activation energies. Furthermore, the difference in the activation energies of complex chemical reactions is expressed by a continuously distributed function f (E ). The conversion rate from 0 to a certain time point t can be expressed as:
α=1−
∫0
∞
exp ⎡−k 0 ⎣
∫0
t
E ⎞ dt ⎤ f (E ) dE exp ⎛− ⎝ RT ⎠ ⎦
nd
Fj = min
i=1
⎜
(1)
Fit(%) = 100 F / Np
∫0
(2)
f (E ) dE = 1
3.2. Modeling for the thermal debinding process of the gelcast SiAlON green body 3.2.1. Solid–fluid–thermal–mechanical coupling model The thermal debinding process of gelcast green bodies includes gel pyrolysis to light gases, such as CO2 and H2O; seepage mass and heat transfer problems of pyrolysis gas in the pores of the green body; pressure and stress evolutions under a temperature gradient, and other complex multiphase, multifield coupling procedures. In this section, a 3D solid–fluid–thermal–mechanical coupling model is proposed to describe the thermal debinding process of the gelcast SiAlON green body. In this model, the green body is assumed to be a porous medium, and Darcy's law is used to describe the problems of the porous seepage of pyrolysis gas in the green body. The porous media heat transfer model is applied to identify the temperature gradient from the surface to the center of the green body. The component transfer model is adopted to determine the residual concentration distribution of the gel polymer, and the stress evolution is governed by a structural mechanical equation. The abovementioned governing equations are coupled and solved through finite element methods. The porous seepage problem of the pyrolysis gas is described by Darcy's law [31], which can be defined as follows:
(3)
Thus, the reaction rate equation for the thermal debinding process of the gelcast green body can be described using the following differential equation:
dα (T ) 1 = dT σ 2π −
∫0
∞
k0 E k − 0 exp ⎡− ⎢ β RT β ⎣
∫0
T
E ⎞ dT exp ⎛− ⎝ RT ⎠
(E − E0)2 ⎤ dE 2σ 2 ⎥ ⎦
(4)
where β refers to the heating rate. In this work, the pyrolysis kinetic model of the DMAA/MBAM gel system is established using the T-DAEM. In this model, the gel polymer in the SiAlON green body is assumed to contain three independently reacting pseudocomponents. Therefore, the pyrolysis behavior of the gel polymer throughout the entire debinding process can be regarded as the weighting of the pyrolysis reactions of the three pseudocomponents. The equation utilized by the T-DAEM is expressed as follows [29]: 3
α=1−
∑ c0,i ∫0
∞
i=1
dα (T ) = dT
3
∑ c0,i ∫0 i=1
k 0, i exp ⎡− ψ (E , T ) ⎤ fi (E ) dE ⎢ ⎥ ⎣ β ⎦
∞
(5)
k 0, i E0, i k 0, i ψ (E , T ) ⎤ fi (E ) dE − exp ⎡− ⎢ ⎥ β β 2π σi ⎦ ⎣ RT
⎜
∂εp ⎞ ∂p + ∇⋅(ρu) = qm ρ ⎛⎜εp χf + ⎟ ∂p ⎠ ∂t ⎝
(10)
κ u = − (∇p + ρg∇D) μ
(11)
where u is Darcy's velocity vector, χf is the fluid compressibility, εp is the porosity, ρ is the density, p is the fluid pressure, ∇D is the unit vector in the direction of gravity, g is the acceleration of gravity, μ is the fluid dynamic viscosity, qm is the mass source term of the pyrolysis gas, and κ is the permeability of the green body, which can be expressed as:
(6)
where ψ (E , T ) is the integral of the Boltzmann factor, c0,i is the mass fraction of pseudocomponent i in the gel polymer, and subscript i refers to the values corresponding to the three pseudocomponents (i = 1, 2, and 3). ψ (E , T ) is simplified using the Fong–Hong–Zou approximation algorithm [27], and therefore, the second-order integral defined in Eqs. (5) and (6) is converted into a first-order integral [24,30], thereby reducing calculation capacity. ψ (E , T ) can be rewritten as follows:
E T ⎞ dT ψ (E , T ) = ∫0 exp ⎛− ⎝ RT ⎠ E exp(−u) ⎛ u4 + 18u3 + 86u2 + 96u ⎞ ≈ 2 4 3 2 R u ⎝ u + 20u + 120u + 240u + 120 ⎠
(9)
where Np indicates the number of data points.
⎟
where E0 is the mean activation energy and σ is the standard deviation of the distributed activation energy. According to the definition of activation energy, Eq. (3) can be obtained as follows: ∞
(8)
where subscript i indicates the data points used, nd refers to the number of data points, αexp, ij represents the experimental data at heating rate j (j = 1, 2, and 3 correspond to the heating rates of 10, 15, and 20 °C/ min, respectively), and αcal, ij denotes a series of parameters and data calculated using Eq. (5). The consistency between the model-predicted value and the experimental data can be determined through the following fit quality parameter:
where α refers to the conversion rate, k0 is the pre-exponential factor, E represents the apparent activation energy, R denotes the ideal gas constant, and T indicates the temperature. The activation energy distribution function f (E ) is generally assumed to be a Gaussian distribution function:
−(E − E0)2 ⎞ 1 exp ⎛ f (E ) = 2σ 2 σ 2π ⎝ ⎠
∑ [αexp,ij − αcal,ij]2
κ=
εp3dp2 150(1 − εp)2
(12)
where dp refers to the mean diameter of the powder. The pressure boundary conditions are applied to the surface of the green body, which is expressed as follows:
p=0
(13)
The pyrolysis behavior in the thermal debinding process of the green body is described by the T-DAEM. The mass source term for the gas production per unit volume can be defined as:
⎟
(7)
where u = E /(RT ) . A nonlinear least-squares fitting method based on the Levenberg–Marquardt algorithm [24] is used to obtain the values of kinetic parameters E0,i, k0,i, and σi in Eq. (5) at three heating rates, that is, 10, 15, and 20 °C/min, in order to ensure that the kinetic parameters
3
qm = −ρφβ ∑ c0, i i=0
dαi ci dT ci (T )
(14)
where φ is the content of the gel polymer in the solid phase and ci(T) is 3
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the theoretical residual content of component i when linearly heated to a temperature of T. The energy conservation equation of the green body is presented as follows:
(ρCp)eff
∂T + ρuCp ∇⋅T = ∇⋅(keff ∇T ) + qm ΔH ∂t
(15)
where ΔH refers to the latent heat of pyrolysis, keff represents the effective thermal conductivity, and (ρCp)eff is the effective volumetric heat capacity at a constant pressure, which is defined by an averaging model to comprise solid matrix and fluid properties as (ρCp)eff = εp ρs Cp, s + (1 − εp) ρCp The thermal boundary conditions of a constant temperature are used for the surface of the green body, which can be described as:
T = T0 + βt
(16)
where T0 refers to the initial temperature. The residual contents of the pseudocomponents in the gel polymer are calculated as:
∂ci = qm, i ∂t
Fig. 2. Calculation flowchart of the 3D solid–fluid–thermal–mechanical coupling model.
(17)
The source term of gel pyrolysis can be expressed as:
qm, i = β
dαi ci dT ci (T )
(18)
For the boundary conditions of Eq. (17), the following Neumann boundary condition is used:
∂ci =0 ∂t
(19)
In the thermal debinding process, the green body is subjected to pore pressure and thermal expansion force, which can be expressed by a structural mechanical equation [32].
∇⋅σ + Fv = 0
(20)
σ = Cε + αT ΔT ⋅I + αB pI
(21)
Fig. 3. Schematic illustration of the geometric structure of the green body used in the 3D model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where σ refers to the Cauchy stress tensor, Fv indicates the volume force, p represents the pore pressure, C = C (E , ν ) is the elastic matrix, ε denotes the strain tensor, αB is the Biot–Willis coefficient, αT is the thermal expansion coefficient, and I is the second-order identity tensor. The equivalent stress σi satisfies the Huber–von Mises–Hencky criterion, which can be represented by the following equation: 2 σi2 = 1/6[(σx − σy )2 + (σy − σz )2 + (σz − σx )2] + τxy
Table 1 The physical parameters of the gelcast SiAlON green body used in the present numerical simulation.
(22)
where σx, σy, and σz are the stresses in the x, y, and z directions, respectively, and τxy denotes the shear stress in the xy plane. 3.2.2. Material and process numerical implementation The T-DAEM is implemented using the Python language. The kinetic parameters (k0,i, E0,i, and σi) for gel polymer pyrolysis during the thermal debinding process are calculated by the kinetic model. Moreover, the kinetic model is coupled with the multi-physics field through the interpolation function of COMSOL software and then solved through finite element methods. The calculation flowchart of the solid–fluid–thermal–mechanical coupling model is illustrated in Fig. 2. The geometric structure of the green body used in the finite element simulation is shown in Fig. 3. The green body is 18 mm long, 18 mm wide and 5 mm thick. The solution domain is divided by adopting an automatic mesh generator with 9310 domain elements, 2394 boundary elements and 208 edge elements, and the degrees of freedom are 382,047. The fixed time step is 30 s Table 1 lists the physical parameters of the gelcast SiAlON green body used in the present numerical simulation.
Model parameters
Value
Mass fraction of DMAA/MBAM gel system Volume fraction of gel polymer Thermal expansion coefficient Thermal conduction coefficient Specific heat coefficient Density Poisson's ratio Young's modulus Average grain size of powder mixtures
9.5% 40% 3.4 × 10−5 K−1 1.25 W/m K 342.63 +1.2272T-0.0005T2 J/kg K 2000 kg/m3 0.3 180e9 Pa 1.0 µm
4. Results and discussion 4.1. Thermogravimetric analysis Fig. 4 depicts the mass loss (TG) and reaction rate (dα/dT) curves of the SiAlON green body in the debinding process obtained at three different heating rates (10, 15, and 20 °C/min). Fig. 4(a) and (b) demonstrate that the TG and dα/dT curves have similar trends for the three heating rates. However, the TG and dα/dT curves move to the hightemperature zone with the increase in the heating rate because of the time-consuming polymer pyrolysis and gas precipitation reactions in 4
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Fig. 5. Evolution of the FTIR spectra recorded during TGA of the gelcast SiAlON green body at different temperatures. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 2 Calculated kinetic parameters of the three pseudocomponents. Gel polymer
c0,i (-)
k0,i (s−1)
E0,i (kJ/mol)
σi (kJ/mol)
Pseudocomponent Ⅰ Pseudocomponent Ⅱ Pseudocomponent Ⅲ
0.19 0.56 0.25
9.31e8 9.31e8 9.31e8
116.0 142.4 158.0
3.81 2.19 20.52
Fig. 4. TG and dα/dT curves of the gelcast SiAlON green body at different heating rates. (a) TG and (b) dα/dT. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the green body. When the heating rate increases, the partial gel polymers cannot adequately rapidly pyrolyze and release their volatiles, thereby resulting in hysteresis. The dα/dT curves in Fig. 4(b) exhibit two strong peaks that were found at the temperature ranges of 220–320 °C and 320–600 °C. The two peaks are two important pyrolysis stages of the DMAA polymer, and the mass loss of the SiAlON green body varied dramatically in the two regions. Fig. 5 presents the evolution of the FTIR spectra of a gelcast SiAlON green body at different temperatures with a heating rate of 10 °C/min. Evidently, the carboxyl (–COOH) and other groups will break from the molecular chain of the polymers in the 220–320 °C stage, thereby forming CO2 and other light gases. Accordingly, the first weight loss peak is formed. Moreover, the carbonyl (–C = O–), amide (–NH), methylene (–CH2), associative hydroxyl (–OH), and other groups escape from the polymer chains at the 320–600 °C stage. Thus, the largest weight loss peak of the green body in the entire debinding period is formed. Only a slight mass loss occurs when the temperature is over 600 °C, indicating that most of the gel has substantially pyrolyzed at a temperature below 600 °C.
Fig. 6. Model prediction of the conversion rate (α) compared with experimental data.
experimental data obtained at different heating rates. Evidently, the model-predicted values agree well with the experimental data, and the fit parameter is less than 2.0%. A comparison of the model-predicted dα/dT curves and the experimental findings is provided in Fig. 7. In this figure, two pyrolysis peaks are found in the predicted dα/dT curves, and each pseudocomponent has a certain pyrolysis behavior. The prediction results of dα/dT are basically consistent with the experimental findings. Therefore, the T-DAEM can effectively describe the pyrolysis behavior observed in the entire thermal debinding process of a gelcast SiAlON green body.
4.2. Kinetic analysis using T-DAEM The pyrolysis kinetic parameters of the DMAA/MBAM gel system during the thermal debinding of a gelcast SiAlON green body calculated by the T-DAEM are listed in Table 2. The average activation energies of pseudocomponents I, II, and III are 116.0, 142.4, and 158.0 kJ/mol, respectively, as presented in Table 2. Fig. 6 displays a comparison of the conversion rate (α) curves obtained by using the T-DAEM with the
4.3. Numerical simulation results The abovementioned solid–fluid–thermal–mechanical coupling model describing the thermal debinding process of the gelcast SiAlON green body is used to analyze the distribution characteristics of the 5
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residual concentration of gel polymer, the temperature, the pressure, and the stress during the thermal debinding process of the green body. Moreover, the effects of the heating rate and heat preservation process on various physical fields are discussed to obtain a reasonable debinding technique. The conditions for the numerical simulation of different debinding techniques are summarized in Table 3. 4.3.1. Variations of mass, heat and mechanical during the whole thermal debinding process Fig. 8 depicts the contours and curves of the residual content of the gel polymer (c), the temperature difference between the surface and the center of the body (△T), the pressure (p), and the von Mises stress (σ) inside the green body at a heating rate of 1 °C/min. As demonstrated in Fig. 8(a), the distributions of the residual content of the gel polymer, the temperature, the pressure and the von Mises stress from the surface to the center of the body vary continuously with increasing temperature. As shown in Fig. 8(b), the residual content of the gel polymer gradually decreases during the continuous debinding process, and the gel polymer is completely pyrolyzed at approximately 600 °C. Moreover, the difference in the residual gel polymer concentration between the surface and the center of the green body is minimal at only 0.1%. The pressure curves show that the pressure at the center of the green body increases sharply before 344 °C due to a large amount (approximately 62%) of gel pyrolysis and gas precipitation and reaches a maximum of 6051 Pa at approximately 453 °C just after the pyrolysis peak of pseudocomponent III. Then, the pressure inside the body decreases gradually because of the reduced pyrolysis gas and increased porosity of the green body due to most of the gel being removed. The pressure distribution shows a slight difference between the center and the quarter of the body. The pressure gradient inside the body is mainly concentrated from the quarter to the surface of the body. Fig. 8(c) reveals that a strong stress peak is found in the interval of 270–390 °C during the debinding process. At 344 °C, the temperature difference between the surface and the center of the green body reaches a maximum value of 0.13 °C. In addition, the von Mises stress values on the surface and at the center of the green body reach maximum values of 0.54 and 0.3 MPa, respectively. Therefore, the green body is most likely to crack in this region. Upon heating to 800 °C, the von Mises stress values on the surface and at the center of the body are decreased to 0.37 and 0.19 MPa, respectively. At this point, the temperature difference between the surface and the center of the green body is 0.09 °C, and the pressure at the center of the body is approximately 3710 Pa. 4.3.2. Effect of heating rate Fig. 9 presents curves showing the changes in the residual content of the gel polymer (c), the temperature difference between the surface and center of the body (△T), and the von Mises stress at the center of the body (σc) with temperature at different heating rates. Evidently, the curves of the residual content of the gel polymer shift to the hightemperature zone with the increase in heating rate. This shift occurs because some chains in the gel polymer cannot be sufficiently removed due to the rapid heating rate; thus, the pyrolysis needs to be completed at a higher temperature. Furthermore, the curves of the temperature differences and von Mises stress shift to the high-temperature zone with the increase in heating rate, and the values of the temperature difference and stress also rapidly increase. At a heating rate of 1 °C/min, the temperature difference between the center and the surface of the green body reaches its highest value of 0.13 °C at 344 °C. Simultaneously, the maximum von Mises stress at the center of the green body is 0.3 MPa. At a heating rate of 2 °C/min, the temperature difference between the surface and the center of the green body reaches its highest value of 0.25 °C at 360 °C. At this time, the maximum stress at the center of the green body is 0.59 MPa. At a heating rate of 5 °C/min, the temperature difference between the surface and the center of the green body reaches its highest value of 0.61 °C at 379 °C. Under this condition, the maximum stress at the center of the green body is 1.47 MPa. These results
Fig. 7. Comparison between the experimental data and dα/dT predicted using the T-DAEM at different heating rates: (a) 10 °C/min, (b) 15 °C/min, and (c) 20 °C/min. Table 3 Calculation conditions for the different debinding techniques. Case
Heating rate
Holding temperature
Holding time
1 2 3 4 5 6 7 8
1 °C/min 2 °C/min 5 °C/min 1 °C/min 1 °C/min 1 °C/min 1 °C/min 1 °C/min
– – – 310 °C 310 °C 310 °C 300 °C 350 °C
– – – 0.5 h 1.0 h 2.0 h 2.0 h 2.0 h
6
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Fig. 8. Predicted distributions of the residual gel polymer content (c), pressure (p), temperature difference (△T), and von Mises stress (σ) inside the gelcast SiAlON green body at different times with a heating rate of 1 °C/min. (a) c, T, σc, and p inside of the body when T = 230 °C, 344 °C and 600 °C, (b) c and p, and (c) c, σ and △T. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
preservation process on the residual content of the gel polymer and the temperature, pressure, and stress fields at a low heating rate of 1 °C/ min.
show that a rapid heating rate indicates a high temperature gradient and stress inside the green body and a high possibility of cracking. Therefore, the thermal debinding process must be performed at a low heating rate. The following studies focus on the influence of the heat 7
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Fig. 9. Predicted distributions of the residual gel polymer content (c), temperature difference (△T), and von Mises stress (σc) inside the gelcast SiAlON green body at different heating rates (1, 2, and 5 °C/min). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4.3.3. Effect of thermal insulation process Heat preservation processes must be adopted to further eliminate the strongest stress peak caused by temperature differences from the center to the outer surface of the green body led by polymer pyrolysis and thermal adsorption at 280–390 °C during the debinding process. The selection of a reasonable holding temperature and time is particularly crucial. Fig. 10 displays the curves of the residual content of gel polymer (c) at the center of the body, the temperature difference between the surface and the center of the green body (△T), and the von Mises stress at the center of the green body (σc) at a heating rate of 1 °C/ min under different thermal insulation processes (preservation temperature and time). As seen from Fig. 10(a), the residual gel polymer content decreases rapidly when the temperature is high during the heat preservation period. At a preservation temperature of 300 °C for 2 h, the temperature difference peak between the surface and the center of the green body and the von Mises stress peak at the center of the green body are temporarily eliminated during the preservation time. However, the two peaks appear again, indicating that the 2 h preservation time at 300 °C is insufficient. At a preservation temperature of 310 °C for 2 h, the temperature difference peak between the surface and the center of the green body and the von Mises stress peak at the center of the green body are completely eliminated. Thus, a low preservation temperature requires an extended preservation time. Moreover, if the preservation temperature is kept at 350 °C, then this temperature is in the peak zone of a rapid thermal decomposition reaction of the gel polymer. A high temperature gradient (△T of approximately 0.12 °C) and von Mises stress (0.29 MPa at the center of the green body) have been formed inside the body. At this temperature, the optimal time to adopt certain preservation temperature measures to prevent the cracking of the green body has been missed. As shown in Fig. 10(b), the high temperature difference and stress peak in the range of 270–390 °C cannot be eliminated by holding at 310 °C for 0.5 and 1 h. In fact, the maximum stress at the center of the green body can still reach 0.27 MPa during the heat preservation. At this point, the temperature difference between the surface and the center of the green body can reach 0.11 °C. When the preservation time is increased to 2 h, however, the temperature difference and von Mises stress peaks inside the green body can be completely eliminated. Therefore, 2 h of preservation at 310 °C can prevent the occurrence of cracks in the green body at the highest stress zone during the debinding process.
Fig. 10. Predicted distributions of the residual gel polymer content (c), temperature difference (△T), and von Mises stress (σc) inside the gelcast SiAlON green body at a heating rate of 1 °C/min under different heat insulation measures. (a) holding temperatures of 300 °C, 310 °C, and 350 °C for 2 h and (b) holding temperature of 310 °C for 0.5, 1.0, and 2 h.
5. Conclusion A thorough and fundamental investigation of the thermal debinding of the gelcast SiAlON green body was conducted through experimental investigations and numerical simulations. In the first part, the pyrolysis characteristics of the DMAA/MBAM gel system in the thermal debinding process of the gelcast SiAlON green body were determined using TG and TG-FTIR analysis. The TG and dα/ dT analysis results showed that two weight loss peaks are found at 220–320 °C and 320–600 °C. These two peaks were the most important pyrolysis stages in the debinding process of the gelcast SiAlON green body. In the second part, the pyrolysis kinetic model of the DMAA/MBAM gel system in the thermal debinding process of the gelcast SiAlON green body was established using a T-DAEM. The results showed that the α and dα/dT data obtained by using the kinetic model are consistent with the experimental data and that the fit parameter is less than 2.0%. This result indicated that the T-DAEM can be used to accurately describe the thermal debinding behavior of the gelcast SiAlON green body. In the third part, a 3D solid–fluid–thermal–mechanical coupling model of thermal debinding for the gelcast SiAlON green body was established through the finite element method. The distribution 8
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characteristics of the residual gel polymer content, temperature, pressure, and von Mises stress in the green body during thermal debinding were analyzed using the present model. Evidently, a strong proportional relationship between the values of the von Mises stress at the center of the body (σc) and the temperature difference between the surface and the center of the green body (△T) is observed under a linear heating program. A high heating rate leads to a high temperature gradient and von Mises stress inside the green body. Furthermore, these results showed that the insulation treatment must be implemented below the largest weight loss peak (320–600 °C) to effectively restrain the strong stress peak caused by the temperature gradient inside the green body. In a given DMAA/MBAM gel system, the strongest stress peak can be effectively eliminated by holding at 310 °C for no less than 2 h at a heating rate of 1 °C/min.
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Acknowledgments This work is financially supported by the National Natural Science Foundation of China (No. 51574120). We thank Dr. Huang for his theoretical guidance in the numerical simulations. We are grateful to the technical assistance and use of facilities at the Shanghai Yuyi Analysis & Test Center, China. References [1] G.V. Franks, C. Tallon, A.R. Studart, M.L. Sesso, S. Leo, Colloidal processing: enabling complex shaped ceramics with unique multiscale structures, J. Am. Ceram. Soc. 100 (2017) 458–490, https://doi.org/10.1111/jace.14705. [2] R. Gilissen, J.P. Erauw, A. Smolders, E. Vanswijgenhoven, J. Luyten, Gelcasting, a near net shape technique, Mater. Des. 21 (2000) 251–257, https://doi.org/10. 1016/S0261-3069(99)00075-8. [3] W. Zeng, X. Gan, Z. Li, K. Zhou, The preparation of silicon nitride ceramics by gelcasting and pressureless sintering, Ceram. Int. 42 (2016) 11593–11597, https:// doi.org/10.1016/j.ceramint.2016.04.040. [4] W. Wan, C. Huang, J. Yang, J. Zeng, T. Qiu, Effect of sintering temperature on the properties of fused silica ceramics prepared by gelcasting, J. Electron. Mater. 43 (2014) 2566–2572, https://doi.org/10.1007/s11664-014-3112-7. [5] I. Ganesh, Near-net shape β-Si4Al2O2N6 parts by hydrolysis induced aqueous gelcasting process, Int. J. Appl. Ceram. Technol. 6 (2009) 89–101, https://doi.org/10. 1111/j.1744-7402.2008.02258.x. [6] L. Guo, J. Yang, Y. Feng, T. Qiu, Non-aqueous gelcasting of AlN ceramics using a low-toxicity monomer (DMAA) as gelling agent, Ceram. Int. 44 (2018) 1621–1626, https://doi.org/10.1016/j.ceramint.2017.10.083. [7] Q. Yao, L. Zhang, Z. Jiang, G. Huang, T. Zhou, Y. Ben, S. Wei, R. Sun, H. Chen, Y. Wang, Isobam assisted slurry optimization and gelcasting of transparent YAG ceramics, Ceram. Int. 44 (2018) 1699–1704, https://doi.org/10.1016/j.ceramint. 2017.10.098. [8] J. Zhang, D. Jiang, Q. Lin, Z. Chen, Z. Huang, Properties of silicon carbide ceramics from gelcasting and pressureless sintering, Mater. Des. 65 (2015) 12–16, https:// doi.org/10.1016/j.matdes.2014.08.034. [9] T. Deng, Y. Wang, A. Dufresne, N. Lin, Simultaneous enhancement of elasticity and strength of Al2O3-based ceramics body from cellulose nanocrystals via gel-casting process, Carbohydr. Polym. 181 (2018) 111–118, https://doi.org/10.1016/j. carbpol.2017.10.058. [10] P. Jamshidi, N. Lu, G. Liu, E. Herny, M.M. Attallah, Netshape centrifugal gel-casting of high-temperature sialon ceramics, Ceram. Int. 44 (2018) 3440–3447, https://doi. org/10.1016/j.ceramint.2017.11.143. [11] S. Yin, L. Pan, L. Guo, Y. Liu, Y. Feng, T. Qiu, J. Yang, Fabrication and properties of porous Si3N4 ceramics by aqueous gelcasting using low-toxic DMAA gelling agent, Ceram. Int. 44 (2018) 7569–7579, https://doi.org/10.1016/j.ceramint.2018.01.
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DAEM kinetics analysis and finite element simulation of thermal debinding process for a gelcast SiAlON green body ⁎
Jing Lia, , Chuanfu Zhanga, Ruiming Yinb, Wenhai Zhanga,c a
School of Metallurgy and Environment, Central South University, Changsha, Hunan 410083, China College of Metallurgical Engineering, Hunan University of Technology, Zhuzhou, Hunan 412008, China c China Nerin Engineering Co., Ltd., Nanchang, Jiangxi 330002, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Gelcast SiAlON Thermal debinding Distributed activation energy model Finite element method
Thermal debinding is an important step in a gelcasting process, and the proper selection of the debinding technique is crucial for the quality of a green body. In this work, the pyrolysis characteristics of an N,N-dimethylacrylamide/N,N′-methylenebisacrylamide gel system in a thermal debinding process of a gelcast SiAlON green body were investigated through nonisothermal thermogravimetric analysis and a thermogravimetric analyzer coupled with Fourier transform infrared spectroscopy analysis. In addition, the pyrolysis kinetics of the thermal debinding process of the gelcast SiAlON green body were described by using a three-parallel-distributed activation energy model. The results showed that the conversion (α) and reaction rate (dα/dT) curves predicted by the kinetic model agreed well with the experimental data. The kinetic parameters (E0,i, k0,i, and σi) of the global thermal debinding process were 116.0–158.0 kJ/mol, 9.31 × 108 s−1 and 2.19–20.52 kJ/mol, respectively. Finally, a solid–fluid–thermal–mechanical coupling numerical model of the thermal debinding process was established through finite element methods on the basis of theories of the porous medium seepage and thermodynamics. The distributions of the residual content of the gel polymer, the temperature, the pressure, and the stress in the green body throughout the entire thermal debinding process were evaluated using this model. It was shown that a high heating rate leads to a high temperature gradient and von Mises stress inside the green body. Moreover, a reasonable debinding technique was proposed to effectively prevent and eliminate defects caused by excessive stress or a temperature gradient inside the green body. In a given DMAA/MBAM gel system, the strongest stress peak can be effectively eliminated by holding at 310 °C for no less than 2 h at a heating rate of 1 °C/min. This work aims to provide a theoretical basis for studying thermal debinding kinetics and optimizing debinding techniques for the gelcasting of various materials.
1. Introduction Gelcasting is a novel in situ solidification molding technology for near-net-shaped ceramic bodies that have been invented in recent years [1,2]. Compared to other molding techniques, gelcasting provides ceramic bodies characterized by favorable uniformity, excellent sintering properties, high strength, and easy deep processing and has been gradually applied to the preparation of various ceramic materials [3–8]. In recent years, numerous research reports have focused mainly on the application of gelcasting in various materials [9,10] and the development of new gel systems, such as selecting low-toxicity organic monomers [11–15] and applying environmentally friendly macromolecular gel systems [16–18]. However, there has been a lack of studies on the evolution and control of defects in gelcasting ceramic bodies during drying, debinding, and sintering. These basic theoretical problems are ⁎
essential for controlling and eliminating harmful defects in ceramic materials and improving their reliability. Thermal debinding is an important step in the gelcasting technique. Moreover, the proper selection of a debinding technique is crucial for the quality of a green body. Experimental results indicate that cracks may occur on the surface and inside gelcast green bodies in the debinding stage. This phenomenon is particularly prominent for large green bodies [19]. In fact, the thermal debinding process of the green body is a complex physical and chemical procedure. This procedure includes a heterogeneous pyrolysis reaction of the gel polymer, a gas precipitation reaction, and seepage mass and heat transfers of pyrolysis gases in the green body. The distribution of the residual gel content and the temperature, pressure, and stress fields in the green body constantly change with the continuous pyrolysis of the gel polymers. Excessive pressure or stress inside the body can easily cause cracking and
Correspondence to: School of Metallurgy and Environment, Central South University, Changsha, Hunan 410083, China. E-mail address: [email protected] (J. Li).
https://doi.org/10.1016/j.ceramint.2019.01.118 Received 27 October 2018; Received in revised form 21 December 2018; Accepted 16 January 2019 0272-8842/ © 2019 Elsevier Ltd and Techna Group S.r.l. All rights reserved.
Please cite this article as: Li, J., Ceramics International, https://doi.org/10.1016/j.ceramint.2019.01.118
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deformation. Furthermore, different gel systems possess various molecular structures and elemental compositions, thereby leading to different pyrolysis behavior [10,11]. Moreover, factors such as debinding temperature and time, heating rate, and body size considerably affect the quality of the green body [20]. Therefore, the pyrolysis kinetics characteristics of the thermal debinding process and the seepage mass and heat transfer mechanism of pyrolysis gases in the green body must be explored. This work aims to provide an important theoretical guide for effectively eliminating the internal defects in gelcast bodies caused by the debinding stage. In recent years, thermogravimetric analysis (TGA) has become a common method for studying the thermal stability of organic matter. Pyrolysis kinetics research is an important aspect of thermal analysis [21]. The distributed activation energy model (DAEM) has been proven to be able to describe the pyrolysis behavior of complex organic polymers, which contains infinite parallel first-order reactions. This model can be established by combining the weight loss data at different heating rates, thereby decreasing the influence of heating rate on the solution of kinetic parameters; furthermore, the model has been extensively used in pyrolysis kinetics studies [22]. Wang et al. [23] obtained the kinetic parameters of coal pyrolysis by adopting nonisothermal TGA and DAEM. Chen et al. [24] divided the pyrolysis process of lignocellulose biomass into three stages using the second derivative and determined the main pyrolysis kinetic parameters (i.e., activation energy E and pre-exponential factor k0) by using a threeparallel-distributed activation energy model (T-DAEM). However, few reports on the thermal debinding kinetics of gelcast ceramic bodies are currently available. The finite element method has been reported as an effective technique for studying the thermal debinding process of green bodies by coupling the pyrolysis kinetic equation with thermal stress and strain. Belgacem et al. [25] estimated the kinetic parameters of 316 L stainless steel feedstocks prepared by powder injection molding in the thermal debinding process through the Kissinger and Ozawa method. The authors conducted a numerical simulation of the thermal debinding process using finite element methods. The simulation results agreed well with the experimental data. At present, however, numerical simulation of the thermal debinding process of the gelcast green body that incorporates gel pyrolysis kinetics remains unexplored. In the present work, gelcast SiAlON green bodies were prepared with a low-toxicity N,N-dimethylacrylamide/N,N′-methylenebisacrylamide (DMAA/MBAM) gel system, and the thermal stability of the SiAlON green bodies was investigated through TGA and a thermogravimetric analyzer coupled with Fourier transform infrared spectroscopy (TG-FTIR) analysis. Then, a T-DAEM was adopted to investigate the thermal debinding kinetics and obtain the kinetic parameters. Finally, a 3D solid–fluid–thermal–mechanical coupling numerical model of the thermal debinding process was established through finite element methods on the basis of the theories of porous medium seepage and thermodynamics. The distribution characteristics of the residual gel content, the temperature, the pressure, and the stress in the body throughout the entire thermal debinding process were studied using this model. The proposed numerical model allowed us to obtain a reasonable debinding technique.
Fig. 1. Experimental flowchart of the gelcast SiAlON green body. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
N,N,N′,N′-tetramethylethylene-diamine were used as the dispersant, initiator, and catalyst, respectively. The pH of the slurry was adjusted with ammonia water. 2.2. Preparation of gelcast SiAlON green bodies Fig. 1 illustrates a schematic flowchart of the preparation process of the SiAlON green body by gelcasting. First, deionized water was used as a solvent to prepare a premixed solution (the monomer and crosslinking agent content is 12.4 wt% with a 16:1 ratio of DMAA to MBAM). Ammonia water was utilized to adjust the pH value of the premixed solution to approximately 11. A thermal treatment at 850 °C for 2 h in air was applied for the surface modification of AlN such that a dense alumina film was formed on the surface of AlN powders, thereby resulting in hydrolytic resistance [26]. Then, 22.5 g of Si3N4, 2 g of Al2O3, 2.8 g of AlN, 0.9 g of Y2O3, and 1.8 g of Ce2O3 were added to the premixed solution, with a solid loading of 42 vol%. After 4 h of ball milling with a planetary ball grinder, 1.0 wt% catalyst and initiator were added to the slurry and degassed in a vacuum device for 20 min. Subsequently, the degassed slurry was poured into a designed polyethylene mold and placed at room temperature for 30 min so that the monomers sufficiently polymerized and cured; then, the slurry was demolded after being dried at 80 °C for 24 h. Finally, the green body was transferred to a drying oven at 90 °C for further drying for 24 h to obtain the gelcast SiAlON green bodies required for the subsequent thermogravimetric experiment. 2.3. TGA experiment A thermogravimetric analyzer (STA-449 F3 Jupiter, NETZSCH, Germany) was used for TGA to study the thermal debinding kinetic characteristics of the green body. Nonisothermal runs in a temperature range of 35–900 °C were conducted at heating rates of 10, 15, and 20 °C/min. The loaded gas was highly pure argon (99.999%) with a flow of 20 mL/min. TG-FTIR analysis (TGA 8000-Frontier, PerkinElmer, USA) was utilized to analyze the pyrolysis reactions of the gel polymer in the green body. The loaded gas was highly pure helium (99.999% purity), and the purge-gas-flow and shielding gas rates were 40 and 20 mL/min, respectively. The thermal analysis temperature was determined in the range of 35–900 °C at a heating rate of 10 °C/min. The infrared spectrum range was 500–4000 cm−1, and the detection resolution was 4 cm−1. The scans were conducted 16 times.
2. Experiments and methodologies 2.1. Raw materials Si3N4 (SN-E10, UBE Industries, Ube, Japan), Al2O3 (99.9% purity, 0.5 µm, AKP-50, Sumitomo Chemical, Japan), and AlN (99.5% purity, 2.0 µm, Aladdin Industrial Co., Ltd., China) were used as raw materials. Y2O3 (99.99% purity, grade fine, H.C. Stark, Germany) and Ce2O3 (99.99% purity, 5 µm, Aladdin Industrial Co., Ltd., China) were used as sintering additives. DMAA and MBAM (Aladdin Industrial Co., Ltd., China) were used as the monomer and crosslinker, respectively. Ammonium polyacrylate, ammonium persulfate (self-prepared), and
3. Kinetic model and numerical simulation by finite element methods 3.1. Theory of DAEM and T-DAEM The DAEM is an effective method for studying the complex kinetics of pyrolysis processes [27,28]. In the DAEM, the reaction system is 2
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of the thermal debinding of the SiAlON green body are reasonable and reliable. The objective function is as follows [23]:
assumed to be composed of numerous independent and irreversible chemical reactions. These chemical reactions possess different activation energies. Furthermore, the difference in the activation energies of complex chemical reactions is expressed by a continuously distributed function f (E ). The conversion rate from 0 to a certain time point t can be expressed as:
α=1−
∫0
∞
exp ⎡−k 0 ⎣
∫0
t
E ⎞ dt ⎤ f (E ) dE exp ⎛− ⎝ RT ⎠ ⎦
nd
Fj = min
i=1
⎜
(1)
Fit(%) = 100 F / Np
∫0
(2)
f (E ) dE = 1
3.2. Modeling for the thermal debinding process of the gelcast SiAlON green body 3.2.1. Solid–fluid–thermal–mechanical coupling model The thermal debinding process of gelcast green bodies includes gel pyrolysis to light gases, such as CO2 and H2O; seepage mass and heat transfer problems of pyrolysis gas in the pores of the green body; pressure and stress evolutions under a temperature gradient, and other complex multiphase, multifield coupling procedures. In this section, a 3D solid–fluid–thermal–mechanical coupling model is proposed to describe the thermal debinding process of the gelcast SiAlON green body. In this model, the green body is assumed to be a porous medium, and Darcy's law is used to describe the problems of the porous seepage of pyrolysis gas in the green body. The porous media heat transfer model is applied to identify the temperature gradient from the surface to the center of the green body. The component transfer model is adopted to determine the residual concentration distribution of the gel polymer, and the stress evolution is governed by a structural mechanical equation. The abovementioned governing equations are coupled and solved through finite element methods. The porous seepage problem of the pyrolysis gas is described by Darcy's law [31], which can be defined as follows:
(3)
Thus, the reaction rate equation for the thermal debinding process of the gelcast green body can be described using the following differential equation:
dα (T ) 1 = dT σ 2π −
∫0
∞
k0 E k − 0 exp ⎡− ⎢ β RT β ⎣
∫0
T
E ⎞ dT exp ⎛− ⎝ RT ⎠
(E − E0)2 ⎤ dE 2σ 2 ⎥ ⎦
(4)
where β refers to the heating rate. In this work, the pyrolysis kinetic model of the DMAA/MBAM gel system is established using the T-DAEM. In this model, the gel polymer in the SiAlON green body is assumed to contain three independently reacting pseudocomponents. Therefore, the pyrolysis behavior of the gel polymer throughout the entire debinding process can be regarded as the weighting of the pyrolysis reactions of the three pseudocomponents. The equation utilized by the T-DAEM is expressed as follows [29]: 3
α=1−
∑ c0,i ∫0
∞
i=1
dα (T ) = dT
3
∑ c0,i ∫0 i=1
k 0, i exp ⎡− ψ (E , T ) ⎤ fi (E ) dE ⎢ ⎥ ⎣ β ⎦
∞
(5)
k 0, i E0, i k 0, i ψ (E , T ) ⎤ fi (E ) dE − exp ⎡− ⎢ ⎥ β β 2π σi ⎦ ⎣ RT
⎜
∂εp ⎞ ∂p + ∇⋅(ρu) = qm ρ ⎛⎜εp χf + ⎟ ∂p ⎠ ∂t ⎝
(10)
κ u = − (∇p + ρg∇D) μ
(11)
where u is Darcy's velocity vector, χf is the fluid compressibility, εp is the porosity, ρ is the density, p is the fluid pressure, ∇D is the unit vector in the direction of gravity, g is the acceleration of gravity, μ is the fluid dynamic viscosity, qm is the mass source term of the pyrolysis gas, and κ is the permeability of the green body, which can be expressed as:
(6)
where ψ (E , T ) is the integral of the Boltzmann factor, c0,i is the mass fraction of pseudocomponent i in the gel polymer, and subscript i refers to the values corresponding to the three pseudocomponents (i = 1, 2, and 3). ψ (E , T ) is simplified using the Fong–Hong–Zou approximation algorithm [27], and therefore, the second-order integral defined in Eqs. (5) and (6) is converted into a first-order integral [24,30], thereby reducing calculation capacity. ψ (E , T ) can be rewritten as follows:
E T ⎞ dT ψ (E , T ) = ∫0 exp ⎛− ⎝ RT ⎠ E exp(−u) ⎛ u4 + 18u3 + 86u2 + 96u ⎞ ≈ 2 4 3 2 R u ⎝ u + 20u + 120u + 240u + 120 ⎠
(9)
where Np indicates the number of data points.
⎟
where E0 is the mean activation energy and σ is the standard deviation of the distributed activation energy. According to the definition of activation energy, Eq. (3) can be obtained as follows: ∞
(8)
where subscript i indicates the data points used, nd refers to the number of data points, αexp, ij represents the experimental data at heating rate j (j = 1, 2, and 3 correspond to the heating rates of 10, 15, and 20 °C/ min, respectively), and αcal, ij denotes a series of parameters and data calculated using Eq. (5). The consistency between the model-predicted value and the experimental data can be determined through the following fit quality parameter:
where α refers to the conversion rate, k0 is the pre-exponential factor, E represents the apparent activation energy, R denotes the ideal gas constant, and T indicates the temperature. The activation energy distribution function f (E ) is generally assumed to be a Gaussian distribution function:
−(E − E0)2 ⎞ 1 exp ⎛ f (E ) = 2σ 2 σ 2π ⎝ ⎠
∑ [αexp,ij − αcal,ij]2
κ=
εp3dp2 150(1 − εp)2
(12)
where dp refers to the mean diameter of the powder. The pressure boundary conditions are applied to the surface of the green body, which is expressed as follows:
p=0
(13)
The pyrolysis behavior in the thermal debinding process of the green body is described by the T-DAEM. The mass source term for the gas production per unit volume can be defined as:
⎟
(7)
where u = E /(RT ) . A nonlinear least-squares fitting method based on the Levenberg–Marquardt algorithm [24] is used to obtain the values of kinetic parameters E0,i, k0,i, and σi in Eq. (5) at three heating rates, that is, 10, 15, and 20 °C/min, in order to ensure that the kinetic parameters
3
qm = −ρφβ ∑ c0, i i=0
dαi ci dT ci (T )
(14)
where φ is the content of the gel polymer in the solid phase and ci(T) is 3
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the theoretical residual content of component i when linearly heated to a temperature of T. The energy conservation equation of the green body is presented as follows:
(ρCp)eff
∂T + ρuCp ∇⋅T = ∇⋅(keff ∇T ) + qm ΔH ∂t
(15)
where ΔH refers to the latent heat of pyrolysis, keff represents the effective thermal conductivity, and (ρCp)eff is the effective volumetric heat capacity at a constant pressure, which is defined by an averaging model to comprise solid matrix and fluid properties as (ρCp)eff = εp ρs Cp, s + (1 − εp) ρCp The thermal boundary conditions of a constant temperature are used for the surface of the green body, which can be described as:
T = T0 + βt
(16)
where T0 refers to the initial temperature. The residual contents of the pseudocomponents in the gel polymer are calculated as:
∂ci = qm, i ∂t
Fig. 2. Calculation flowchart of the 3D solid–fluid–thermal–mechanical coupling model.
(17)
The source term of gel pyrolysis can be expressed as:
qm, i = β
dαi ci dT ci (T )
(18)
For the boundary conditions of Eq. (17), the following Neumann boundary condition is used:
∂ci =0 ∂t
(19)
In the thermal debinding process, the green body is subjected to pore pressure and thermal expansion force, which can be expressed by a structural mechanical equation [32].
∇⋅σ + Fv = 0
(20)
σ = Cε + αT ΔT ⋅I + αB pI
(21)
Fig. 3. Schematic illustration of the geometric structure of the green body used in the 3D model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where σ refers to the Cauchy stress tensor, Fv indicates the volume force, p represents the pore pressure, C = C (E , ν ) is the elastic matrix, ε denotes the strain tensor, αB is the Biot–Willis coefficient, αT is the thermal expansion coefficient, and I is the second-order identity tensor. The equivalent stress σi satisfies the Huber–von Mises–Hencky criterion, which can be represented by the following equation: 2 σi2 = 1/6[(σx − σy )2 + (σy − σz )2 + (σz − σx )2] + τxy
Table 1 The physical parameters of the gelcast SiAlON green body used in the present numerical simulation.
(22)
where σx, σy, and σz are the stresses in the x, y, and z directions, respectively, and τxy denotes the shear stress in the xy plane. 3.2.2. Material and process numerical implementation The T-DAEM is implemented using the Python language. The kinetic parameters (k0,i, E0,i, and σi) for gel polymer pyrolysis during the thermal debinding process are calculated by the kinetic model. Moreover, the kinetic model is coupled with the multi-physics field through the interpolation function of COMSOL software and then solved through finite element methods. The calculation flowchart of the solid–fluid–thermal–mechanical coupling model is illustrated in Fig. 2. The geometric structure of the green body used in the finite element simulation is shown in Fig. 3. The green body is 18 mm long, 18 mm wide and 5 mm thick. The solution domain is divided by adopting an automatic mesh generator with 9310 domain elements, 2394 boundary elements and 208 edge elements, and the degrees of freedom are 382,047. The fixed time step is 30 s Table 1 lists the physical parameters of the gelcast SiAlON green body used in the present numerical simulation.
Model parameters
Value
Mass fraction of DMAA/MBAM gel system Volume fraction of gel polymer Thermal expansion coefficient Thermal conduction coefficient Specific heat coefficient Density Poisson's ratio Young's modulus Average grain size of powder mixtures
9.5% 40% 3.4 × 10−5 K−1 1.25 W/m K 342.63 +1.2272T-0.0005T2 J/kg K 2000 kg/m3 0.3 180e9 Pa 1.0 µm
4. Results and discussion 4.1. Thermogravimetric analysis Fig. 4 depicts the mass loss (TG) and reaction rate (dα/dT) curves of the SiAlON green body in the debinding process obtained at three different heating rates (10, 15, and 20 °C/min). Fig. 4(a) and (b) demonstrate that the TG and dα/dT curves have similar trends for the three heating rates. However, the TG and dα/dT curves move to the hightemperature zone with the increase in the heating rate because of the time-consuming polymer pyrolysis and gas precipitation reactions in 4
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Fig. 5. Evolution of the FTIR spectra recorded during TGA of the gelcast SiAlON green body at different temperatures. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 2 Calculated kinetic parameters of the three pseudocomponents. Gel polymer
c0,i (-)
k0,i (s−1)
E0,i (kJ/mol)
σi (kJ/mol)
Pseudocomponent Ⅰ Pseudocomponent Ⅱ Pseudocomponent Ⅲ
0.19 0.56 0.25
9.31e8 9.31e8 9.31e8
116.0 142.4 158.0
3.81 2.19 20.52
Fig. 4. TG and dα/dT curves of the gelcast SiAlON green body at different heating rates. (a) TG and (b) dα/dT. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the green body. When the heating rate increases, the partial gel polymers cannot adequately rapidly pyrolyze and release their volatiles, thereby resulting in hysteresis. The dα/dT curves in Fig. 4(b) exhibit two strong peaks that were found at the temperature ranges of 220–320 °C and 320–600 °C. The two peaks are two important pyrolysis stages of the DMAA polymer, and the mass loss of the SiAlON green body varied dramatically in the two regions. Fig. 5 presents the evolution of the FTIR spectra of a gelcast SiAlON green body at different temperatures with a heating rate of 10 °C/min. Evidently, the carboxyl (–COOH) and other groups will break from the molecular chain of the polymers in the 220–320 °C stage, thereby forming CO2 and other light gases. Accordingly, the first weight loss peak is formed. Moreover, the carbonyl (–C = O–), amide (–NH), methylene (–CH2), associative hydroxyl (–OH), and other groups escape from the polymer chains at the 320–600 °C stage. Thus, the largest weight loss peak of the green body in the entire debinding period is formed. Only a slight mass loss occurs when the temperature is over 600 °C, indicating that most of the gel has substantially pyrolyzed at a temperature below 600 °C.
Fig. 6. Model prediction of the conversion rate (α) compared with experimental data.
experimental data obtained at different heating rates. Evidently, the model-predicted values agree well with the experimental data, and the fit parameter is less than 2.0%. A comparison of the model-predicted dα/dT curves and the experimental findings is provided in Fig. 7. In this figure, two pyrolysis peaks are found in the predicted dα/dT curves, and each pseudocomponent has a certain pyrolysis behavior. The prediction results of dα/dT are basically consistent with the experimental findings. Therefore, the T-DAEM can effectively describe the pyrolysis behavior observed in the entire thermal debinding process of a gelcast SiAlON green body.
4.2. Kinetic analysis using T-DAEM The pyrolysis kinetic parameters of the DMAA/MBAM gel system during the thermal debinding of a gelcast SiAlON green body calculated by the T-DAEM are listed in Table 2. The average activation energies of pseudocomponents I, II, and III are 116.0, 142.4, and 158.0 kJ/mol, respectively, as presented in Table 2. Fig. 6 displays a comparison of the conversion rate (α) curves obtained by using the T-DAEM with the
4.3. Numerical simulation results The abovementioned solid–fluid–thermal–mechanical coupling model describing the thermal debinding process of the gelcast SiAlON green body is used to analyze the distribution characteristics of the 5
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residual concentration of gel polymer, the temperature, the pressure, and the stress during the thermal debinding process of the green body. Moreover, the effects of the heating rate and heat preservation process on various physical fields are discussed to obtain a reasonable debinding technique. The conditions for the numerical simulation of different debinding techniques are summarized in Table 3. 4.3.1. Variations of mass, heat and mechanical during the whole thermal debinding process Fig. 8 depicts the contours and curves of the residual content of the gel polymer (c), the temperature difference between the surface and the center of the body (△T), the pressure (p), and the von Mises stress (σ) inside the green body at a heating rate of 1 °C/min. As demonstrated in Fig. 8(a), the distributions of the residual content of the gel polymer, the temperature, the pressure and the von Mises stress from the surface to the center of the body vary continuously with increasing temperature. As shown in Fig. 8(b), the residual content of the gel polymer gradually decreases during the continuous debinding process, and the gel polymer is completely pyrolyzed at approximately 600 °C. Moreover, the difference in the residual gel polymer concentration between the surface and the center of the green body is minimal at only 0.1%. The pressure curves show that the pressure at the center of the green body increases sharply before 344 °C due to a large amount (approximately 62%) of gel pyrolysis and gas precipitation and reaches a maximum of 6051 Pa at approximately 453 °C just after the pyrolysis peak of pseudocomponent III. Then, the pressure inside the body decreases gradually because of the reduced pyrolysis gas and increased porosity of the green body due to most of the gel being removed. The pressure distribution shows a slight difference between the center and the quarter of the body. The pressure gradient inside the body is mainly concentrated from the quarter to the surface of the body. Fig. 8(c) reveals that a strong stress peak is found in the interval of 270–390 °C during the debinding process. At 344 °C, the temperature difference between the surface and the center of the green body reaches a maximum value of 0.13 °C. In addition, the von Mises stress values on the surface and at the center of the green body reach maximum values of 0.54 and 0.3 MPa, respectively. Therefore, the green body is most likely to crack in this region. Upon heating to 800 °C, the von Mises stress values on the surface and at the center of the body are decreased to 0.37 and 0.19 MPa, respectively. At this point, the temperature difference between the surface and the center of the green body is 0.09 °C, and the pressure at the center of the body is approximately 3710 Pa. 4.3.2. Effect of heating rate Fig. 9 presents curves showing the changes in the residual content of the gel polymer (c), the temperature difference between the surface and center of the body (△T), and the von Mises stress at the center of the body (σc) with temperature at different heating rates. Evidently, the curves of the residual content of the gel polymer shift to the hightemperature zone with the increase in heating rate. This shift occurs because some chains in the gel polymer cannot be sufficiently removed due to the rapid heating rate; thus, the pyrolysis needs to be completed at a higher temperature. Furthermore, the curves of the temperature differences and von Mises stress shift to the high-temperature zone with the increase in heating rate, and the values of the temperature difference and stress also rapidly increase. At a heating rate of 1 °C/min, the temperature difference between the center and the surface of the green body reaches its highest value of 0.13 °C at 344 °C. Simultaneously, the maximum von Mises stress at the center of the green body is 0.3 MPa. At a heating rate of 2 °C/min, the temperature difference between the surface and the center of the green body reaches its highest value of 0.25 °C at 360 °C. At this time, the maximum stress at the center of the green body is 0.59 MPa. At a heating rate of 5 °C/min, the temperature difference between the surface and the center of the green body reaches its highest value of 0.61 °C at 379 °C. Under this condition, the maximum stress at the center of the green body is 1.47 MPa. These results
Fig. 7. Comparison between the experimental data and dα/dT predicted using the T-DAEM at different heating rates: (a) 10 °C/min, (b) 15 °C/min, and (c) 20 °C/min. Table 3 Calculation conditions for the different debinding techniques. Case
Heating rate
Holding temperature
Holding time
1 2 3 4 5 6 7 8
1 °C/min 2 °C/min 5 °C/min 1 °C/min 1 °C/min 1 °C/min 1 °C/min 1 °C/min
– – – 310 °C 310 °C 310 °C 300 °C 350 °C
– – – 0.5 h 1.0 h 2.0 h 2.0 h 2.0 h
6
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Fig. 8. Predicted distributions of the residual gel polymer content (c), pressure (p), temperature difference (△T), and von Mises stress (σ) inside the gelcast SiAlON green body at different times with a heating rate of 1 °C/min. (a) c, T, σc, and p inside of the body when T = 230 °C, 344 °C and 600 °C, (b) c and p, and (c) c, σ and △T. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
preservation process on the residual content of the gel polymer and the temperature, pressure, and stress fields at a low heating rate of 1 °C/ min.
show that a rapid heating rate indicates a high temperature gradient and stress inside the green body and a high possibility of cracking. Therefore, the thermal debinding process must be performed at a low heating rate. The following studies focus on the influence of the heat 7
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Fig. 9. Predicted distributions of the residual gel polymer content (c), temperature difference (△T), and von Mises stress (σc) inside the gelcast SiAlON green body at different heating rates (1, 2, and 5 °C/min). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4.3.3. Effect of thermal insulation process Heat preservation processes must be adopted to further eliminate the strongest stress peak caused by temperature differences from the center to the outer surface of the green body led by polymer pyrolysis and thermal adsorption at 280–390 °C during the debinding process. The selection of a reasonable holding temperature and time is particularly crucial. Fig. 10 displays the curves of the residual content of gel polymer (c) at the center of the body, the temperature difference between the surface and the center of the green body (△T), and the von Mises stress at the center of the green body (σc) at a heating rate of 1 °C/ min under different thermal insulation processes (preservation temperature and time). As seen from Fig. 10(a), the residual gel polymer content decreases rapidly when the temperature is high during the heat preservation period. At a preservation temperature of 300 °C for 2 h, the temperature difference peak between the surface and the center of the green body and the von Mises stress peak at the center of the green body are temporarily eliminated during the preservation time. However, the two peaks appear again, indicating that the 2 h preservation time at 300 °C is insufficient. At a preservation temperature of 310 °C for 2 h, the temperature difference peak between the surface and the center of the green body and the von Mises stress peak at the center of the green body are completely eliminated. Thus, a low preservation temperature requires an extended preservation time. Moreover, if the preservation temperature is kept at 350 °C, then this temperature is in the peak zone of a rapid thermal decomposition reaction of the gel polymer. A high temperature gradient (△T of approximately 0.12 °C) and von Mises stress (0.29 MPa at the center of the green body) have been formed inside the body. At this temperature, the optimal time to adopt certain preservation temperature measures to prevent the cracking of the green body has been missed. As shown in Fig. 10(b), the high temperature difference and stress peak in the range of 270–390 °C cannot be eliminated by holding at 310 °C for 0.5 and 1 h. In fact, the maximum stress at the center of the green body can still reach 0.27 MPa during the heat preservation. At this point, the temperature difference between the surface and the center of the green body can reach 0.11 °C. When the preservation time is increased to 2 h, however, the temperature difference and von Mises stress peaks inside the green body can be completely eliminated. Therefore, 2 h of preservation at 310 °C can prevent the occurrence of cracks in the green body at the highest stress zone during the debinding process.
Fig. 10. Predicted distributions of the residual gel polymer content (c), temperature difference (△T), and von Mises stress (σc) inside the gelcast SiAlON green body at a heating rate of 1 °C/min under different heat insulation measures. (a) holding temperatures of 300 °C, 310 °C, and 350 °C for 2 h and (b) holding temperature of 310 °C for 0.5, 1.0, and 2 h.
5. Conclusion A thorough and fundamental investigation of the thermal debinding of the gelcast SiAlON green body was conducted through experimental investigations and numerical simulations. In the first part, the pyrolysis characteristics of the DMAA/MBAM gel system in the thermal debinding process of the gelcast SiAlON green body were determined using TG and TG-FTIR analysis. The TG and dα/ dT analysis results showed that two weight loss peaks are found at 220–320 °C and 320–600 °C. These two peaks were the most important pyrolysis stages in the debinding process of the gelcast SiAlON green body. In the second part, the pyrolysis kinetic model of the DMAA/MBAM gel system in the thermal debinding process of the gelcast SiAlON green body was established using a T-DAEM. The results showed that the α and dα/dT data obtained by using the kinetic model are consistent with the experimental data and that the fit parameter is less than 2.0%. This result indicated that the T-DAEM can be used to accurately describe the thermal debinding behavior of the gelcast SiAlON green body. In the third part, a 3D solid–fluid–thermal–mechanical coupling model of thermal debinding for the gelcast SiAlON green body was established through the finite element method. The distribution 8
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characteristics of the residual gel polymer content, temperature, pressure, and von Mises stress in the green body during thermal debinding were analyzed using the present model. Evidently, a strong proportional relationship between the values of the von Mises stress at the center of the body (σc) and the temperature difference between the surface and the center of the green body (△T) is observed under a linear heating program. A high heating rate leads to a high temperature gradient and von Mises stress inside the green body. Furthermore, these results showed that the insulation treatment must be implemented below the largest weight loss peak (320–600 °C) to effectively restrain the strong stress peak caused by the temperature gradient inside the green body. In a given DMAA/MBAM gel system, the strongest stress peak can be effectively eliminated by holding at 310 °C for no less than 2 h at a heating rate of 1 °C/min.
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