First-principles calculations to investigate the thermal response of the ZrC(1−x)Nx ceramics at extreme conditions
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Hassan Alipour
Abstract
We present the thermodynamic properties of ZrC(1−x)N x ceramics at elevated temperature (0–1,000 K) and pressure (0–150 GPa) conditions, explored by density functional theory. We implemented the Debye–Grüneisen quasi-harmonic model in our calculations. In our investigation, we cover elastic constants, elastic moduli, compressibility, ductility/brittleness, hardness, sound velocities, minimum thermal conductivity, melting temperature, anisotropy indices, isothermal bulk modulus, heat capacities, entropy, Debye temperature, Grüneisen parameter, thermal expansion coefficient, and thermal pressure. We address the effect of the structural anisotropy and bonding nature of ZrC(1−x)N x compounds on their thermal response to extreme conditions. Considering ZrC(1−x)N x with the x in the range of 0.0, 0.25, 0.5, 0.75, and 1.0, ZrC0.50N0.50 stands out in the response to the applied conditions. At higher temperatures, the thermal expansion of the ZrC0.50N0.50 shows a smaller increase, which makes it a favorable candidate for coating material in cutting tools against commonly used ZrN and ZrC ceramics. Similar behavior is observed for the heat capacity by increasing pressure at higher temperatures, where a smaller reduction is observed. It could be interpreted as a more stable response regarding the application-specific design conditions.
1 Introduction
Over the past two decades, transition metal carbides (TMCs) and transition metal nitrides (TMNs) have been utilized in various technological applications due to their appealing properties such as excellent mechanical stability, hardness, conductivity, and erosion resistance. Drill bits, golf shoe spikes, and snow tires are examples of the applications where their remarkable mechanical properties come into play. Their interesting electronic, optical, and magnetic properties have also given them applications in manufacturing optoelectronic devices, such as electrical contacts and optical coatings [1]. In addition, the excellent catalytic activity of TMCs makes them a proper choice for applications where higher electrochemical performance is demanded [2]. The interesting properties of TMCs and TMNs originate from their peculiar bonding nature; their structures contain covalent, metallic, and ionic bonds [3,4,5,6].
Zirconium carbide (ZrC) and zirconium nitride (ZrN) are two widely used TMCs and TMNs that meet the strict requirements of different industrial applications. Owing to their hardness and good resistance at high pressure and high temperature conditions, they are commonly used in manufacturing cutting tools, tool bits, and machinery [7]. Rock-salt structure is the ground state structure of ZrC and ZrN ceramics, a crystalline face-centered cubic (FCC) structure with the Fm3m-225 space group [8]. In this structure, carbon and nitrogen are located in the octahedral interstitial positions. The structure contains covalent, metallic, and ionic bonds, among which the covalent bonds are the dominant type.
Vibrational thermal effects are essential in the calculation of the thermodynamic properties of a solid as a function of pressure and temperature, f(p, T) [9]. This function can be represented through Debye–Grüneisen quasi-harmonic model that provides vibrational Helmholtz free energy under given conditions [10]. Equation of states (EOSs) is then obtained by minimizing non-equilibrium Gibbs free energy at arbitrary temperatures and pressures [11,12]. Thermodynamic properties provide insight into the atomic interactions, interatomic bond strength, and stability of the alloys, properties that are affected by the lattice vibrations [13,14,15].
The thermodynamic characteristics of ZrC and ZrN have been extensively investigated, both experimentally and theoretically. X-ray radiation, ultrasonic and electromotive force, Knudsen cell, differential thermal analysis, cryostat, and calorimeter are among the experimental methods implemented to better understand these materials’ behaviors under extreme conditions [3,8,16]. Ab initio and classical molecular dynamics are theoretical methods that provide an atomic-level description of the underlying physics. This feature can describe and justify the experimental observations or even build a predictive model. First-principles calculations, developed based on the pseudopotential and full potential approaches, are broadly used in physics, mechanics, and materials science to calculate a wide range of material properties. In context with these powerful simulation tools, various theoretical models, such as the augmented plane wave method [17,18], muffin-tin orbital techniques [17,19,20], and their linearized versions, the linearized augmented-plane-wave (LAPW) and the linear muffin-tin orbital, have been developed [17,19,21]. Moreover, different approximations, such as local density approximation (LDA) [22], generalized gradient approximation (GGA) [23], or Perdew, Burke, and Ernzerhof for solid (PBEsol) [24], have been developed in the framework of the density functional theory (DFT) method [25].
Numerous studies have been conducted to investigate the thermodynamic properties of ZrC and ZrN. Guillermet et al. analyzed the available thermodynamic information and calculated the cohesive energy of carbides and nitrates of 4d metals. They reached an order of precision that made a detailed comparison between ab initio and experimental results became possible [26]. Using the first-principles calculations, Hao et al. calculated various features of ZrC and ZrN, including pressure-induced structure transition, elastic constants, heat capacity, and Debye temperature as a function of temperature and pressure [27]. Zhu et al. examined the elastic and thermodynamic properties of ZrC by ab initio pseudopotential plane-wave calculations. They also used quasi-harmonic Debye models to calculate the temperature/pressure dependence of some of these elastic and thermodynamic properties [6]. Kim et al. studied the temperature- and pressure-dependent thermodynamic properties of ZrC, ZrN, and ZrC0.50N0.50 using first-principles calculations and Debye–Grüneisen theory. They concluded that except for the bulk modulus, ZrC0.50N0.50 has the highest elastic moduli [28]. Jing et al. provided a combination of the quasi-harmonic Debye model and thermal electronic excitation to study the thermodynamic properties of ZrC for the stable rock-salt structure. One of their quantitative results was that below the critical point of V/V 0 = 0.570, the transition of the B1 to B2 structure has the highest probability at higher pressures [29]. Varshney and Shriya presented the thermodynamic properties of some TMCs at high pressure and temperature within the framework of a quasi-harmonic Debye model. They reported that the heat capacity decreases non-linearly with the applied pressure at different temperatures [30]. Yang et al. studied the thermodynamic properties of different transient metal carbides and nitrides with DFT based on the quasi-harmonic Debye–Grüneisen model. One of their reported results is that carbides and nitrides have higher heat capacity at higher pressure and temperature than metals [31]. Using the full-potential LAPW (FP-LAPW) method, implemented in WIEN2k, various properties of materials, including electronic, thermoelectric, optical, mechanical, magnetic, and magneto-electronic properties of different materials, were investigated in the literature [32,33,34,35,36,37,38,39,40,41,42,43]. Although a few studies have addressed the thermodynamic properties of ZrN and ZrC at elevated pressures and temperatures, little attention has been paid to the comparative study of the thermodynamic properties of their ternary compounds. In this article, we perform first-principles calculations using the FP-LAPW method to study the structural and mechanical properties and the temperature- and pressure-dependence of the thermodynamic properties of ZrC, ZrN, and their ternary compounds, ZrC(1−x)N x . We elucidate how the bonding in ZrC(1−x)N x structure affects its properties. As for the mechanical properties, we cover elastic constants, bulk, shear, and Young’s moduli, compressibility, Poisson’s ratio, Cauchy pressure, Pugh’s ratio, and hardness. We also explore different thermodynamic properties in our investigation, including sound velocities, minimum thermal conductivity, melting temperature, anisotropy indices, isothermal bulk modulus, heat capacities, entropy, Debye temperature, Grüneisen parameter, thermal expansion coefficient, and thermal pressure. Our results could facilitate selecting the ZrC(1−x)N x compound, from which a superior response is expected in the given harsh conditions. In addition, our results could provide a theoretical basis for the experimental investigations. By investigating a range of C and N concentrations, we show how structural properties could be manipulated to achieve the desired outcomes in a specific application. The structure of the article is as follows. After presenting the computational details in Section 1, we describe the theoretical basis of the calculated properties in Section 2. Results and the associated discussion are presented in Section 3. Section 4 contains the conclusion and outlook.
2 Computational details
DFT calculations were performed in the FP-LAPW framework as implemented in the WIEN2k package [19,44,45]. The PBE GGA exchange-correlation functional [44] was used in all simulations. The wave function inside the muffin-tin spheres was extended as spherical harmonics, for which the maximum angular momentum was considered to be l max = 10. The proper convergence was obtained by setting the plane wave cutoff for kinetic energy as R MT K max = 7. The expansion of the plane wave outside of the sphere is called the interstitial region. The convergence of the total energy and zero charge leakage from particles was considered in the configuration of the muffin-tin radiuses (R MT) [32]. The R MT values for C, N, and Zr atoms were set to 1.89, 1.94, and 2.31, respectively. The convergence criterion in self-consistent calculations is the energy convergence with the tolerance of 10−4 Ry. The experimental lattice parameter of ZrC was initially used to create its crystal structures. The ternary ZrC(1−x)N x compounds, x being 0.25, 0.5, 0.75, and 1, were built by substituting the corresponding number (to x) of C atoms with N atoms. After structure minimization, the optimum lattice constants were obtained, and the structures were rebuilt to be used in the calculations. It should be noted that due to the existence of different bonds and deviation from cubic symmetry, ternary structures have anisotropy similar to the orthorhombic structure, and they took different a, b, and c values upon optimization. Moreover, 5,000 k-points were used for sampling the Brillouin zone. As mentioned earlier, the Bravais lattice of ZrC and ZrN crystals is a FCC known by the Fm-3m structural group. In order to create structures with the desired content of N and C atoms, a 1 × 1 × 1 cell containing four Zr atoms and four C or N atoms was created. Monitoring energy-vs-volume was used to obtain the optimized lattice parameter and bulk modulus. The structure with lower internal energy was taken as the stable structure. The elastic constants were calculated using the IRelast package [46], and the Voigt–Reuss–Hill (VRH) approximation was used to account for the polycrystalline elasticity (elastic moduli) at zero pressures [37,41]. In addition, the thermodynamic parameters were calculated using the Debye–Grüneisen quasi-harmonic model [47] as implemented in the Gibbs2 package [11,48]. Based on the obtained results, the temperature range of 0–1,000 K (the temperature near the Debye temperature) and pressure range of 0–150 GPa (the pressure at which the structures are still stable in their phases) were considered for the representation of all thermal parameters.
2.1 Equilibrium structure and mechanical properties
The energy–volume relationship is an isothermal EOS describing the thermodynamic correlation of energy, pressure, and volume at a constant temperature. One of the EOSs widely used to estimate the optimum volume against energy is the Birch–Murnaghan (B–M) isothermal equation. This equation is derived by taking the relationship between the volume of the crystal and the applied pressure into account. B–M isothermal equation describes the system’s internal energy as a function of volume as in the following equation (1):
Here, E is internal energy, V
0 is the volume of the reference structure, V is the volume of the deformed structure, B
0 is the bulk modulus, and
Different parameters were taken into account to investigate the mechanical behavior of the structure. First, we calculated the elastic moduli (bulk, shear, and Young’s modulus) using the VRH approximations, which establish a relationship between the elastic constants of anisotropic single crystalline and isotropic polycrystalline elastic moduli [7]. The brittle or ductile nature of ZrC x N(1−x) structures was described by evaluating the Poisson’s ratio, the Pugh’s ratio, and the Cauchy pressure, based on the method introduced in ref. [50]. In addition, the hardness, the sound velocities, the minimum thermal conductivity, the melting temperature, and the anisotropy indices of ZrC x N(1−x) were calculated using the corresponding relations explained in refs [21,50,51]. All the mechanical parameters were calculated at atmospheric pressure.
2.2 Thermodynamic parameters
A solid’s thermodynamic properties are related to its lattice’s phonon vibrations. Calculating these properties at high temperatures and pressures can help us better understand how they behave in environments with extreme conditions. This work explores different thermodynamic properties, including isothermal bulk modulus, specific heat capacities (at constant pressure and volume), entropy, Debye temperature, Grüneisen parameter, thermal expansion coefficient, and thermal pressure. We chose a range of temperature and pressure at which the Debye–Grüneisen quasi-harmonic model is valid. This model extends the framework of harmonic phonons to high temperatures by considering the effects of thermal expansion against a bulk modulus.
Further information about this model can be found elsewhere [52]. Therefore, by implementing this model, phonon vibrations at high temperature/pressure conditions are considered. This provides better resolution in quantifying how thermodynamic properties are affected in extreme conditions [11,52]. The Gibbs non-equilibrium function obtained from this model is expressed in equation (2) as follows:
where E(V) is internal energy per unit cell, P(V) is constant hydrostatic pressure, T is temperature, and θ(V) is Debye temperature. F vib is the vibrational Helmholtz free energy, which is related to the phonon vibrations through the Debye model and is defined in equation (3) as follows [53]:
In equation (3), k B is the Boltzmann constant, D(θ D/T) is the Debye function, and n is the number of atoms per unit cell. Debye temperature, θ D, is also an important thermophysical property that reflects the phonon contributions to the entropy and provides a reasonable estimation of the vibrational entropy [52]. This property is related to a solid’s maximum frequency of thermal vibration, in which the corresponding wavelength equals the lattice constant [54]. It is known that the Debye temperature is also a reflection of the strength of the chemical bonding and the hardness of materials. So, in some sense, the Debye temperature indicates the rigidity of the material, i.e., the higher the Debye temperature, the harder the material (stronger covalent bonds and greater hardness/rigidity) [28,31,55]. The Debye temperature (θ D) generally relates to properties like lattice vibration, specific heat, thermal conductivity, melting temperature, thermal expansion coefficient, and elastic constants. Therefore, it is an essential property for describing well-known phenomena in solid-state physics [56,57]. Increasing the number of constituent elements of the structure can also lead to an alternation in Debye temperature, which could be inferred as a variation of the material’s rigidity [58]. Debye temperature is usually obtained empirically from heat capacity measurements. However, since acoustic vibrations are the only source of vibrational excitations at low temperatures, the Debye temperature is theoretically calculated from elastic constants. This is the same as its determination from specific heat capacity measurements [30,52]. Since the velocity of the elastic waves, which propagate through the lattice, can be determined from elastic constants, the Debye temperature can also be correlated with the elastic constants [52]. For an isotropic solid, equation (4) describes the Debye temperature in terms of system volume θ D(V) as follows [59]:
where M is the atomic mass per unit cell, ħ is the plank constant, σ is Poisson’s ratio, and B s is the adiabatic bulk modulus, approximated at the equilibrium atomic volume V 0 by equation (5) as follows [11]:
Also, the value of f(σ) is calculated in terms of Poisson’s ratio as in equation (6) [11].
By deriving the non-equilibrium Gibbs function as a function of volume, the thermal EOS as a function of temperature and pressure can be obtained. We used Gibbs2 code to explore the temperature and pressure dependence of the thermodynamic properties of the target ZrC, ZrN, and ZrC(1−x)N x materials. This code uses the quasi-harmonic approximation to quantify this dependency. As a result, several thermodynamic parameters can be obtained by specifying the equilibrated volume and thermal EOS that are presented later [11].
2.2.1 Isothermal bulk modulus
Investigating the behavior of bulk modules at different temperatures and pressures provides an understanding of the thermoelastic and non-harmonic properties of a crystal [31]. Isothermal bulk modulus (B) is an intensive property, having a constant value at the normal ambient condition. It is viewed as a manifestation of the elastic response of a solid to the hydrostatic compression or, in other words, resistance to the volume change [60]. This parameter is one of the basic quantities that determines the elasticity of an isotropic material and is usually used to estimate the structural strength of the material. As an important engineering parameter, bulk modulus has intimate relation with the performance characteristics of a material, e.g., the expansibility, elasticity, Debye temperature, thermal conductivity, elastic wave velocity, and heat capacity. A higher bulk modulus in a mechanically stiffened material is related to the compression and strengthening of bonds or, lower lattice vibrations. In fact, the pressure-induced stiffening of the material is ascribed to an increased cohesive energy. Hence, bulk modulus can be tuned by applying pressure and temperature [61]. Using the proposed EOS, the behavior of the isothermal bulk modulus at different temperatures can be obtained from equation (7) as follows [37]:
2.2.2 Entropy
Atomic properties such as atomic size and mass, electron-to-atom ratio, and electronegativity are often used to justify the vibrational entropy profile since subatomic particles such as electrons also contribute to the vibrational entropy of solids. Variation of interatomic force constants due to the temperature change can explain the dependence of the vibrational entropy on the temperature (at constant volume) [31]. In the quasi-harmonic approximation, the vibrational entropy of the crystal (S) in conditions far from phase transition is calculated from equation (8) as follows [62]:
2.2.3 Grüneisen parameter
The Grüneisen parameter, denoted by γ, describes the effect of the volume change of the lattice on its vibrational properties. So, it reflects the anharmonic effects on the properties of the solid and could represent the effects of temperature/pressure change on the atomic displacements and dynamics of the lattice [63]. Any change in the lattice vibration that comes from variations in temperature or pressure (which eventually results in volume change) can be described by the Grüneisen parameter [31,54]. For most solids, the Grüneisen parameter has a value of 1.5–2.5 [30]. This parameter is generally written as a dimensionless combination of thermal expansion coefficient, isothermal bulk modulus, density, and specific heat capacity. Therefore, it provides a qualitative relationship between the thermal and mechanical properties of the element, as expressed in equation (9) [64]:
2.2.4 Volumetric thermal expansion coefficient
An increase in temperature causes the atoms to vibrate at a higher frequency (in higher energy states), giving rise to the higher contribution of vibration-related anharmonic behavior. This could also be related to the variation of interatomic distances, as energetic atoms can move further from each other. Bonding strength determines the extent to which the atoms can oscillate; the stronger the bonds, the lower the thermal expansion, and vice versa [65]. When heating a ceramic whose constituent elements have different thermal expansions, crack generation will be unavoidable [8]. This is a challenge in many technological applications, like thin film coatings and cutting tools. A mismatch of the thermal expansions generates thermal stresses at the interface of the two materials and reduces their adherence to the substrate [31]. Taking the definition of the thermal expansion coefficient (α) into consideration (equation (10)), any expansion/contraction of the lattice due to the variation of the temperature causes the frequency of the vibrations to change [30]. Although the value of thermal expansion has a directional dependency in the crystal, this parameter can also be considered to be isotropic by the assumption of isotropic structure (single crystal with cubic symmetry). Isotropic thermal expansion can be obtained from equation (10). It is also worth noting that there is a similarity between the temperature dependence of the thermal expansion coefficient and specific heat capacity in the crystals as follows [65]:
2.2.5 Heat capacity
Heat capacity plays a vital role in linking thermodynamics with lattice dynamics. The knowledge of the material’s specific heat capacity provides an insight into its vibrational properties [31]. Heat capacity correlates with the amount of thermal energy added to the material and the resulting temperature change. Increasing the heat capacity upon rising the temperature implies phonon thermal softening in the structure [66]. Einstein’s formula [67], derived from the vibration of the molecular bonds, provides a reasonable estimation of the heat capacity. According to this relation, heat capacity reaches zero as T approaches 0 K, and at sufficiently low temperatures, it is proportional to T 3 [68]. At high temperatures, heat capacity reaches a saturation plateau, the Dulong–Petit limit, which is caused by the energy saturation of oscillating bonds [8,69]. In addition, the heat capacity of materials decreases non-linearly with pressure at different temperatures, meaning that the vibration frequency of the particles in materials is influenced by temperature and pressure. However, heat capacity varies weakly with pressure for higher temperatures, e.g., close to the Debye temperature [30]. Equations (11) and (12) describe the heat capacity at constant pressure (C p) and the heat capacity at constant volume (C v) as a function of temperature and show the interconnection of these parameters as follows [69]:
2.2.6 Thermal pressure
Thermal pressure is a physical quantity important for investigating a material’s thermoelastic properties at high temperatures [70]. Thermal pressure represents the expansion induced by the atomic vibrations and collective excitations of phonon vibrational modes in the crystal. The volumetric expansion of solids due to the temperature increase directly depends on that material’s thermal pressure [71]. Having thermal pressure, an isothermal EOS can be converted to a temperature-dependent EOS. Thus, a more accurate EOS could be expected from a well-described thermal pressure as a function of temperature [72]. Thermal pressure is obtained from lattice dynamics and thermodynamics through a relationship that correlates thermal pressure, in an isobar condition, to the isothermal bulk modulus and thermal expansion [73]. From another point of view, thermal pressure is a measure of the pressure change resulting from temperature change at a constant volume [74]. Equation (13) provides thermal pressure derived by quasi-harmonic approximation as follows [75]:
3 Results
3.1 Structural and mechanical properties
Structural parameters of a crystalline material, such as its bulk modulus, its first pressure derivative, and equilibrium lattice constant, can be obtained via monitoring the system’s total energy in a range of system volumes around the ground state. We calculated the total energy of ZrC(1−x)N x structures with the x being 0, 0.25, 0.5, 0.75, 1. Then, the third-order Birch–Murnaghan EOS was fitted to the energy-volume data points. Table 1 compares the values of the properties obtained from our calculations with the experimental data and those reported in the literature. The values of the ground state energy (E 0) and corresponding volume (V 0), lattice constant (a), and the first pressure derivative of bulk modulus (B′) are listed in Table 1. As seen, the lattice constants of the ternary ceramics differ slightly in three directions, indicating a slight deviation of these compounds from the symmetric cubic structure. Figure 1 presents a variation of the lattice constant for different nitrogen contents in the structure. As seen, a good agreement with ref. [76] has been acquired, but both the simulated results are ∼0.03 off with respect to the experimental value. It should be noted that the experimental values are determined by the neutron scattering measurements. The elastic constants determined by this method have about (10–15%) uncertainty [77], which could be the reason for the discrepancy between the simulated and experimental results. Increasing the N content leads to a decrease in lattice constant and, subsequently, the volume of the structure; this trend is also shown as a function of temperature and pressure in Figure 2.
Volume (V 0), energy (E 0), and lattice constant of the ground state structure (a), the first pressure derivative of the bulk modulus (B) calculated from the Birch–Murnaghan equation and the experimental and other calculated theoretical values
| V 0 (Å3) | B′ | E 0 (Ry) | a (Å) | b (Å) | c (Å) | |
|---|---|---|---|---|---|---|
| ZrC | 104.8909 | 4.1162 | −29,099 | 4.71605 | ||
| Expt.a | 4.68 | |||||
| Others (GGA)b | 4.724 | |||||
| Others (LDA)c | 4.69 | |||||
| ZrC 0.75 N 0.25 | 102.9033 | 4.1828 | −29,132 | 4.68599 | 4.68599 | 4.68626 |
| Othersb | 4.699 | |||||
| ZrC 0.50 N 0.50 | 101.0927 | 4.0841 | −29,166 | 4.65856 | 4.65854 | 4.65820 |
| Othersb | 4.671 | |||||
| ZrC 0.25 N 0.75 | 99.29783 | 4.3817 | −29,199 | 4.63083 | 4.63082 | 4.63048 |
| Othersb | 4.642 | |||||
| ZrN | 97.74682 | 4.4624 | −29,232 | 4.60649 | ||
| Expt.d | 4.537 | |||||
| Others (GGA)b | 4.619 | |||||
| Others (LDA)e | 4.53 |
lattice constant of ZrC(1−x)N(x) as a function of nitrogen content (x).
The volume of ZrC(1−x)N x as a function of nitrogen content (x), (a) pressures, and (b) temperatures.
Mechanical properties of materials and the stability of their crystal structures can be perceived through the derivation of elastic constants, which for the cubic structure are only C 11, C 12, and C 44 due to the structural symmetry. In Table 2, elastic constants (C ij ), elastic moduli (B, E, and G), compressibility (ρ), and ductility/brittleness criteria for ZrC and ZrN and their ternary ceramics are presented. An increase in N-content has various influences on the different elastic constants. Structural stability of the materials can be investigated by the Born mechanical stability criteria (for cubic structure: C 44 > 0, C 11 + 2 C 12 > 0, C 11 – C 12 > 0) [50], using the elastic constants. According to Table 2, all structures are mechanically stable. In the ternary compounds, increase in N-content leads to an increase in C 11, indicating the strengthening against the length change. The C 12 value of ZrC0.50N0.50 indicates its highest resistance to the deformation resulting from the axial stress in the plane (100). On the other hand, ZrC0.25N0.75 has a higher resistance to deformation due to the applied tangential shear stress than the other ceramics; this is perceived from its C 44 value. The calculated bulk modulus is 224.38 GPa, in a good agreement with the experimental value of 223 GPa, and the reported value in ref. [76], 220.6 GPa. For ZrN, the bulk modulus calculated from our simulations is 253.36 GPa, which has a fair agreement with 246.6 GPa, reported in ref. [76], and a considerable difference from the experimental value of 215 GPa. The highest Young’s and shear moduli are obtained for ZrC0.25N0.75. The values of υ = 0.26, B/G = 1.75, and P Cauchy = 0 are the boundary values of the ductility and brittleness of the material. As seen, the structures are in the ductile region, and the ternary ceramics have higher brittleness than ZrC and ZrN; this is also true for the compressibility of structures. To compare the mechanical performance of the compound ceramics (ZrC1−x N x ) with other Zr-based alloys, we bring in the elastic moduli of ZrAlC. As seen, the mechanical behavior of the ZrC1−x N x is notably superior to that of the ZrAlC, which makes it more favorable for high-pressure applications like cutting tool materials.
Elastic constants (C ij ), elastic moduli (B, E, G), compressibility (ρ), and ductility/brittleness criteria for ZrC x N(1−x)
| Elastic constants (GPa) | Elastic moduli (GPa) | ρ (1/GPa) | Ductility/brittleness criterion | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| C 11 | C 12 | C 44 | B | E | G | υ | B/G | P Cauchy (GPa) | ||
| ZrC | 457 | 106 | 149 | 224 | 386 | 159 | 0.00628 | 0.211 | 1.401 | −43 |
| Expt. | 472a | 98a | 159a | 223b | 390b | 162b | ||||
| Others (GGA) | 462d | 102d | 154d | 220c | 375c | 155c | ||||
| Others (LDA) | 499e | 93e | 170e | 228e | 435e | 185e | 0.18e | |||
| ZrC 0.75 N 0.25 | 503 | 99 | 158 | 232 | 419 | 174 | 0.00574 | 0.201 | 1.340 | −59 |
| Others | 228d | |||||||||
| ZrC 0.50 N 0.50 | 525 | 103 | 159 | 238 | 427 | 177 | 0.00565 | 0.206 | 1.371 | −56 |
| Others | 235c | 399c | ||||||||
| ZrC 0.25 N 0.75 | 575 | 98 | 167 | 246 | 462 | 193 | 0.00519 | 0.200 | 1.335 | −69 |
| Others | 241d | |||||||||
| ZrN | 561 | 112 | 132 | 253 | 406 | 163 | 0.00612 | 0.241 | 1.603 | −19 |
| Expt. | 471g | 88g | 138g | 215h | 390c | 147c | ||||
| Others(GGA) | 523h | 107h | 121g | 246f | 376c | 135c | ||||
| Others(LDA) | 548i | 91i | 117i | 243i | 434i | 142i | ||||
| ZrAlC | 125j | 222j | 92j | 0.20j | 0.74j | −24j | ||||
Table 3 lists hardness, sound velocities, minimum thermal conductivity, melting temperature, and anisotropy indices for ZrC x N(1−x ). The hardness of the material is a measure of its resistance to plastic deformation. As expected (based on the values of ductility/brittleness criteria), the ternary compounds have a higher hardness. Also, due to the higher Young’s modulus of ternary compounds, sound waves propagate faster in these compounds; this could be the reason for their higher thermal conductivity. The Debye temperature and melting point for the ZrC x N(1−x) were calculated using the Anderson method [83] and Fine model [84,85,86], respectively. Based on this method, the ZrC0.75N0.25 has the highest Debye temperature and melting point among the structures. The anisotropy of structures increases, as the N-content increases, so that the ZrN features have the highest anisotropy.
Hardness, sound velocities, minimum thermal conductivity, melting temperature, and anisotropy indices for ZrC x N(1−x)
| H v (GPa) | v t (m/s) | v l (m/s) | v m (m/s) | k min (W/m.K) | T Debye (K) | T melt (K) | A L | A U | |
|---|---|---|---|---|---|---|---|---|---|
| ZrC | 23.18 | 4,939 | 8,167 | 5,459 | 1.660 | 689.1 | 3,253 | 0.0138 | 0.0309 |
| ZrC 75 N 25 | 26.05 | 5,103 | 8,344 | 5,634 | 1.748 | 715.8 | 3,524 | 0.0324 | 0.0724 |
| ZrC 50 N 50 | 25.57 | 5,083 | 8,359 | 5,616 | 1.757 | 718.1 | 3,657 | 0.0502 | 0.1135 |
| ZrC 75 C 25 | 27.95 | 5,269 | 8,606 | 5,817 | 1.843 | 747.9 | 3,950 | 0.0678 | 0.1539 |
| ZrN | 19.70 | 4,777 | 8,186 | 5,298 | 1.723 | 685.3 | 3,869 | 0.1494 | 0.3454 |
3.2 Volume
Figure 2 presents the system’s volume as a function of pressure and temperature (ZrC represents x = 0 and ZrN refers to x = 1). At higher pressures (at constant temperature), the volume decreases at a lower rate, signaling that the bulk modulus of structures increases by applying extra pressure. Similarly, as the nitrogen content increases, the structure shows more resistance to the volume change by adding to the applied pressure and temperature. For example, the volume change in ZrC (x = 0) and ZrN (x = 1) structures at 300 K, in a pressure range of 0–15 GPa, is 31.29 and 29.29%, respectively. Also, at zero pressure, with the temperature range of 0–1,000 K, ZrC, and ZrN show a volume change of 2.35 and 2.28%, respectively. This could be related to the lower compressibility, greater bulk modulus, and volume reduction of the structure in higher nitrogen content. Although the effect of applied pressure on the volume change might sound greater than the effect of thermal energy, since the temperature change and the applied pressure are of different orders, comparing the degree that volume has been affected by variation of each of these parameters might not be reasonable.
The resistance to the volume change could be related to the bonding strength of the structure, such that the structure with shorter interatomic bonds shows higher resistance to the volume change. On the other hand, as stated earlier, the bulk modulus is a measure of the structure’s resistance to volume change. Hence, the structure with shorter bonds is expected to have a higher bulk modulus. N atoms have a smaller radius than C atoms; therefore, the Zr–N bond is stronger than the Zr–C bond. This could explain the higher bulk modulus of ZrN compared to ZrC. Applying the same analogy might help describe the behavior of the ZrC(1−x)N x compounds. By increasing x, the contribution of Zr–N bonds becomes stronger, which could explain the increasing bulk modulus with x. In Figure 3, the isothermal bulk modulus in different pressures (a) and temperatures (b) has been presented as a function of nitrogen content. In Figure 3a, the temperature has been fixed at 300 K. As seen, in all structures, the isothermal bulk modulus increases with increasing pressure. This could be ascribed to the contraction of bonds, greater repulsion of atoms, and hence stronger bonding in the crystal. At 150 GPa, the isothermal bulk modulus of ZrC and ZrN has been increased by 234.17 and 256.04%, respectively. In Figure 3b, the applied pressure equals to the ambient pressure. With increasing the temperature to 150 K, the isothermal bulk modulus slightly decreases, but a higher reduction is observed for higher temperatures. In lower temperatures, the magnitude of the phonon vibrations is not high enough to increase the length of the bonds and hence weaken their strength. However, in the temperature range of >150 K, due to the induced thermal stress, phonon vibrations become intense enough that smaller temperature increment could affect (increase) the bond length and their softening rate. Increasing the temperature up to 1,000 K results in a 9.69 and 12.42% decrease in the isothermal bulk modulus of ZrC and ZrN, respectively. As seen in Figure 3, the bulk modulus increases by increasing the nitrogen content. The ZrC0.50N0.50 structure has been influenced differently than the other compounds, as it shows a lower increase in pressure (Figure 3a) and a smaller decrease in temperature (Figure 3b). The coexistence of different types of interatomic forces, resembles a mechanical stress and leads to the structural anisotropy of the system. This anisotropy has its highest value in the ZrC0.50N0.50 structure.
Isothermal bulk modulus of ZrC(1−x)N x as a function of nitrogen content (x) for (a) different pressures at 300 K and (b) different temperatures at zero pressure.
3.3 Entropy
We calculated the entropy of ZrC(1−x)N x against temperature, pressure, and nitrogen content (x), as shown in Figure 4. Each parameter impacts the entropy through its associated mechanism; temperature via affecting the interatomic force constants, pressure through volume change, and nitrogen content by the difference in atomic size and mass, electronegativity, electron-to-atom ratio, effective mass, or band filling phenomenon in Zr–C and Zr–N bonds. As seen in Figure 4a, by increasing the pressure (at 300 K), the entropy drops in all structures, which is attributed to the reduction of system volume and the enhancement of the resistance of bonds to the length change. Reduction of volume diminishes the degree of disorder, or the phonon vibrations take place with limited amplitude. In other words, increasing the pressure limits/decreasing the frequency of vibrations is equivalent to driving the system toward a higher order level. The effect of increasing temperature on entropy is shown in Figure 4b. Temperature increase results in volume expansion, bond softening, and the atoms’ excitement toward vibration with higher frequency. With the expansion of volume, the disorder’s degree also increases. Similarly, increasing the energy increases the density of interactions. Softening also leads to a higher frequency of vibrations. As mentioned earlier, increasing the nitrogen content could have a multifaceted effect on the entropy, possibly counterbalancing each other. Replacing the C atoms with N atoms increases the number of electrons in the system, which could be interpreted as higher entropy. However, ZrN bonds added to the system (as x increases) have a shorter length than the pre-existing ZrC bonds. So, phonon vibrations decline, and the entropy decreases. Two competing effects of N atoms on the entropy were introduced; however, taking Figure 4 into account, we conclude that the electronegativity and reduction of bonding length is the dominant effect, and the entropy of ZrC(1−x)N x structures decrease with x [31]. Nevertheless, there exists an exclusion at x = 0.5, becoming more noticeable at higher pressures. This compound has the lowest entropy drop among the other compounds, which has appeared as a peak in the plot. Table 4 contains the Gibbs free energy and the contribution of phonon vibrations in free Helmholtz energy of structures at the pressures of 0 and 150 GPa. The pressure-induced increase in anisotropy has affected the reducing rate of the Gibbs energy in the ZrC0.50N0.50 structure, signaling lower stability of this compound compared to the others. Moreover, the amount of reduction in vibration-induced energy due to applying a pressure of 150 GPa is the lowest for ZrC0.50N0.50. It is also worth noting that the influence of the pressure change on the entropy is significantly lower than the temperature change.
Entropy of ZrC(1−x)N x as a function of nitrogen content (x) for (a) different pressures at 300 K and (b) different temperatures at zero pressures.
Gibbs free energy values and the contribution of the vibration in G at the pressure of 150 GPa
| P (GPa) | ZrC | ZrC0.75N0.25 | ZrC0.50C0.50 | ZrC0.25N0.75 | ZrN | |
|---|---|---|---|---|---|---|
| G (kJ·mol−1) | 0 | −9,549,936 | −9,560,920 | −9,571,897 | −9,582,869 | −9,593,835 |
| 150 | −9,547,977 | −9,558,987 | −9,569,995 | −9,580,985 | −9,591,972 | |
| ΔG (kJ·mol−1) | 1959.334 | 1932.938 | 1901.551 | 1883.922 | 1863.147 | |
| F vib (kJ·mol−1) | 0 | 7.565 | 7.772 | 7.905 | 8.059 | 8.229 |
| 150 | 19.165 | 19.347 | 19.119 | 19.766 | 19.944 | |
| ΔF vib (kJ·mol−1) | 11.601 | 11.575 | 11.214 | 11.707 | 11.714 |
3.4 Grüneisen
Figure 5 presents a variation of the Grüneisen parameter by temperature, pressure, and nitrogen content (x) for ZrC(1−x)N x structures. This parameter describes the effect of volume change on the vibrations and dynamics of the crystal lattice. A change in temperature, pressure, or content of constituent atoms could cause this volume change. The dependency of Grüneisen on the volume change could be explained from two aspects. First, by the change in the compressibility of the structure, and second, by altering the phonon vibrations (i.e., entropy). As shown in Figure 5a, by increasing pressure, Grüneisen continuously decreases. Given the discussion about the variation of the volume and entropy by pressure, it could be stated that volume drop dominates the entropy drop by pressure increase. Therefore, Grüneisen drops constantly by pressure, and equation (9) confirms this trend. Moreover, as the applied pressure increases, the rate at which the Grüneisen declines also decreases, which could be related to the reduction of compressibility of the crystal by pressure. In Figure 3b, in which the variation of the Grüneisen parameter with temperature has been presented, the opposite profile is observed; Grüneisen increases with temperature. Temperature increase leads to volume expansion, bond softening, and an increase in entropy. The rate of the changes also grows with the temperature, which is due to the lower resistance of the lattice to the volume change and the higher extent of disorder in the system. Nitrogen content affects both the volume change and the entropy. Still, the behavior of the ZrC0.50N0.50 structure differs from the others as it has the lowest Grüneisen parameter in Figure 3. The nitrogen content (x) is directly related to the Grüneisen; however, this observation is not justified with the obtained results.
Grüneisen of ZrC(1−x)N x as a function of nitrogen content (x) for (a) different pressures at 300 K and (b) different temperatures at zero pressures.
3.5 Thermal expansion
Any change in the lattice condition, e.g., applying pressure, changing temperature, or content of the elements, could affect bond characterizations, such as length, strength, and the vibrational response of the atoms in the bonds. Figure 6 presents the effect of temperature, pressure, and nitrogen content on thermal expansion. As seen in Figure 6a, pressure increase has a similar, reducing effect on the thermal expansion. The temperature increase, however, in all structures results in an enhancement of the thermal expansion coefficient. The effect of pressure comes from the reduction of compressibility and the entropy of the structure. Conversely, the effect of temperature originates from the structure’s softening and increasing entropy. In lower pressures and temperatures, thermal expansion has been affected considerably higher. Literature has reported a T 3 correlation in the intermediate temperature range and a linear relation at higher temperatures [63]. Another affecting parameter on the volumetric expansion coefficient is the nitrogen content. As seen, the nitrogen content has a negligible effect on the thermal expansion at high pressures and low temperatures.
Volumetric expansion coefficient of ZrC(1−x)N x as a function of nitrogen content (x) for (a) different pressures at 300 K and (b) different temperatures at zero pressure.
3.6 Heat capacity
Since the heat capacity mostly depends on the long-wave lattice vibrations, variation of the heat capacity due to external factors (e.g., temperature and pressure) and compositional changes could be related to the alternation in the number of excited phonon modes [31,59]. We report the analysis of C v and C p in Figures 7 and 8. The difference between the two parameters arises from the anharmonic effects of thermal expansion, which could be inferred from equations (11) and (12) [31,59]. Increasing the pressure at a temperature of 300 K causes the C v to decrease, where the reduction rate drops with pressure. A similar trend is visible for the C p at different temperatures. It should be noted that at higher temperatures, the effect of the pressure becomes weaker, which could be ascribed to the suppression of anharmonic effects on the heat capacity [63,87]; in other words, dependency on the temperature and pressure becomes weaker. The temperature has a higher impact on the heat capacity than the pressure. Increasing temperature increases heat capacity, which originates from phonon thermal softening [68]. The extent of the influence is a function of temperature, for which some descriptive models have been developed. The Debye model is one of these models that is valid in the low-temperature range. Generally, the temperature-dependent behavior of the heat capacity at low and high temperatures has quantum and classical aspects, respectively [27,88]. In the quantum approach, the low-temperature behavior of the heat capacity is affected by the density of states of the acoustic phonons at low phonon frequencies. In this temperature range, heat capacity has a T 3 correlation with the temperature due to the long-wavelength acoustic phonons [27,89]. By increasing the temperature toward the Debye temperature, the classic behavior takes over. The classic model resembles Einstein’s model in which heat capacity is exponentially related to the temperature. Finally, heat capacity reaches a constant value called the Dulong–Petit limit in all structures at temperatures far above the Debye temperature. This indicates that the supplied thermal energy has excited all phonon modes, leading to saturation of vibrational bonds [8,27,90,91]. In Figure 7b, it is shown that at 0 K, C v amounts to zero. By increasing the temperature, C v increases with a T 3 profile (the Debye model) up to 200 K. After reaching the Debye temperature, it follows an exponential profile and finally approaches the Dulong–Petit constant value. The temperature-wise behavior of C p is shown in Figure 8. In all structures, C p shows an increasing trend with the temperature (as with C v), but when a pressure increase accompanies it, the increasing rate becomes smaller. Another point to be mentioned is that the heat capacity increases by adding N atoms to the ZrC structure. Furthermore, for ZrC0.50N0.50, the reduction of heat capacity by pressure is lower.
Constant volume specific heat of ZrC(1−x)N x as a function of nitrogen content (x) for (a) different pressures at 300 K and (b) different temperatures at zero pressures.
Constant pressure heat capacity as a function of temperature and for different pressures for (a) ZrC, (b) ZrC0.75N0.25, (c) ZrC0.50N0.50, (d) ZrC0.25N0.75, and (e) ZrN.
3.7 Debye temperature
Given 0 K as the onset point with no vibrational mode, the vibration of atoms increases by temperature. Debye temperature is a temperature at which the atomic vibrational modes are maximum in magnitude [92]. Temperature, pressure, and material composition are among the parameters that affect the Debye temperature. Figure 9 presents the Debye temperature of ZrC(1−x)N x structures as a function of pressure (a) and temperature (b). As seen in Figure 9a, Debye temperature continuously increases with pressure, which is mostly because of the reduction of lattice constant. By reducing the lattice constant, the maximum frequency of thermal vibrations and corresponding equivalent temperature increases [54]. The values of T Debye for the compounds calculated via the Anderson method are also depicted in Figure 9a, agreeing with the values obtained from Debye–Grüneisen quasi-harmonic model. In Figure 9b, on the contrary, in all structures, the Debye temperature decreases with temperature. This stems from the volume expansion and increase of the interatomic distances, which weaken the bonding of atoms. As previously mentioned, since Zr–N bonds are shorter than Zr–C bonds, the volume of the structure becomes continuously smaller by adding the N atoms to the ZrC crystal. So, as seen in Figure 9a and b, the Debye temperature decreases by nitrogen content. The reduction, however, is lower in the ZrC0.50N0.50 structure, such that it has the lowest value at high pressures. The same is valid with the temperature increase, as this structure has the lowest step-wise drop and, accordingly, highest Debye temperature (compared to the other ternary compounds) in the high-temperature range. Debye temperature has a different behavior from the isothermal bulk modulus. The Debye temperature of the ZrC0.50N0.50 structure is less affected upon increasing pressure and temperature.
Debye temperature for structures of ZrC(1−x)N x as a function of nitrogen content (x) for (a) different pressures at 300 K and (b) different temperatures at zero pressure.
3.8 Thermal pressure
Figure 10 shows the variation of thermal pressure of ZrC(1−x)N x structures as a function of pressure and temperature. In Figure 10a, the temperature is fixed at 300 K, and the values in Figure 10b were obtained at zero pressure. Thermal pressure increases with temperature. Thermal pressure is a component of total pressure representing the effect of phonon vibrations caused by temperature increase. At ambient pressure conditions, this part is canceled out by the negative static pressure so that the net applied pressure becomes zero. By increasing the temperature, phonon energy is at a higher level, resulting in higher thermal pressure. Overall, the dependency of thermal pressure on temperature is stronger than the pressure. As seen in Figure 10a, it has been slightly affected by pressure increase. Taking the ZrC0.50N0.50 structure as an exclusion, nitrogen content has an increasing effect on thermal pressure in different temperatures and pressures (in the low-pressure range, the rate is lower). ZrC0.50N0.50 structure has the lowest thermal pressure among all ternary compounds in all pressures and temperatures.
Thermal pressure of ZrC(1−x)N x as a function of nitrogen content (x) for (a) different pressures at 300 K and (b) different temperatures at zero pressures.
4 Conclusion
We performed DFT calculations to investigate the thermal properties of ZrC(1−x)N x ceramics at extreme conditions, with x being 0.0, 0.25, 0.5, 0.75, 1.0. The temperature varied from 0 to 1,000 K, and the pressure range was chosen to be 0–150 GPa. In our investigation, we covered elastic constants, elastic moduli, compressibility, ductility/brittleness, hardness, sound velocities, minimum thermal conductivity, melting temperature, anisotropy indices, isothermal bulk modulus, heat capacities, entropy, Debye temperature, Grüneisen parameter, thermal expansion coefficient, and thermal pressure. We used the Debye–Grüneisen quasi-harmonic model to account for temperature and pressure effects. We consider the anisotropy and the bonding nature of the structure in analyzing the compounds’ behavior. Pressure has a direct effect on the bulk modulus and thermal pressure and inversely affects the heat capacity and thermal expansion. Predictably, temperature affects these properties oppositely. However, the extent to which each property is affected by temperature and pressure increase, is different. ZrC0.50N0.50 shows a similar qualitative but different quantitative response in all properties, such that the effect of applied conditions is different with respect to the other compounds. Considering the thermal expansion and heat capacity, we conclude that ZrC0.50N0.50 could perform better than the other compounds in high temperature and pressure conditions. This feature could make the ZrC0.50N0.50 ternary compound more favorable over frequently used ZrC and ZrN binaries in industrial coating applications. The increase of thermal expansion at elevated temperature negatively impacts the adhesion of the coating material to the target substrate. Moreover, a smaller variation of heat capacity by pressure at high temperatures is desired in confronting off-design conditions; namely the operational efficiency of the coatings of the gas turbine blades.
Acknowledgment
The support from Shahid Beheshti University is gratefully acknowledged.
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Funding information: The authors state no funding involved.
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Author contributions: Hassan Alipour performed the simulations, analyzed the results, and wrote the main manuscript text. Ali. Hamedani has a contribution in analyzing the results and writing the main manuscript text. Ghasem Alahyarizadeh supervised and conducted the whole work and the manuscript. All authors reviewed the manuscript.
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Conflict of interest: The authors state no conflict of interest.
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Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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Code availability: There was not any code data related to the research.
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Articles in the same Issue
- Research Articles
- First-principles investigation of phase stability and elastic properties of Laves phase TaCr2 by ruthenium alloying
- Improvement and prediction on high temperature melting characteristics of coal ash
- First-principles calculations to investigate the thermal response of the ZrC(1−x)Nx ceramics at extreme conditions
- Study on the cladding path during the solidification process of multi-layer cladding of large steel ingots
- Thermodynamic analysis of vanadium distribution behavior in blast furnaces and basic oxygen furnaces
- Comparison of data-driven prediction methods for comprehensive coke ratio of blast furnace
- Effect of different isothermal times on the microstructure and mechanical properties of high-strength rebar
- Analysis of the evolution law of oxide inclusions in U75V heavy rail steel during the LF–RH refining process
- Simultaneous extraction of uranium and niobium from a low-grade natural betafite ore
- Transfer and transformation mechanism of chromium in stainless steel slag in pedosphere
- Effect of tool traverse speed on joint line remnant and mechanical properties of friction stir welded 2195-T8 Al–Li alloy joints
- Technology and analysis of 08Cr9W3Co3VNbCuBN steel large diameter thick wall pipe welding process
- Influence of shielding gas on machining and wear aspects of AISI 310–AISI 2205 dissimilar stainless steel joints
- Effect of post-weld heat treatment on 6156 aluminum alloy joint formed by electron beam welding
- Ash melting behavior and mechanism of high-calcium bituminous coal in the process of blast furnace pulverized coal injection
- Effect of high temperature tempering on the phase composition and structure of steelmaking slag
- Numerical simulation of shrinkage porosity defect in billet continuous casting
- Influence of submerged entry nozzle on funnel mold surface velocity
- Effect of cold-rolling deformation and rare earth yttrium on microstructure and texture of oriented silicon steel
- Investigation of microstructure, machinability, and mechanical properties of new-generation hybrid lead-free brass alloys
- Soft sensor method of multimode BOF steelmaking endpoint carbon content and temperature based on vMF-WSAE dynamic deep learning
- Mechanical properties and nugget evolution in resistance spot welding of Zn–Al–Mg galvanized DC51D steel
- Research on the behaviour and mechanism of void welding based on multiple scales
- Preparation of CaO–SiO2–Al2O3 inorganic fibers from melting-separated red mud
- Study on diffusion kinetics of chromium and nickel electrochemical co-deposition in a NaCl–KCl–NaF–Cr2O3–NiO molten salt
- Enhancing the efficiency of polytetrafluoroethylene-modified silica hydrosols coated solar panels by using artificial neural network and response surface methodology
- High-temperature corrosion behaviours of nickel–iron-based alloys with different molybdenum and tungsten contents in a coal ash/flue gas environment
- Characteristics and purification of Himalayan salt by high temperature melting
- Temperature uniformity optimization with power-frequency coordinated variation in multi-source microwave based on sequential quadratic programming
- A novel method for CO2 injection direct smelting vanadium steel: Dephosphorization and vanadium retention
- A study of the void surface healing mechanism in 316LN steel
- Effect of chemical composition and heat treatment on intergranular corrosion and strength of AlMgSiCu alloys
- Soft sensor method for endpoint carbon content and temperature of BOF based on multi-cluster dynamic adaptive selection ensemble learning
- Evaluating thermal properties and activation energy of phthalonitrile using sulfur-containing curing agents
- Investigation of the liquidus temperature calculation method for medium manganese steel
- High-temperature corrosion model of Incoloy 800H alloy connected with Ni-201 in MgCl2–KCl heat transfer fluid
- Investigation of the microstructure and mechanical properties of Mg–Al–Zn alloy joints formed by different laser welding processes
- Effect of refining slag compositions on its melting property and desulphurization
- Effect of P and Ti on the agglomeration behavior of Al2O3 inclusions in Fe–P–Ti alloys
- Cation-doping effects on the conductivities of the mayenite Ca12Al14O33
- Modification of Al2O3 inclusions in SWRH82B steel by La/Y rare-earth element treatment
- Possibility of metallic cobalt formation in the oxide scale during high-temperature oxidation of Co-27Cr-6Mo alloy in air
- Multi-source microwave heating temperature uniformity study based on adaptive dynamic programming
- Round-robin measurement of surface tension of high-temperature liquid platinum free of oxygen adsorption by oscillating droplet method using levitation techniques
- High-temperature production of AlN in Mg alloys with ammonia gas
- Review Article
- Advances in ultrasonic welding of lightweight alloys: A review
- Topical Issue on High-temperature Phase Change Materials for Energy Storage
- Compositional and thermophysical study of Al–Si- and Zn–Al–Mg-based eutectic alloys for latent heat storage
- Corrosion behavior of a Co−Cr−Mo−Si alloy in pure Al and Al−Si melt
- Al–Si–Fe alloy-based phase change material for high-temperature thermal energy storage
- Density and surface tension measurements of molten Al–Si based alloys
- Graphite crucible interaction with Fe–Si–B phase change material in pilot-scale experiments
- Topical Issue on Nuclear Energy Application Materials
- Dry synthesis of brannerite (UTi2O6) by mechanochemical treatment
- Special Issue on Polymer and Composite Materials (PCM) and Graphene and Novel Nanomaterials - Part I
- Heat management of LED-based Cu2O deposits on the optimal structure of heat sink
- Special Issue on Recent Developments in 3D Printed Carbon Materials - Part I
- Porous metal foam flow field and heat evaluation in PEMFC: A review
- Special Issue on Advancements in Solar Energy Technologies and Systems
- Research on electric energy measurement system based on intelligent sensor data in artificial intelligence environment
- Study of photovoltaic integrated prefabricated components for assembled buildings based on sensing technology supported by solar energy
- Topical Issue on Focus of Hot Deformation of Metaland High Entropy Alloys - Part I
- Performance optimization and investigation of metal-cored filler wires for high-strength steel during gas metal arc welding
- Three-dimensional transient heat transfer analysis of micro-plasma arc welding process using volumetric heat source models
Articles in the same Issue
- Research Articles
- First-principles investigation of phase stability and elastic properties of Laves phase TaCr2 by ruthenium alloying
- Improvement and prediction on high temperature melting characteristics of coal ash
- First-principles calculations to investigate the thermal response of the ZrC(1−x)Nx ceramics at extreme conditions
- Study on the cladding path during the solidification process of multi-layer cladding of large steel ingots
- Thermodynamic analysis of vanadium distribution behavior in blast furnaces and basic oxygen furnaces
- Comparison of data-driven prediction methods for comprehensive coke ratio of blast furnace
- Effect of different isothermal times on the microstructure and mechanical properties of high-strength rebar
- Analysis of the evolution law of oxide inclusions in U75V heavy rail steel during the LF–RH refining process
- Simultaneous extraction of uranium and niobium from a low-grade natural betafite ore
- Transfer and transformation mechanism of chromium in stainless steel slag in pedosphere
- Effect of tool traverse speed on joint line remnant and mechanical properties of friction stir welded 2195-T8 Al–Li alloy joints
- Technology and analysis of 08Cr9W3Co3VNbCuBN steel large diameter thick wall pipe welding process
- Influence of shielding gas on machining and wear aspects of AISI 310–AISI 2205 dissimilar stainless steel joints
- Effect of post-weld heat treatment on 6156 aluminum alloy joint formed by electron beam welding
- Ash melting behavior and mechanism of high-calcium bituminous coal in the process of blast furnace pulverized coal injection
- Effect of high temperature tempering on the phase composition and structure of steelmaking slag
- Numerical simulation of shrinkage porosity defect in billet continuous casting
- Influence of submerged entry nozzle on funnel mold surface velocity
- Effect of cold-rolling deformation and rare earth yttrium on microstructure and texture of oriented silicon steel
- Investigation of microstructure, machinability, and mechanical properties of new-generation hybrid lead-free brass alloys
- Soft sensor method of multimode BOF steelmaking endpoint carbon content and temperature based on vMF-WSAE dynamic deep learning
- Mechanical properties and nugget evolution in resistance spot welding of Zn–Al–Mg galvanized DC51D steel
- Research on the behaviour and mechanism of void welding based on multiple scales
- Preparation of CaO–SiO2–Al2O3 inorganic fibers from melting-separated red mud
- Study on diffusion kinetics of chromium and nickel electrochemical co-deposition in a NaCl–KCl–NaF–Cr2O3–NiO molten salt
- Enhancing the efficiency of polytetrafluoroethylene-modified silica hydrosols coated solar panels by using artificial neural network and response surface methodology
- High-temperature corrosion behaviours of nickel–iron-based alloys with different molybdenum and tungsten contents in a coal ash/flue gas environment
- Characteristics and purification of Himalayan salt by high temperature melting
- Temperature uniformity optimization with power-frequency coordinated variation in multi-source microwave based on sequential quadratic programming
- A novel method for CO2 injection direct smelting vanadium steel: Dephosphorization and vanadium retention
- A study of the void surface healing mechanism in 316LN steel
- Effect of chemical composition and heat treatment on intergranular corrosion and strength of AlMgSiCu alloys
- Soft sensor method for endpoint carbon content and temperature of BOF based on multi-cluster dynamic adaptive selection ensemble learning
- Evaluating thermal properties and activation energy of phthalonitrile using sulfur-containing curing agents
- Investigation of the liquidus temperature calculation method for medium manganese steel
- High-temperature corrosion model of Incoloy 800H alloy connected with Ni-201 in MgCl2–KCl heat transfer fluid
- Investigation of the microstructure and mechanical properties of Mg–Al–Zn alloy joints formed by different laser welding processes
- Effect of refining slag compositions on its melting property and desulphurization
- Effect of P and Ti on the agglomeration behavior of Al2O3 inclusions in Fe–P–Ti alloys
- Cation-doping effects on the conductivities of the mayenite Ca12Al14O33
- Modification of Al2O3 inclusions in SWRH82B steel by La/Y rare-earth element treatment
- Possibility of metallic cobalt formation in the oxide scale during high-temperature oxidation of Co-27Cr-6Mo alloy in air
- Multi-source microwave heating temperature uniformity study based on adaptive dynamic programming
- Round-robin measurement of surface tension of high-temperature liquid platinum free of oxygen adsorption by oscillating droplet method using levitation techniques
- High-temperature production of AlN in Mg alloys with ammonia gas
- Review Article
- Advances in ultrasonic welding of lightweight alloys: A review
- Topical Issue on High-temperature Phase Change Materials for Energy Storage
- Compositional and thermophysical study of Al–Si- and Zn–Al–Mg-based eutectic alloys for latent heat storage
- Corrosion behavior of a Co−Cr−Mo−Si alloy in pure Al and Al−Si melt
- Al–Si–Fe alloy-based phase change material for high-temperature thermal energy storage
- Density and surface tension measurements of molten Al–Si based alloys
- Graphite crucible interaction with Fe–Si–B phase change material in pilot-scale experiments
- Topical Issue on Nuclear Energy Application Materials
- Dry synthesis of brannerite (UTi2O6) by mechanochemical treatment
- Special Issue on Polymer and Composite Materials (PCM) and Graphene and Novel Nanomaterials - Part I
- Heat management of LED-based Cu2O deposits on the optimal structure of heat sink
- Special Issue on Recent Developments in 3D Printed Carbon Materials - Part I
- Porous metal foam flow field and heat evaluation in PEMFC: A review
- Special Issue on Advancements in Solar Energy Technologies and Systems
- Research on electric energy measurement system based on intelligent sensor data in artificial intelligence environment
- Study of photovoltaic integrated prefabricated components for assembled buildings based on sensing technology supported by solar energy
- Topical Issue on Focus of Hot Deformation of Metaland High Entropy Alloys - Part I
- Performance optimization and investigation of metal-cored filler wires for high-strength steel during gas metal arc welding
- Three-dimensional transient heat transfer analysis of micro-plasma arc welding process using volumetric heat source models
