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Experiments and calculations on refining mechanism of NbC on primary M7C3 carbide in hypereutectic Fe-Cr-C alloy
Journal of Alloys and Compounds 713 (2017) 108e118
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Experiments and calculations on refining mechanism of NbC on primary M7C3 carbide in hypereutectic Fe-Cr-C alloy Sha Liu a, Zhijie Wang a, Zhijun Shi a, Yefei Zhou a, b, **, Qingxiang Yang a, * a b
State Key Laboratory of Metastable Materials Science & Technology, Yanshan University, Qinhuangdao 066004, PR China College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 March 2017 Accepted 14 April 2017 Available online 15 April 2017
Two hypereutectic Fe-Cr-C hardfacing alloys were manufactured, whose Nb contents are 0 and 1.2 wt% respectively. Their phase precipitation curves were calculated by Thermal-calc software and their phase structures were detected by X-ray diffractometer, which indicate that NbC precipitates previously to the nucleation of primary M7C3 carbide. The alloys were observed by metallographic microscope, which finds that the primary M7C3 carbides in the alloy with 1.2 wt% Nb are obviously refined. The chemical compositions of carbides were further measured by energy diffraction spectrum, which finds that NbC particles are distributed in the primary M7C3 carbides. The melting process of M7C3 carbide was simulated by molecular dynamics method and the melting point of M7C3 carbide is identified to be 1625±5 K. The melting behaviors of M7C3 embryos with different radii were simulated, and the solid-liquid interfacial energy of M7C3 carbide was calculated to be 3.312 J/m2 based on the classical nucleation theory. Two kinds of M7C3(0001)/NbC(111) interface models with different atomic stacking modes were built. By means of first-principles calculations, M7C3(0001)/NbC(111) interface I is combined by polar covalent/ionic bonds and metallic bonds, while M7C3(0001)/NbC(111) interface II is combined by ionic bonds and metallic bonds. The works of adhesion are 1.23 J/m2 and 2.24 J/m2 respectively. There exists a region where the interfacial energy of interface II is lower than the solid-liquid interfacial energy of M7C3 carbide. The above experimental and calculation results demonstrate that NbC particle is the heterogeneous nucleus of primary M7C3 carbide and thereby refines it. © 2017 Elsevier B.V. All rights reserved.
Keywords: M7C3 carbide Grain refinement Molecular dynamics First-principles Heterogeneous nucleus
1. Introduction With the excellent wear-resistance, hypereutectic Fe-Cr-C alloys have been extensively applied in the wear-resistance improvement of work-pieces by arc-welding on their surfaces [1]. Currently, additive manufacturing industry aims hypereutectic Fe-Cr-C alloys at satisfying the needs of variously shaped work-pieces with wearable surfaces by means of 3D printing [2e5]. There exist a large amount of primary M7C3 carbides in the hypereutectic Fe-Cr-C alloys [6], which serve as the hardening constituent [7]. The abrasive resistance of primary M7C3 carbides determines the performance of the hypereutectic Fe-Cr-C alloys. However, the primary M7C3 carbides which precipitate directly from the Fe-Cr-C melt are coarse in size. As a result, the large * Corresponding author. State Key Laboratory of Metastable Materials Science & Technology, Yanshan University, Qinhuangdao 066004, PR China. ** Corresponding author. College of Mechanical Engineering, State Key Laboratory of Metastable Materials Science & Technology, Yanshan University, Qinhuangdao 066004, PR China. E-mail addresses: [email protected] (Y. Zhou), [email protected] (Q. Yang). http://dx.doi.org/10.1016/j.jallcom.2017.04.167 0925-8388/© 2017 Elsevier B.V. All rights reserved.
primary M7C3 carbides will wreck the continuity of the base and accelerate the forming and expanding of the crack, so that the service life of the work-pieces will be reduced [8,9]. With the rapid development of additive manufacturing industry, the demands for long-lived hypereutectic Fe-Cr-C alloys are higher and higher. Therefore, the key issue of the widespread of hypereutectic Fe-Cr-C alloys is the improvement of their antistrip performance as well as wear resistance [10]. Refined microstructure even ultra-refined microstructure can be achieved by microalloying treatment (mainly Nb, Ti, V elements and so on) [1,8,11e13], which draws a great deal of attention to the study on primary M7C3 carbides [14,15]. X.J. Wu et al. [1] researched on the effect of Ti additive on the microstructure of hypereutectic Fe-Cr-C alloy. The results showed that TiC precipitates firstly and the primary M7C3 carbide are refined. X.W. Qi et al. [8] investigated the effect of V additive on the microstructure and mechanical property of hypereutectic Fe-Cr-C alloy, which found that V additive can refine the primary M7C3 carbide and the wear-resistance of the alloy are effectively improved. However, to the authors' knowledge, the reports on the effect of Nb additive on the microstructural
S. Liu et al. / Journal of Alloys and Compounds 713 (2017) 108e118
refinement of the primary M7C3 carbide in hypereutectic Fe-Cr-C alloys are quite few. In the above literature, the refinement of primary M7C3 carbide is ascribed to the MC carbide (namely TiC, VC and so on), which precipitates previously to primary M7C3 carbide. The MC carbide plays a key role as the heterogeneous nucleus of primary M7C3 carbide. However based on the classical nucleation theory, in the solidification process of the hypereutectic Fe-Cr-C melts, the MC carbide can act as heterogeneous nucleus only if the interfacial energy between MC carbide and primary M7C3 carbide is smaller than the solid-liquid interfacial energy of primary M7C3 carbide [16]. While, the solid-liquid interfacial energy of primary M7C3 carbide is difficult to measure by means of experiments for now. So the theoretical mechanism of the refining effect on primary M7C3 carbide is still unaccountable. In the recent years, molecular dynamics simulation and firstprinciples calculation, which are of high accuracy, appeal much attention to the study on solidification process and interfacial behaviors [17e19]. Y. Shibuta et al. [20] investigated the nucleation and solidification of the undercooled iron melt from the atomistic point of view by molecular dynamics simulation, which successfully linked the empirical interpretation in metallurgy with the atomistic behavior of nucleation and solidification. A.A. Potter et al. [21] computed the crystal-melt interfacial free energy of the ternary Cu-Ag-Au alloy system by molecular dynamics simulation, and the results suggested that solute species exhibiting a large enthalpy of mixing will tend to promote a transition of the dendrite crystallographic growth directions from <100> to <110>. J. Yang et al. [22] revealed the electronic property and bonding configuration of NbC(111)/NbN(111) interface by first-principles calculation. The first-principles calculation has also been applied in investigating the effects of solute atom segregation on grain boundary strengths [23]. While, combined molecular dynamics simulation and first-principles calculation analysis of the heterogeneous nucleus of primary M7C3 carbide has never been reported. In this paper, the study object is the hypereutectic Fe-Cr-C alloys with Nb additive, which is manufactured by hardfacing. The phase precipitation process and phase component are calculated and detected, and the refining phenomenon of primary M7C3 carbide is observed. On this basis, the crystal model of M7C3 carbide is built and the solid-liquid interfacial energy of M7C3 carbide is acquired by molecular dynamics simulation. Besides, the interfacial properties between M7C3 and NbC are analyzed by first-principles calculation. Thereby, the theoretical mechanism of the refining effect of NbC on primary M7C3 carbide is revealed, which provides theoretical basis for producing hypereutectic Fe-Cr-C alloys with better antistrip performance and higher wear resistance. 2. Experimental and calculation methods 2.1. Experimental methods In order to verify the effects of NbC on primary M7C3 carbides, two hypereutectic Fe-Cr-C alloys were manufactured by hardfacing. The chemical compositions are listed in Table 1, which shows that except for the Nb contents (0 and 1.20 wt% respectively), other element contents of them are almost the same. On the basis of the chemical compositions, the phase precipitation curves were Table 1 Chemical composition of Fe-Cr-C alloys (wt.%).
Without Nb additive With Nb additive
C
Cr
Si
Mn
V
Nb
Mo
Fe
3.69 3.71
26.30 26.31
0.93 0.96
1.42 1.39
0.11 0.10
0 1.20
0.16 0.15
Bal. Bal.
109
calculated by Thermal-calc 4.0 software. The phase components were detected by D/max-2500/PC X-ray diffractometer (XRD) with Cu Ka radiation (20e120 , 1 /s). The alloys were etched by 35% FeCl3þ6%HNO3þ2%HClþ57%C2H5OH solution and then observed by Axiovert 200 MAT metallographic microscope. And the chemical compositions of carbides were measured by EMAX energy diffraction spectrum (EDS). 2.2. Calculation methods The molecular dynamics simulations of M7C3 carbide were fulfilled by using the Large-scale Atomic-molecular Massively Parallel Simulator (LAMMPS) with timestep of 0.1fs in an NPT ensemble. The Berendsen method [24] was used to control the temperature and the pressure. In molecular dynamics simulations, the trajectories of atoms and molecules (namely the velocity and position of each atom/molecule at each time) are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are calculated by using interatomic potentials or molecular mechanics force fields. In this paper, the interactions between Fe, Cr, C atoms were described by the EAM (embedded atom method) potential function [25], in which every atom is embedded in the matrix of other atoms. The total energy is described as the sum of embedding energy and pair potential, and given in the following form [26]:
Etot ¼
X
Fi ðri Þ þ
i
XX i
4ij rij
(1)
j
where Etot is the total energy for the system; ri is the background electron density for atom i (namely the electron density of the matrix assuming atom i is out of existence); Fi ðri Þ is the embedding energy for atom i embedded in a background electron density; rij is the distance between atoms i and j; 4ij ðrij Þ is the pair potential between atoms i and j. It also has the following relationship:
ri ¼
X fj rij
(2)
jsi
where fj is the electron density contribution of atom j to atom i. The first-principles calculations on NbC and M7C3 bulk properties, surface properties and the interfacial properties were fulfilled by using the density functional theory (DFT) as implemented in the Cambridge Sequential Total Energy Package (CASTEP). The exchange-correlation energy is described by the generalized gradient approximation (GGA) of Perdew and Wang (PW91) [27]. In the calculations on M7C3 and NbC bulk properties, the plane-wave cutoff energies are both 350eV and the Brillouin zone sampling are performed using 4 4 6 and 6 6 6 Monkhorst-Pack respectively. In the calculations of M7C3 and NbC surface properties, the plane-wave cutoff energies are both 350eV and the Brillouin zone sampling are performed using 4 4 1 and 6 6 1 Monkhorst-Pack respectively. In the calculations of interfacial properties, the plane-wave cutoff energy is 350eV and the Brillouin zone sampling is performed using 4 4 1 Monkhorst-Pack. 3. Effect of NbC on primary M7C3 carbide in hypereutectic FeCr-C alloy 3.1. Phase structure analysis The phase precipitation curves of the hypereutectic Fe-Cr-C alloys with 0 Nb and 1.20 wt% Nb are shown in Fig. 1. It is found from
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Fig. 1a that for the hypereutectic Fe-Cr-C alloy with 0 Nb, the primary M7C3 carbide starts to precipitate at 1330 C, and the eutectic reaction (liquid/eutectic M7C3 carbide þ eutectic austenite) occurs at 1300 C. It is found from Fig. 1b that for the hypereutectic FeCr-C alloy with 1.20 wt% Nb, MC carbide precipitates at 1350 C previously and then the primary M7C3 carbide starts to precipitate at 1330 C. The eutectic reaction (liquid/eutectic M7C3 carbide þ eutectic austenite) also occurs at 1300 C. Except for the precipitation of MC, the phase transition processes and the phase structures of the two hypereutectic Fe-Cr-C alloys are almost the same. Fig. 2 is the XRD patterns of the hypereutectic Fe-Cr-C alloys. The alloy with 0 Nb mainly consists of M7C3 carbide and g-Fe, while the alloy with 1.2 wt% Nb consists of M7C3 carbide, g-Fe as well as a small amount of NbC. Therefore, it can be deduced that MC carbide precipitate at 1350 C in Fig. 1b is NbC. 3.2. Microstructural observation The metallographic images of the hypereutectic Fe-Cr-C alloys are shown in Fig. 3a and b. The white polygons are primary M7C3 carbides. It is found that the primary M7C3 carbides in the hypereutectic Fe-Cr-C hardfacing alloy with 0 Nb are about 30e50 mm in diameter, while those in the hypereutectic Fe-Cr-C hardfacing alloy with 1.2 wt% Nb are only 10e20 mm in diameter, which indicates that Nb additive can effectively refine the primary M7C3 carbide. In order to investigate the reason, the primary M7C3 carbides were observed by secondary electron diffraction, which is shown in Fig. 3c. There exist some small particles in the primary M7C3 carbides with different contrast. The EDS analysis of particle A is shown in Fig. 3d. The major elements are Nb and C, which illustrates that the small particles are NbC. 4. Calculation on solid-liquid interfacial energy of M7C3 carbide 4.1. Crystal model of M7C3 carbide The primary M7C3 carbide in Fe-Cr-C alloys has hexagonal structure, whose lattice parameters are similar with Fe7C3 [28,29]. Because the Fe/Cr ratio of primary M7C3 carbide is uncertain, it can be expressed as Fe7-xCrxC3. Elements Fe and Cr are very close in the periodic table. It has been proved in literature [28] that atomic size effects of Fe and Cr on the lattice parameters and atomic positions are rather weak for the mixed M7C3 carbide. So Fe7-xCrxC3 can be
Fig. 2. XRD patterns of Fe-Cr-C hardfacing alloys with (a) 0 Nb and (b) 1.2 wt% Nb.
considered as the acquisition of Cr atoms partially replacing the Fe positions in Fe7C3. There are three kinds of Wyckoff positions for Fe atoms in Fe7C3, which are site 2b FeI (0.3333, 0.6667, 0.818), site 6c FeII (0.4563, 0.5437, 0.318) and site 6c FeIII (0.1219, 0.8781, 0) respectively. Six kinds of mixed Fe7-xCrxC3 carbides can be obtained by replacing one or two kinds of Fe atoms with Cr atoms, which are Fe6Cr1(I)C3, Fe4Cr3(II)C3, Fe4Cr3(III)C3, Fe3Cr4(I,II)C3, Fe3Cr4(I,III)C3 and Fe1Cr6(II,III)C3 respectively. By means of first-principles calculation, the six kinds of Fe7-xCrxC3 carbides were fully structural relaxed, and their formation energies DEfor ðFe7x Crx C3 Þ were acquired by:
DEfor ðFe7x Crx C3 Þ ¼ Etot ðFe7x Crx C3 Þ ð7 xÞEtot ðFeÞ xEtot ðCrÞ 3Etot ðCÞ
(3)
where Etot ðFe7x Crx C3 Þ is the total energy of Fe7-xCrxC3; Etot ðFeÞ, Etot ðCrÞ and Etot ðCÞ are the energies of a single Fe, Cr and C atom. The values of DEfor ðFe7x Crx C3 Þ are listed in Table 2, which is in coincidence with the calculation results of P.F. Zhang et al. [30]. DEfor ðFe7x Crx C3 Þ of Fe3Cr4(I,III)C3 and Fe1Cr6(II,III)C3 are lower than that of other carbides, which illustrates they are more stable from thermodynamics. In addition, the measured atomic ratio of Fe:Cr:C in primary M7C3 carbide by EDS is about 3:4:3. So the follow-up calculations in this paper is conducted by Fe3Cr4(I,III)C3. The crystal model of Fe3Cr4(I,III)C3 is shown in Fig. 4, whose lattice parameters are a ¼ b ¼ 6.999A, c ¼ 4.268A, which is close to the measurement value in literature [31].
Fig. 1. Phase fraction vs. temperature curves of Fe-Cr-C hardfacing alloys with (a) 0 Nb and (b) 1.2 wt% Nb.
S. Liu et al. / Journal of Alloys and Compounds 713 (2017) 108e118
111
Fig. 3. Microstructures of the hypereutectic Fe-Cr-C hardfacing alloys with (a) 0 Nb and (b) 1.2 wt% Nb. (c) SEM image of Fe-Cr-C hardfacing alloy with 1.2 wt% Nb. (d) EDS analysis of particle A.
Table 2 Calculated formation energy of Fe7-xCrxC3. Carbide types
Fe6Cr1(I)C3
Fe4Cr3(II)C3
Fe4Cr3(III)C3
Fe3Cr4(I,II)C3
Fe3Cr4(I,III)C3
Fe1Cr6(II,III)C3
DEfor ðFe7x Crx C3 Þ (eV/cell)
0.02
0.64
0.37
0.72
0.89
0.90
4.2. Melting point of M7C3 carbide To get the critical nucleation undercooling degree of M7C3 embryos with different radii, the melting point of M7C3 carbide was calculated firstly. On the basis of the crystal model in Fig. 4, 320000 atoms were placed into a periodic rectangular simulation box with size 23.8399 nm 13.764 nm 9.08 nm. The system was kept at 1000 K until reached energy equilibrium. The system is still of hexagonal structure, as shown in Fig. 5a. Then the system was slowly heated to 2200 K until reached energy equilibrium. The final structure of the system is shown in Fig. 5b. It can be seen that the
atoms in the system are disordered at 2200 K, which illustrates that M7C3 carbide has been melted. In addition, it can be seen from the radial distribution functions at the lower right corners of Fig. 5a and b that the structure at 1000 K is long-range ordered, while that at 2200 K is not ordered, which also illustrates the transformation from M7C3 crystal to M7C3 melt. Fig. 6 shows the internal energy of the system versus temperature during the heating process. It can be seen that there exists a singularity between 1620 K and 1630 K, which means the system energy changes abruptly. The internal energy changes continuously before and after the abrupt change, which characterizes the M7C3 crystal and the M7C3 melt respectively. Therefore, the melting point Tm of M7C3 carbide is corresponding to the singularity, namely Tm z (1625 ± 5)K.
4.3. Melting behavior of M7C3 embryo
Fig. 4. Crystal structure of Fe3Cr4(I,III)C3.
According to the classical nucleation theory [32], an undercooling degree is necessary for the solidification of metal whose solid-liquid interface is of curvature. This phenomenon is called Gibbs-Thomson effect. The change in Gibbs free energy to form a sphere crystal embryo with radius r from the undercooled melt is expressed as:
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Fig. 6. Internal energy of M7C3 carbide versus temperature.
Fig. 5. (a) M7C3 crystal configuration in a periodic orthorhombic box and its radial distribution function; (b) M7C3 melt configuration in a periodic orthorhombic box and its radial distribution function.
4 3
DGr ¼ pr3 DGV þ 4pr2 gSL
(4)
where DGV is the Gibbs free energy difference per unit volume between the crystal and melt at the same temperature; gSL is the solid-liquid interfacial energy. DGV can be approximated as:
DGV ¼ LV
DT Tm
melt, namely the critical nucleation undercooling degree for the critical radius was reached. The melting behaviors of the crystal embryo with a radius of 3.5 nm cooled to different temperatures are displayed in Fig. 7. It can be seen that the outer shell of the embryo is melted at 1360 K, while the embryo is almost unchanged at 1345 K. Therefore, the critical nucleation temperature for the M7C3 embryo with a radius of 3.5 nm is 1345 K, namely the critical nucleation undercooling degree is 280 K. Numerous simulations of various embryo radii have been done, which resulted in the relationship of critical nucleation undercooling degrees to the reciprocal of critical radii in Fig. 8. The equation of the fitting line in Fig. 8 is as follow:
DT ¼ 70 þ 1:243 106
1 R
(7)
(5)
where LV is the melting enthalpy per unit volume at the melting point Tm . DT is the undercooling degree. The critical radius R is the value of r when Eq. (4) takes the maximum value and can be expressed as:
R¼
2gSL 2gSL Tm 1 ¼ DGV LV DT
(6)
It can be known from Eq. (6) that, for a given DT, there exists a specific R. The embryos with r > R will grow while those with r < R will melt. In other words, the values of R have a one-to-one relationship with the values of DT. On the basis of the M7C3 crystal model in Fig. 4, 320000 atoms were placed into a periodic rectangular simulation box with size 23.8399 nm 13.764 nm 9.08 nm. The system was kept at 1625 K until reached energy equilibrium. At this temperature, the system still remains hexagonal structure. A crystal embryo with critical radius R was built by fixing the coordinates of the atoms in it. Then the system was kept at 2000K so that the atoms outside the embryo could reach melt. By this means, a M7C3 embryo embedded in the Fe-Cr-C melt is formed, which is displayed in Fig. 7a. Then the system was rapidly cooled to a certain degree, and the melting behavior of the embryos was observed. If the embryo melts, it indicates that the temperature does not meet the undercooling degree requirement for the critical radius. Numerous attempts have been made to find the temperature that the embryo just did not
Fig. 7. (a) A M7C3 embryo with a radius of 3.5 nm in Fe-Cr-C melt. (b) The embryo cooled to 1360 K. (c) The embryo cooled to 1345 K.
S. Liu et al. / Journal of Alloys and Compounds 713 (2017) 108e118
113
5. Interfacial bonding and electronic structure between M7C3 and NbC 5.1. Bulk properties of M7C3 and NbC
Fig. 8. Critical nucleation undercooling degrees versus the reciprocal of critical radii.
4.4. Solid-liquid interfacial energy of M7C3 carbide The transformation formula of Eq. (6) is:
DT ¼
2G gSL Tm 2 ¼ R LV R
Fig. 9 shows the band structures as well as the density of states (DOS) of M7C3 carbide and NbC by first-principles calculation, in which the Fermi level (EF) is marked by the dotted lines. Fig. 9a is the band structure of M7C3 carbide, which shows that it is of metallic characteristics overall because its electron bands cross the EF. Fig. 9b is the DOS of M7C3 carbide, in which the bands of Fe, Cr, C atoms are overlapping. Between 7.8eV and 6eV, DOS mainly consist of Cr-d, Fe-d and C-p bands. The Cr-d bands and Fe-d bands resonate significantly with the C-p bands between 7.8eV and 4.2eV, which illustrates p-d hybrid orbitals are formed between Cr and C as well as Fe and C, namely covalent bonds are formed. However, the center-of-gravity of Cr-d bands, Fe-d bands and C-p bands are not consistent, which shows Cr-C bond and Fe-C bond are of ionic characteristics. Therefore, the bonding characteristics of M7C3 carbide can be expressed as the combination of metallic bonds, covalent bonds and ionic bonds. From the band structure of NbC in Fig. 9c, NbC also shows metallic characteristics overall. Fig. 9d is the DOS of NbC, in which the bands of Nb and C atoms are overlapping. Between 7.5eV and 7eV, DOS mainly consists of Nb-d and C-p bands, which resonate significantly. So pd hybrid orbitals are formed between Nb and C, namely Nb-C covalent bonds are formed. However, the center-of-gravity of Nb-
(8)
where G ¼ gSLLVTm is the Gibbs-Thomson coefficient.
According to Eq. (8), the theoretical value of DT is 0 as the critical radius R is infinitely great. However, by comparing Eqs. (7) and (8), it can be found that a constant term 70 is excessed in Eq. (7), which means that the calculation result in this paper is 70 K lower than the theoretical value. On account that the potential functions in molecular dynamics simulation are empirical potentials, such small deviation can be neglected. Moreover, it can be acquired from the slope k of the line in Fig. 8 that G ¼ 12 k ¼ 6:213 107 K$m. So gSL of M7C3 carbide can be obtained if the value of LV is known.
LV ¼ Tm $S
(9)
where DS is the melting entropy per unit volume. The relationship between DS and melting enthalpy DH is:
DS ¼
rS Tm
DH
(10)
where rS is the number density of M7C3 crystal. The lattice parameter of M7C3 carbide is a ¼ b ¼ 6.999A, c ¼ 4.268A, and 20 atoms are contained in a unit cell. So it is acquired that rS ¼ 1:105 1029 atom=m3 . The enthalpy of M7C3 crystal at Tm is acquired by holding the system at 1625 K, which is 1243946.8eV. The enthalpy of M7C3 melt at Tm is acquired by holding the system at 2200 K until it is melted, and then holding the system at 1625 K, which is 1087152.7eV. So the value of melting enthalpy can be derived as DH ¼ 7:840 1020 J=atom. Then put the values of rS , DH and Tm into Eq. (10), and DS ¼ 5:331 106 J=K$m3 is obtained. Finally, the solid-liquid interfacial energy of M7C3 carbide is derived as gSL ¼ 3:312J=m2 .
Fig. 9. (a) Electronic band structure (b) and density of states of M7C3 carbide. (c) Electronic band structure and (d) density of states of NbC.
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d bands and C-p bands are not consistent, which suggests the ionic characteristics of NbC. Therefore, the bonding characteristics of NbC can be expressed as the combination of metallic bonds, covalent bonds and ionic bonds too.
In order to reduce the lattice misfit, a 1 1 M7C3(0001) slab and a 2 2 NbC(111) slab are chosen to conduct the interface models. Before building the interface models, convergence tests of the slicing layers for M7C3(0001) slab and NbC(111) slab are necessary to ensure that the interiors of M7C3(0001) slab and NbC(111) slab can achieve the bulk properties. Generally, the upper and lower surfaces of the slab should be the same to eliminate the dipole effect. On account of the crystal structure complexity of M7C3 carbide, the M7C3(0001) slab is conducted by stacking unit cells. By this means, M7C3(0001) slab has three kinds of surfaces, which are Cr-terminated, Cr&Fe-terminated and C-terminated. While according to our calculation, the surface energy of the Cr-terminated M7C3(0001) slab is the smallest, which means that it is the most stable from thermodynamics. So the follow-up calculation of M7C3(0001) slab is only based on the Crterminated one. The surface energy sM7 C3 ð0001Þ of M7C3(0001) slab is expressed as:
sM7 C3 ð0001Þ
(11)
where A is the surface area; Eslab is the total energy of the fully relaxed slab; NFe , NCr , NC are the numbers of Fe, Cr and C atoms in the slab; mFe , mCr , mC are the chemical potentials of Fe, Cr and C atoms in M7C3 carbide. The total energy mbulk M7 C3 of bulk M7C3 is:
(12)
In the M7C3(0001) slab which is conducted by stacking unit cells, the relationship between NFe and NC is as follow:
NFe ¼ NC
(13)
Integrating Eqs. (11)e(13), it can be derived that:
1 N 4 Eslab Fe mbulk N þ N Cr mCr 2A 3 Fe 3 M7 C 3
(14)
To evaluate the convergence, herein mCr zmbulk Cr ¼ 2467:7eV is
is the chemical potential of Cr atom in pure taken, where mbulk Cr chromium. The values of sM7 C3 ð0001Þ are listed in Table 3, from which, sM7 C3 ð0001Þ of the M7C3(0001) slabs tend to be constant when the layer number reaches 17. It is indicated that 17 slicing layers can meet the convergence requirement for M7C3(0001) slab. Similarly, in order to ensure the upper and lower surfaces are the same, the NbC(111) slab is conducted by odd number of slicing layers. NbC(111) slab has two kinds of surfaces, which are Nbterminated and C-terminated. The surface energy sNbCð111Þ of NbC(111) slab is expressed as:
where NNb and NC are the numbers of Nb and C atoms in the slab; mNb and mC are the chemical potentials of Nb and C atoms in NbC.
mbulk NbC ¼ mNb þ mC
(16)
Integrating Eqs. (15) and (16), it can be derived that:
sNbCð111Þ ¼
i 1 h Eslab NNb mbulk þ ðNNb NC ÞmC NbC 2A
Table 3 Calculated surface energy of M7C3(0001).
sM7 C3 ð0001Þ
3.174 J/m
17-layers 2
3.031 J/m
25-layers 2
(17)
To evaluate the convergence, herein mC zmbulk ¼ 155:03eV is C
is the chemical potential of C atom in pure cartaken, where mbulk C bon. The values of sNbCð111Þ are listed in Table 4, from which, sNbCð111Þ of the NbC(111) slabs tend to be constant when the layer number reaches 9. It indicates that 9 slicing layers can meet the convergence requirement for NbC(111) slabs. 5.3. Surface energies of M7C3 and NbC slab In Section 5.2, the approximate treatments of the chemical potentials are made to judge the convergence of slicing layers. Actubulk are variables. It has been mentioned above that ally, mbulk Cr and mC 17 slicing layers can meet the convergence requirement for NbC(111) slab. Put the numbers of Fe, Cr, C atoms into Eq. (14) and the following formula can be obtained:
sM7 C3 ð0001Þ ¼
i 1 h E 4mbulk M7 C3 3mCr 2A slab
(18)
bulk bulk bulk DH0f ðM7 C3 Þ ¼ mbulk M7 C3 4mCr 3 mFe þ mC
2.997 J/m
2
2
2.958 J/m
(19)
and (12), it can be derived that:
1 4
0 mCr mbulk Cr ¼ DHf ðM7 C3 Þ
3 3 mC mbulk mFe mbulk Fe C 4 4 (20)
bulk 0, Because mCr mbulk Cr 0, mC mC following relationship is obtained:
mFe mbulk Fe 0, the
1 0 DH ðM7 C3 Þ mCr mbulk Cr 0 4 f
(21)
Integrating Eqs. (18) and (21), the relationship between
sM7 C3 ð0001Þ for the 17-layered M7C3(0001) slab and mCr mbulk is Cr displayed Fig. 10a. From Fig. 10a, sM7 C3 ð0001Þ varies in 3.03 J/ m2~3.22 J/m2 over the entire range of mCr mbulk Cr . In the same way, the numbers of Nb, C atoms in the 9-layered NbC(111) slabs are put into Eq. (17) and the following formulas can be obtained:
sNbCð111Þ ðC terminatedÞ ¼
33-layers
It is obtained that DHf0 ðM7 C3 Þ ¼ 1:32eV. Integrating Eqs. (19)
sNbCð111Þ ðNb terminatedÞ ¼
9-layers
(15)
The heat of formation DH0f ðM7 C3 Þ for bulk M7C3 carbide is:
mbulk M7 C3 ¼ 3mFe þ 4mCr þ 3mC
sM7 C3 ð0001Þ ¼
1 ½E NNb mNb NC mC 2A slab
of bulk NbC is: The total energy mbulk NbC
5.2. Convergence tests of M7C3 and NbC slicing layers
1 ½E ¼ NFe mFe NCr mCr NC mC 2A slab
sNbCð111Þ ¼
i 1 h Eslab 20mbulk þ 4mC NbC 2A
i 1 h Eslab 16mbulk 4mC NbC 2A
The heat of formation DH0f ðNbCÞ for bulk NbC is:
(22)
(23)
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115
Table 4 Calculated surface energy of NbC(111).
Nb-terminated C-terminated
3-layers
5-layers
7-layers
9-layers
11-layers
13-layers
3.044 J/m2 5.763 J/m2
2.988 J/m2 5.708 J/m2
2.786 J/m2 5.533 J/m2
2.785 J/m2 5.558 J/m2
2.761 J/m2 5.561 J/m2
2.766 J/m2 5.556 J/m2
bulk Fig. 10. (a) Surface energy of 17-layered M7C3(0001) slab as a function of mCr mbulk Cr ; (b) surface energy of 9-layered Nb-terminated NbC(111) slab as a function of mC mC .
bulk bulk DH0f ðNbCÞ ¼ mbulk NbC mNb mC
(24)
It is obtained that DH0f ðNbCÞ ¼ 1:46eV. Integrating Eqs. (24) and (16), it can be derived that:
0 bulk mNb mbulk Nb ¼ DHf ðNbCÞ mC mC
(25)
Because mNb mbulk 0, mC mbulk 0, the following relationC Nb ship is obtained:
DHf0 ðNbCÞ mC mbulk 0 C
(26)
Integrating Eqs. (22), (23) and (26), the relationship between
sNbCð111Þ for the 9-layered NbC(111) slabs and mC mbulk are disC played Fig. 10b. From Fig. 10b, sNbCð111Þ ðNb terminatedÞ is smaller than sNbCð111Þ ðC terminatedÞ over the entire range of mC mbulk C , which means that the Nb-terminated slab is more stable from thermodynamics. Therefore, the follow-up interface models are conducted by the Nb-terminated NbC(111) slab and M7C3(0001) slab.
5.4. Properties of M7C3(0001)/NbC(111) interface 5.4.1. Interface models Based on the different atoms on the sub-surface of the M7C3(0001) slab, two kinds of M7C3(0001)/NbC(111) interface models are conducted, which are shown in Fig. 11a and c. The interfaces are marked by the dotted lines. In order to clearly display the stacking modes, Fig. 11b and d schematically show the sites of interfacial and sub-interfacial atoms from positive z-axis and the atoms are numbered separately(the equivalent atoms are numbered as the same). On account of the crystal structure complexity of M7C3 carbide, the electronic structures on ð1120Þ
and ð0110Þ planes are analyzed for comprehensive understanding, which are marked by the dotted lines in Fig. 11b and d. 5.4.2. Interfacial electronic structures Fig. 12 shows the electron densities and electron density differences on ð1120Þ plane and ð0110Þ plane for M7C3(0001)/ NbC(111) interface I, where the interface is marked by the dotted lines. From the electron density on ð1120Þ plane in Fig. 12a, the atomic spacing between Nb4 and C5 atoms is small, and strong bond between them is formed. It can be found that Nb1 and Cr1 atoms also form bond, while it is weaker than the Nb4-C5 bond. Besides, Nb1 atom and Fe atom on the Fe3Cr4C3 side also form bond. The electron density difference on ð1120Þ plane in Fig. 12b illustrates the localized interfacial charge redistribution for M7C3(0001)/ NbC(111) interface I. The population of Nb4-C5 bond is 0.31, which suggests the covalent characteristic of Nb4-C5 bond. While, the obvious charge depletion of Nb4 atom and charge accumulation of C5 atom further suggest the ionic characteristic of Nb4-C5 bond, so it is polar covalent/ionic bond. The charge of Nb1 atom and Cr1 atom transfers medially, suggesting the metallic characteristic of Nb1-Cr1 bond. Similarly, the Nb1-Fe bond is also metallic bond. From the electron density on ð0110Þ plane in Fig. 12c, the atomic spacing between Nb3 and C6 atoms, Nb3 and Cr3 atoms, Nb4 and C5 atoms as well as Nb4 and Cr3 atoms are small, and bonds between them are formed. The electron density difference on ð0110Þ plane in Fig. 12d displays obvious charge depletion of Nb3 atom and charge accumulation of C6 atom. Besides, the population of Nb3-C6 bond is 0.31, which illustrates its polar covalent/ionic characteristic. The charge of Nb3 atom and Cr3 atom transfers medially, suggesting the metallic characteristic of Nb3-Cr3 bond. Similarly, it also can be known that the Nb4-Cr3 bond is metallic bond. Fig. 13 shows the electron densities and electron density differences on ð1120Þ plane and ð0110Þ plane for M7C3(0001)/ NbC(111) interface II. From the electron density on ð1120Þ plane in
116
S. Liu et al. / Journal of Alloys and Compounds 713 (2017) 108e118
Fig. 11. (a) M7C3(0001)/NbC(111) interface I and (b) schematic illustrations of the site of interfacial atoms before relaxation. (c) Fe3Cr4C3(0001)/NbC(111) interface II and (d) schematic illustrations of the site of interfacial atoms before relaxation.
Fig. 12. Electron density and electron density difference for M7C3(0001)/NbC(111) interface I. (a) Electron density on ð1120Þ plane; (b) electron density difference on ð1120Þ plane; (c) electron density on ð0110Þ plane; (d) electron density difference on ð0110Þ plane.
Fig. 13a, the atomic spacing between Nb1 and Cr1 atoms, Nb1 and Cr4 atoms as well as Nb1 and Fe1 atoms are small, and strong bonds between them are formed. C1 atom on NbC side and Cr4 on M7C3 side also form bond. The electron density difference on ð1120Þ plane in Fig. 13b also illustrates the localized interfacial charge redistribution for M7C3(0001)/NbC(111) interface II. The medially transferred charge illustrates the metallic characteristic of Nb1-Cr1 bond, Nb1-Cr4 bond and Nb1-Fe1 bond. The obvious charge depletion of Cr4 atom and charge accumulation of C1 atom suggests the ionic characteristic of C1-Cr4 bond. Together with the fact that the population of C1-Cr4 bond is 0, the C1-Cr4 bond is pure ionic bond.
From the electron density on ð0110Þ plane in Fig. 13c, bonds between Nb3 and Fe2 atoms, Nb3 and Cr3 atoms, Nb4 and Fe2 atoms as well as Nb4 and Cr3 atoms are formed. It can be known from the medially transferred charge in Fig. 13d that the Nb3-Fe2 bond, Nb3Cr3 bond, Nb4-Fe2 bond and Nb4-Cr2 bond are all metallic bonds. 5.4.3. Work of adhesions and interfacial energies The work of adhesion Wad is defined as the reversible work of separating an interface into two free surfaces. A large value of Wad implies that the binding strength of the interface is strong. The Wad of M7C3(0001)/NbC(111) interface is expressed as:
S. Liu et al. / Journal of Alloys and Compounds 713 (2017) 108e118
117
Fig. 13. Electron density and electron density difference for M7C3(0001)/NbC(111) interface II. (a) Electron density on ð1120Þ plane; (b) electron density difference on ð1120Þ plane; (c) electron density on ð0110Þ plane; (d) electron density difference on ð0110Þ plane.
bulk Fig. 14. Interface energy of M7C3(0001)/NbC(111) interface I (a) and interface II (b) versus mCr mbulk Cr and mC mC .
Wad
1 ENbC þ EM7 C3 EM7 C3 =NbC ¼ A
(27)
where EM7 C3 and ENbC are the total energies for M7C3(0001) and NbC(111) slab after fully relaxation, namely the Eslab in Eqs. (11) and (15); EM7 C3 =NbC is the total energy for M7C3(0001)/NbC(111) interface model; A is the interfacial area. By putting the corresponding values into Eq. (27), Wad for the M7C3(0001)/NbC(111) interface model I and interface model II are 1.23 J/m2 and 2.24 J/m2, which means that bonding of interface II is stronger than that of interface I. The interfacial energy gM7 C3 =NbC of M7C3(0001)/NbC(111) interface is defined as the additional internal energy of increasing unit interfacial area, which is expressed as follow:
gM7 C3 =NbC ¼ sM7 C3 ð0001Þ þ sNbCð111Þ Wad
(28)
Integrating Eqs. (18), (22), (28) and Wad , gM7 C3 =NbC of the two
interface models as a function of mCr mbulk and mC mbulk can be Cr C drawn in Fig. 14, in which gSL of M7C3 carbide is marked by the yellow planes. From Fig. 14b, it can be seen that there exists a region where gM7 C3 =NbC of interface II is lower than gSL of M7C3 carbide. Therefore, there exists the possibility that NbC can act as the heterogeneous nucleus of M7C3 carbide. Cooperating with the experimental results that NbC precipitates previously to primary M7C3 carbide and exists in the primary M7C3 carbide, the refinement mechanism of NbC on primary M7C3 carbides is approved. 6. Conclusion (1) In order to verify the effects of NbC on primary M7C3 carbides, two hypereutectic Fe-Cr-C alloys with 0 Nb and 1.20 wt % Nb are manufactured. By experimental methods, the primary M7C3 carbides in the hypereutectic Fe-Cr-C alloy with 1.20 wt% Nb are obviously smaller than those in the hypereutectic Fe-Cr-C alloy with 0 Nb. Besides, NbC is found to
118
S. Liu et al. / Journal of Alloys and Compounds 713 (2017) 108e118
precipitate previously to primary M7C3 carbide and exist in the primary M7C3 carbide. (2) By molecular dynamics simulations, it is acquired that the melting point of M7C3 carbide is Tmz(1625 ± 5)K. According to the classical nucleation theory, the solid-liquid interfacial energy of M7C3 carbide is calculated as gSL ¼ 3:312J=m2 . (3) By first-principles calculations, the interfacial electronic structures of the two kinds of M7C3(0001)/NbC(111) interface models with different atomic stacking modes are analyzed. M7C3(0001)/NbC(111) interface I is combined by polar covalent/ionic bonds and metallic bonds, while M7C3(0001)/ NbC(111) interface II is combined by ionic bonds and metallic bonds. (4) The works of adhesion for M7C3(0001)/NbC(111) interface I and II are 1.23 J/m2 and 2.24 J/m2 respectively, which means that the bonding of interface II is stronger than that of interface I. There exists a region where interfacial energy of interface II is lower than the solid-liquid interfacial energy of M7C3 carbide. Therefore, the refining mechanism of NbC on primary M7C3 carbide is approved theoretically, which is the heterogeneous nucleus theory.
[11]
[12]
[13]
[14] [15]
[16]
[17]
[18] [19]
[20]
Acknowledgements The authors would like to express their gratitude for projects supported by the National Natural Science Foundation of China (No. 51471148), the Hebei province Basic Research Foundation of China (No. 16961008D) and the Innovation Fund for Graduate Students of Hebei Province (No. CXZZBS2017046).
[21]
[22]
[23]
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Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
Experiments and calculations on refining mechanism of NbC on primary M7C3 carbide in hypereutectic Fe-Cr-C alloy Sha Liu a, Zhijie Wang a, Zhijun Shi a, Yefei Zhou a, b, **, Qingxiang Yang a, * a b
State Key Laboratory of Metastable Materials Science & Technology, Yanshan University, Qinhuangdao 066004, PR China College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 March 2017 Accepted 14 April 2017 Available online 15 April 2017
Two hypereutectic Fe-Cr-C hardfacing alloys were manufactured, whose Nb contents are 0 and 1.2 wt% respectively. Their phase precipitation curves were calculated by Thermal-calc software and their phase structures were detected by X-ray diffractometer, which indicate that NbC precipitates previously to the nucleation of primary M7C3 carbide. The alloys were observed by metallographic microscope, which finds that the primary M7C3 carbides in the alloy with 1.2 wt% Nb are obviously refined. The chemical compositions of carbides were further measured by energy diffraction spectrum, which finds that NbC particles are distributed in the primary M7C3 carbides. The melting process of M7C3 carbide was simulated by molecular dynamics method and the melting point of M7C3 carbide is identified to be 1625±5 K. The melting behaviors of M7C3 embryos with different radii were simulated, and the solid-liquid interfacial energy of M7C3 carbide was calculated to be 3.312 J/m2 based on the classical nucleation theory. Two kinds of M7C3(0001)/NbC(111) interface models with different atomic stacking modes were built. By means of first-principles calculations, M7C3(0001)/NbC(111) interface I is combined by polar covalent/ionic bonds and metallic bonds, while M7C3(0001)/NbC(111) interface II is combined by ionic bonds and metallic bonds. The works of adhesion are 1.23 J/m2 and 2.24 J/m2 respectively. There exists a region where the interfacial energy of interface II is lower than the solid-liquid interfacial energy of M7C3 carbide. The above experimental and calculation results demonstrate that NbC particle is the heterogeneous nucleus of primary M7C3 carbide and thereby refines it. © 2017 Elsevier B.V. All rights reserved.
Keywords: M7C3 carbide Grain refinement Molecular dynamics First-principles Heterogeneous nucleus
1. Introduction With the excellent wear-resistance, hypereutectic Fe-Cr-C alloys have been extensively applied in the wear-resistance improvement of work-pieces by arc-welding on their surfaces [1]. Currently, additive manufacturing industry aims hypereutectic Fe-Cr-C alloys at satisfying the needs of variously shaped work-pieces with wearable surfaces by means of 3D printing [2e5]. There exist a large amount of primary M7C3 carbides in the hypereutectic Fe-Cr-C alloys [6], which serve as the hardening constituent [7]. The abrasive resistance of primary M7C3 carbides determines the performance of the hypereutectic Fe-Cr-C alloys. However, the primary M7C3 carbides which precipitate directly from the Fe-Cr-C melt are coarse in size. As a result, the large * Corresponding author. State Key Laboratory of Metastable Materials Science & Technology, Yanshan University, Qinhuangdao 066004, PR China. ** Corresponding author. College of Mechanical Engineering, State Key Laboratory of Metastable Materials Science & Technology, Yanshan University, Qinhuangdao 066004, PR China. E-mail addresses: [email protected] (Y. Zhou), [email protected] (Q. Yang). http://dx.doi.org/10.1016/j.jallcom.2017.04.167 0925-8388/© 2017 Elsevier B.V. All rights reserved.
primary M7C3 carbides will wreck the continuity of the base and accelerate the forming and expanding of the crack, so that the service life of the work-pieces will be reduced [8,9]. With the rapid development of additive manufacturing industry, the demands for long-lived hypereutectic Fe-Cr-C alloys are higher and higher. Therefore, the key issue of the widespread of hypereutectic Fe-Cr-C alloys is the improvement of their antistrip performance as well as wear resistance [10]. Refined microstructure even ultra-refined microstructure can be achieved by microalloying treatment (mainly Nb, Ti, V elements and so on) [1,8,11e13], which draws a great deal of attention to the study on primary M7C3 carbides [14,15]. X.J. Wu et al. [1] researched on the effect of Ti additive on the microstructure of hypereutectic Fe-Cr-C alloy. The results showed that TiC precipitates firstly and the primary M7C3 carbide are refined. X.W. Qi et al. [8] investigated the effect of V additive on the microstructure and mechanical property of hypereutectic Fe-Cr-C alloy, which found that V additive can refine the primary M7C3 carbide and the wear-resistance of the alloy are effectively improved. However, to the authors' knowledge, the reports on the effect of Nb additive on the microstructural
S. Liu et al. / Journal of Alloys and Compounds 713 (2017) 108e118
refinement of the primary M7C3 carbide in hypereutectic Fe-Cr-C alloys are quite few. In the above literature, the refinement of primary M7C3 carbide is ascribed to the MC carbide (namely TiC, VC and so on), which precipitates previously to primary M7C3 carbide. The MC carbide plays a key role as the heterogeneous nucleus of primary M7C3 carbide. However based on the classical nucleation theory, in the solidification process of the hypereutectic Fe-Cr-C melts, the MC carbide can act as heterogeneous nucleus only if the interfacial energy between MC carbide and primary M7C3 carbide is smaller than the solid-liquid interfacial energy of primary M7C3 carbide [16]. While, the solid-liquid interfacial energy of primary M7C3 carbide is difficult to measure by means of experiments for now. So the theoretical mechanism of the refining effect on primary M7C3 carbide is still unaccountable. In the recent years, molecular dynamics simulation and firstprinciples calculation, which are of high accuracy, appeal much attention to the study on solidification process and interfacial behaviors [17e19]. Y. Shibuta et al. [20] investigated the nucleation and solidification of the undercooled iron melt from the atomistic point of view by molecular dynamics simulation, which successfully linked the empirical interpretation in metallurgy with the atomistic behavior of nucleation and solidification. A.A. Potter et al. [21] computed the crystal-melt interfacial free energy of the ternary Cu-Ag-Au alloy system by molecular dynamics simulation, and the results suggested that solute species exhibiting a large enthalpy of mixing will tend to promote a transition of the dendrite crystallographic growth directions from <100> to <110>. J. Yang et al. [22] revealed the electronic property and bonding configuration of NbC(111)/NbN(111) interface by first-principles calculation. The first-principles calculation has also been applied in investigating the effects of solute atom segregation on grain boundary strengths [23]. While, combined molecular dynamics simulation and first-principles calculation analysis of the heterogeneous nucleus of primary M7C3 carbide has never been reported. In this paper, the study object is the hypereutectic Fe-Cr-C alloys with Nb additive, which is manufactured by hardfacing. The phase precipitation process and phase component are calculated and detected, and the refining phenomenon of primary M7C3 carbide is observed. On this basis, the crystal model of M7C3 carbide is built and the solid-liquid interfacial energy of M7C3 carbide is acquired by molecular dynamics simulation. Besides, the interfacial properties between M7C3 and NbC are analyzed by first-principles calculation. Thereby, the theoretical mechanism of the refining effect of NbC on primary M7C3 carbide is revealed, which provides theoretical basis for producing hypereutectic Fe-Cr-C alloys with better antistrip performance and higher wear resistance. 2. Experimental and calculation methods 2.1. Experimental methods In order to verify the effects of NbC on primary M7C3 carbides, two hypereutectic Fe-Cr-C alloys were manufactured by hardfacing. The chemical compositions are listed in Table 1, which shows that except for the Nb contents (0 and 1.20 wt% respectively), other element contents of them are almost the same. On the basis of the chemical compositions, the phase precipitation curves were Table 1 Chemical composition of Fe-Cr-C alloys (wt.%).
Without Nb additive With Nb additive
C
Cr
Si
Mn
V
Nb
Mo
Fe
3.69 3.71
26.30 26.31
0.93 0.96
1.42 1.39
0.11 0.10
0 1.20
0.16 0.15
Bal. Bal.
109
calculated by Thermal-calc 4.0 software. The phase components were detected by D/max-2500/PC X-ray diffractometer (XRD) with Cu Ka radiation (20e120 , 1 /s). The alloys were etched by 35% FeCl3þ6%HNO3þ2%HClþ57%C2H5OH solution and then observed by Axiovert 200 MAT metallographic microscope. And the chemical compositions of carbides were measured by EMAX energy diffraction spectrum (EDS). 2.2. Calculation methods The molecular dynamics simulations of M7C3 carbide were fulfilled by using the Large-scale Atomic-molecular Massively Parallel Simulator (LAMMPS) with timestep of 0.1fs in an NPT ensemble. The Berendsen method [24] was used to control the temperature and the pressure. In molecular dynamics simulations, the trajectories of atoms and molecules (namely the velocity and position of each atom/molecule at each time) are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are calculated by using interatomic potentials or molecular mechanics force fields. In this paper, the interactions between Fe, Cr, C atoms were described by the EAM (embedded atom method) potential function [25], in which every atom is embedded in the matrix of other atoms. The total energy is described as the sum of embedding energy and pair potential, and given in the following form [26]:
Etot ¼
X
Fi ðri Þ þ
i
XX i
4ij rij
(1)
j
where Etot is the total energy for the system; ri is the background electron density for atom i (namely the electron density of the matrix assuming atom i is out of existence); Fi ðri Þ is the embedding energy for atom i embedded in a background electron density; rij is the distance between atoms i and j; 4ij ðrij Þ is the pair potential between atoms i and j. It also has the following relationship:
ri ¼
X fj rij
(2)
jsi
where fj is the electron density contribution of atom j to atom i. The first-principles calculations on NbC and M7C3 bulk properties, surface properties and the interfacial properties were fulfilled by using the density functional theory (DFT) as implemented in the Cambridge Sequential Total Energy Package (CASTEP). The exchange-correlation energy is described by the generalized gradient approximation (GGA) of Perdew and Wang (PW91) [27]. In the calculations on M7C3 and NbC bulk properties, the plane-wave cutoff energies are both 350eV and the Brillouin zone sampling are performed using 4 4 6 and 6 6 6 Monkhorst-Pack respectively. In the calculations of M7C3 and NbC surface properties, the plane-wave cutoff energies are both 350eV and the Brillouin zone sampling are performed using 4 4 1 and 6 6 1 Monkhorst-Pack respectively. In the calculations of interfacial properties, the plane-wave cutoff energy is 350eV and the Brillouin zone sampling is performed using 4 4 1 Monkhorst-Pack. 3. Effect of NbC on primary M7C3 carbide in hypereutectic FeCr-C alloy 3.1. Phase structure analysis The phase precipitation curves of the hypereutectic Fe-Cr-C alloys with 0 Nb and 1.20 wt% Nb are shown in Fig. 1. It is found from
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Fig. 1a that for the hypereutectic Fe-Cr-C alloy with 0 Nb, the primary M7C3 carbide starts to precipitate at 1330 C, and the eutectic reaction (liquid/eutectic M7C3 carbide þ eutectic austenite) occurs at 1300 C. It is found from Fig. 1b that for the hypereutectic FeCr-C alloy with 1.20 wt% Nb, MC carbide precipitates at 1350 C previously and then the primary M7C3 carbide starts to precipitate at 1330 C. The eutectic reaction (liquid/eutectic M7C3 carbide þ eutectic austenite) also occurs at 1300 C. Except for the precipitation of MC, the phase transition processes and the phase structures of the two hypereutectic Fe-Cr-C alloys are almost the same. Fig. 2 is the XRD patterns of the hypereutectic Fe-Cr-C alloys. The alloy with 0 Nb mainly consists of M7C3 carbide and g-Fe, while the alloy with 1.2 wt% Nb consists of M7C3 carbide, g-Fe as well as a small amount of NbC. Therefore, it can be deduced that MC carbide precipitate at 1350 C in Fig. 1b is NbC. 3.2. Microstructural observation The metallographic images of the hypereutectic Fe-Cr-C alloys are shown in Fig. 3a and b. The white polygons are primary M7C3 carbides. It is found that the primary M7C3 carbides in the hypereutectic Fe-Cr-C hardfacing alloy with 0 Nb are about 30e50 mm in diameter, while those in the hypereutectic Fe-Cr-C hardfacing alloy with 1.2 wt% Nb are only 10e20 mm in diameter, which indicates that Nb additive can effectively refine the primary M7C3 carbide. In order to investigate the reason, the primary M7C3 carbides were observed by secondary electron diffraction, which is shown in Fig. 3c. There exist some small particles in the primary M7C3 carbides with different contrast. The EDS analysis of particle A is shown in Fig. 3d. The major elements are Nb and C, which illustrates that the small particles are NbC. 4. Calculation on solid-liquid interfacial energy of M7C3 carbide 4.1. Crystal model of M7C3 carbide The primary M7C3 carbide in Fe-Cr-C alloys has hexagonal structure, whose lattice parameters are similar with Fe7C3 [28,29]. Because the Fe/Cr ratio of primary M7C3 carbide is uncertain, it can be expressed as Fe7-xCrxC3. Elements Fe and Cr are very close in the periodic table. It has been proved in literature [28] that atomic size effects of Fe and Cr on the lattice parameters and atomic positions are rather weak for the mixed M7C3 carbide. So Fe7-xCrxC3 can be
Fig. 2. XRD patterns of Fe-Cr-C hardfacing alloys with (a) 0 Nb and (b) 1.2 wt% Nb.
considered as the acquisition of Cr atoms partially replacing the Fe positions in Fe7C3. There are three kinds of Wyckoff positions for Fe atoms in Fe7C3, which are site 2b FeI (0.3333, 0.6667, 0.818), site 6c FeII (0.4563, 0.5437, 0.318) and site 6c FeIII (0.1219, 0.8781, 0) respectively. Six kinds of mixed Fe7-xCrxC3 carbides can be obtained by replacing one or two kinds of Fe atoms with Cr atoms, which are Fe6Cr1(I)C3, Fe4Cr3(II)C3, Fe4Cr3(III)C3, Fe3Cr4(I,II)C3, Fe3Cr4(I,III)C3 and Fe1Cr6(II,III)C3 respectively. By means of first-principles calculation, the six kinds of Fe7-xCrxC3 carbides were fully structural relaxed, and their formation energies DEfor ðFe7x Crx C3 Þ were acquired by:
DEfor ðFe7x Crx C3 Þ ¼ Etot ðFe7x Crx C3 Þ ð7 xÞEtot ðFeÞ xEtot ðCrÞ 3Etot ðCÞ
(3)
where Etot ðFe7x Crx C3 Þ is the total energy of Fe7-xCrxC3; Etot ðFeÞ, Etot ðCrÞ and Etot ðCÞ are the energies of a single Fe, Cr and C atom. The values of DEfor ðFe7x Crx C3 Þ are listed in Table 2, which is in coincidence with the calculation results of P.F. Zhang et al. [30]. DEfor ðFe7x Crx C3 Þ of Fe3Cr4(I,III)C3 and Fe1Cr6(II,III)C3 are lower than that of other carbides, which illustrates they are more stable from thermodynamics. In addition, the measured atomic ratio of Fe:Cr:C in primary M7C3 carbide by EDS is about 3:4:3. So the follow-up calculations in this paper is conducted by Fe3Cr4(I,III)C3. The crystal model of Fe3Cr4(I,III)C3 is shown in Fig. 4, whose lattice parameters are a ¼ b ¼ 6.999A, c ¼ 4.268A, which is close to the measurement value in literature [31].
Fig. 1. Phase fraction vs. temperature curves of Fe-Cr-C hardfacing alloys with (a) 0 Nb and (b) 1.2 wt% Nb.
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Fig. 3. Microstructures of the hypereutectic Fe-Cr-C hardfacing alloys with (a) 0 Nb and (b) 1.2 wt% Nb. (c) SEM image of Fe-Cr-C hardfacing alloy with 1.2 wt% Nb. (d) EDS analysis of particle A.
Table 2 Calculated formation energy of Fe7-xCrxC3. Carbide types
Fe6Cr1(I)C3
Fe4Cr3(II)C3
Fe4Cr3(III)C3
Fe3Cr4(I,II)C3
Fe3Cr4(I,III)C3
Fe1Cr6(II,III)C3
DEfor ðFe7x Crx C3 Þ (eV/cell)
0.02
0.64
0.37
0.72
0.89
0.90
4.2. Melting point of M7C3 carbide To get the critical nucleation undercooling degree of M7C3 embryos with different radii, the melting point of M7C3 carbide was calculated firstly. On the basis of the crystal model in Fig. 4, 320000 atoms were placed into a periodic rectangular simulation box with size 23.8399 nm 13.764 nm 9.08 nm. The system was kept at 1000 K until reached energy equilibrium. The system is still of hexagonal structure, as shown in Fig. 5a. Then the system was slowly heated to 2200 K until reached energy equilibrium. The final structure of the system is shown in Fig. 5b. It can be seen that the
atoms in the system are disordered at 2200 K, which illustrates that M7C3 carbide has been melted. In addition, it can be seen from the radial distribution functions at the lower right corners of Fig. 5a and b that the structure at 1000 K is long-range ordered, while that at 2200 K is not ordered, which also illustrates the transformation from M7C3 crystal to M7C3 melt. Fig. 6 shows the internal energy of the system versus temperature during the heating process. It can be seen that there exists a singularity between 1620 K and 1630 K, which means the system energy changes abruptly. The internal energy changes continuously before and after the abrupt change, which characterizes the M7C3 crystal and the M7C3 melt respectively. Therefore, the melting point Tm of M7C3 carbide is corresponding to the singularity, namely Tm z (1625 ± 5)K.
4.3. Melting behavior of M7C3 embryo
Fig. 4. Crystal structure of Fe3Cr4(I,III)C3.
According to the classical nucleation theory [32], an undercooling degree is necessary for the solidification of metal whose solid-liquid interface is of curvature. This phenomenon is called Gibbs-Thomson effect. The change in Gibbs free energy to form a sphere crystal embryo with radius r from the undercooled melt is expressed as:
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Fig. 6. Internal energy of M7C3 carbide versus temperature.
Fig. 5. (a) M7C3 crystal configuration in a periodic orthorhombic box and its radial distribution function; (b) M7C3 melt configuration in a periodic orthorhombic box and its radial distribution function.
4 3
DGr ¼ pr3 DGV þ 4pr2 gSL
(4)
where DGV is the Gibbs free energy difference per unit volume between the crystal and melt at the same temperature; gSL is the solid-liquid interfacial energy. DGV can be approximated as:
DGV ¼ LV
DT Tm
melt, namely the critical nucleation undercooling degree for the critical radius was reached. The melting behaviors of the crystal embryo with a radius of 3.5 nm cooled to different temperatures are displayed in Fig. 7. It can be seen that the outer shell of the embryo is melted at 1360 K, while the embryo is almost unchanged at 1345 K. Therefore, the critical nucleation temperature for the M7C3 embryo with a radius of 3.5 nm is 1345 K, namely the critical nucleation undercooling degree is 280 K. Numerous simulations of various embryo radii have been done, which resulted in the relationship of critical nucleation undercooling degrees to the reciprocal of critical radii in Fig. 8. The equation of the fitting line in Fig. 8 is as follow:
DT ¼ 70 þ 1:243 106
1 R
(7)
(5)
where LV is the melting enthalpy per unit volume at the melting point Tm . DT is the undercooling degree. The critical radius R is the value of r when Eq. (4) takes the maximum value and can be expressed as:
R¼
2gSL 2gSL Tm 1 ¼ DGV LV DT
(6)
It can be known from Eq. (6) that, for a given DT, there exists a specific R. The embryos with r > R will grow while those with r < R will melt. In other words, the values of R have a one-to-one relationship with the values of DT. On the basis of the M7C3 crystal model in Fig. 4, 320000 atoms were placed into a periodic rectangular simulation box with size 23.8399 nm 13.764 nm 9.08 nm. The system was kept at 1625 K until reached energy equilibrium. At this temperature, the system still remains hexagonal structure. A crystal embryo with critical radius R was built by fixing the coordinates of the atoms in it. Then the system was kept at 2000K so that the atoms outside the embryo could reach melt. By this means, a M7C3 embryo embedded in the Fe-Cr-C melt is formed, which is displayed in Fig. 7a. Then the system was rapidly cooled to a certain degree, and the melting behavior of the embryos was observed. If the embryo melts, it indicates that the temperature does not meet the undercooling degree requirement for the critical radius. Numerous attempts have been made to find the temperature that the embryo just did not
Fig. 7. (a) A M7C3 embryo with a radius of 3.5 nm in Fe-Cr-C melt. (b) The embryo cooled to 1360 K. (c) The embryo cooled to 1345 K.
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5. Interfacial bonding and electronic structure between M7C3 and NbC 5.1. Bulk properties of M7C3 and NbC
Fig. 8. Critical nucleation undercooling degrees versus the reciprocal of critical radii.
4.4. Solid-liquid interfacial energy of M7C3 carbide The transformation formula of Eq. (6) is:
DT ¼
2G gSL Tm 2 ¼ R LV R
Fig. 9 shows the band structures as well as the density of states (DOS) of M7C3 carbide and NbC by first-principles calculation, in which the Fermi level (EF) is marked by the dotted lines. Fig. 9a is the band structure of M7C3 carbide, which shows that it is of metallic characteristics overall because its electron bands cross the EF. Fig. 9b is the DOS of M7C3 carbide, in which the bands of Fe, Cr, C atoms are overlapping. Between 7.8eV and 6eV, DOS mainly consist of Cr-d, Fe-d and C-p bands. The Cr-d bands and Fe-d bands resonate significantly with the C-p bands between 7.8eV and 4.2eV, which illustrates p-d hybrid orbitals are formed between Cr and C as well as Fe and C, namely covalent bonds are formed. However, the center-of-gravity of Cr-d bands, Fe-d bands and C-p bands are not consistent, which shows Cr-C bond and Fe-C bond are of ionic characteristics. Therefore, the bonding characteristics of M7C3 carbide can be expressed as the combination of metallic bonds, covalent bonds and ionic bonds. From the band structure of NbC in Fig. 9c, NbC also shows metallic characteristics overall. Fig. 9d is the DOS of NbC, in which the bands of Nb and C atoms are overlapping. Between 7.5eV and 7eV, DOS mainly consists of Nb-d and C-p bands, which resonate significantly. So pd hybrid orbitals are formed between Nb and C, namely Nb-C covalent bonds are formed. However, the center-of-gravity of Nb-
(8)
where G ¼ gSLLVTm is the Gibbs-Thomson coefficient.
According to Eq. (8), the theoretical value of DT is 0 as the critical radius R is infinitely great. However, by comparing Eqs. (7) and (8), it can be found that a constant term 70 is excessed in Eq. (7), which means that the calculation result in this paper is 70 K lower than the theoretical value. On account that the potential functions in molecular dynamics simulation are empirical potentials, such small deviation can be neglected. Moreover, it can be acquired from the slope k of the line in Fig. 8 that G ¼ 12 k ¼ 6:213 107 K$m. So gSL of M7C3 carbide can be obtained if the value of LV is known.
LV ¼ Tm $S
(9)
where DS is the melting entropy per unit volume. The relationship between DS and melting enthalpy DH is:
DS ¼
rS Tm
DH
(10)
where rS is the number density of M7C3 crystal. The lattice parameter of M7C3 carbide is a ¼ b ¼ 6.999A, c ¼ 4.268A, and 20 atoms are contained in a unit cell. So it is acquired that rS ¼ 1:105 1029 atom=m3 . The enthalpy of M7C3 crystal at Tm is acquired by holding the system at 1625 K, which is 1243946.8eV. The enthalpy of M7C3 melt at Tm is acquired by holding the system at 2200 K until it is melted, and then holding the system at 1625 K, which is 1087152.7eV. So the value of melting enthalpy can be derived as DH ¼ 7:840 1020 J=atom. Then put the values of rS , DH and Tm into Eq. (10), and DS ¼ 5:331 106 J=K$m3 is obtained. Finally, the solid-liquid interfacial energy of M7C3 carbide is derived as gSL ¼ 3:312J=m2 .
Fig. 9. (a) Electronic band structure (b) and density of states of M7C3 carbide. (c) Electronic band structure and (d) density of states of NbC.
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d bands and C-p bands are not consistent, which suggests the ionic characteristics of NbC. Therefore, the bonding characteristics of NbC can be expressed as the combination of metallic bonds, covalent bonds and ionic bonds too.
In order to reduce the lattice misfit, a 1 1 M7C3(0001) slab and a 2 2 NbC(111) slab are chosen to conduct the interface models. Before building the interface models, convergence tests of the slicing layers for M7C3(0001) slab and NbC(111) slab are necessary to ensure that the interiors of M7C3(0001) slab and NbC(111) slab can achieve the bulk properties. Generally, the upper and lower surfaces of the slab should be the same to eliminate the dipole effect. On account of the crystal structure complexity of M7C3 carbide, the M7C3(0001) slab is conducted by stacking unit cells. By this means, M7C3(0001) slab has three kinds of surfaces, which are Cr-terminated, Cr&Fe-terminated and C-terminated. While according to our calculation, the surface energy of the Cr-terminated M7C3(0001) slab is the smallest, which means that it is the most stable from thermodynamics. So the follow-up calculation of M7C3(0001) slab is only based on the Crterminated one. The surface energy sM7 C3 ð0001Þ of M7C3(0001) slab is expressed as:
sM7 C3 ð0001Þ
(11)
where A is the surface area; Eslab is the total energy of the fully relaxed slab; NFe , NCr , NC are the numbers of Fe, Cr and C atoms in the slab; mFe , mCr , mC are the chemical potentials of Fe, Cr and C atoms in M7C3 carbide. The total energy mbulk M7 C3 of bulk M7C3 is:
(12)
In the M7C3(0001) slab which is conducted by stacking unit cells, the relationship between NFe and NC is as follow:
NFe ¼ NC
(13)
Integrating Eqs. (11)e(13), it can be derived that:
1 N 4 Eslab Fe mbulk N þ N Cr mCr 2A 3 Fe 3 M7 C 3
(14)
To evaluate the convergence, herein mCr zmbulk Cr ¼ 2467:7eV is
is the chemical potential of Cr atom in pure taken, where mbulk Cr chromium. The values of sM7 C3 ð0001Þ are listed in Table 3, from which, sM7 C3 ð0001Þ of the M7C3(0001) slabs tend to be constant when the layer number reaches 17. It is indicated that 17 slicing layers can meet the convergence requirement for M7C3(0001) slab. Similarly, in order to ensure the upper and lower surfaces are the same, the NbC(111) slab is conducted by odd number of slicing layers. NbC(111) slab has two kinds of surfaces, which are Nbterminated and C-terminated. The surface energy sNbCð111Þ of NbC(111) slab is expressed as:
where NNb and NC are the numbers of Nb and C atoms in the slab; mNb and mC are the chemical potentials of Nb and C atoms in NbC.
mbulk NbC ¼ mNb þ mC
(16)
Integrating Eqs. (15) and (16), it can be derived that:
sNbCð111Þ ¼
i 1 h Eslab NNb mbulk þ ðNNb NC ÞmC NbC 2A
Table 3 Calculated surface energy of M7C3(0001).
sM7 C3 ð0001Þ
3.174 J/m
17-layers 2
3.031 J/m
25-layers 2
(17)
To evaluate the convergence, herein mC zmbulk ¼ 155:03eV is C
is the chemical potential of C atom in pure cartaken, where mbulk C bon. The values of sNbCð111Þ are listed in Table 4, from which, sNbCð111Þ of the NbC(111) slabs tend to be constant when the layer number reaches 9. It indicates that 9 slicing layers can meet the convergence requirement for NbC(111) slabs. 5.3. Surface energies of M7C3 and NbC slab In Section 5.2, the approximate treatments of the chemical potentials are made to judge the convergence of slicing layers. Actubulk are variables. It has been mentioned above that ally, mbulk Cr and mC 17 slicing layers can meet the convergence requirement for NbC(111) slab. Put the numbers of Fe, Cr, C atoms into Eq. (14) and the following formula can be obtained:
sM7 C3 ð0001Þ ¼
i 1 h E 4mbulk M7 C3 3mCr 2A slab
(18)
bulk bulk bulk DH0f ðM7 C3 Þ ¼ mbulk M7 C3 4mCr 3 mFe þ mC
2.997 J/m
2
2
2.958 J/m
(19)
and (12), it can be derived that:
1 4
0 mCr mbulk Cr ¼ DHf ðM7 C3 Þ
3 3 mC mbulk mFe mbulk Fe C 4 4 (20)
bulk 0, Because mCr mbulk Cr 0, mC mC following relationship is obtained:
mFe mbulk Fe 0, the
1 0 DH ðM7 C3 Þ mCr mbulk Cr 0 4 f
(21)
Integrating Eqs. (18) and (21), the relationship between
sM7 C3 ð0001Þ for the 17-layered M7C3(0001) slab and mCr mbulk is Cr displayed Fig. 10a. From Fig. 10a, sM7 C3 ð0001Þ varies in 3.03 J/ m2~3.22 J/m2 over the entire range of mCr mbulk Cr . In the same way, the numbers of Nb, C atoms in the 9-layered NbC(111) slabs are put into Eq. (17) and the following formulas can be obtained:
sNbCð111Þ ðC terminatedÞ ¼
33-layers
It is obtained that DHf0 ðM7 C3 Þ ¼ 1:32eV. Integrating Eqs. (19)
sNbCð111Þ ðNb terminatedÞ ¼
9-layers
(15)
The heat of formation DH0f ðM7 C3 Þ for bulk M7C3 carbide is:
mbulk M7 C3 ¼ 3mFe þ 4mCr þ 3mC
sM7 C3 ð0001Þ ¼
1 ½E NNb mNb NC mC 2A slab
of bulk NbC is: The total energy mbulk NbC
5.2. Convergence tests of M7C3 and NbC slicing layers
1 ½E ¼ NFe mFe NCr mCr NC mC 2A slab
sNbCð111Þ ¼
i 1 h Eslab 20mbulk þ 4mC NbC 2A
i 1 h Eslab 16mbulk 4mC NbC 2A
The heat of formation DH0f ðNbCÞ for bulk NbC is:
(22)
(23)
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Table 4 Calculated surface energy of NbC(111).
Nb-terminated C-terminated
3-layers
5-layers
7-layers
9-layers
11-layers
13-layers
3.044 J/m2 5.763 J/m2
2.988 J/m2 5.708 J/m2
2.786 J/m2 5.533 J/m2
2.785 J/m2 5.558 J/m2
2.761 J/m2 5.561 J/m2
2.766 J/m2 5.556 J/m2
bulk Fig. 10. (a) Surface energy of 17-layered M7C3(0001) slab as a function of mCr mbulk Cr ; (b) surface energy of 9-layered Nb-terminated NbC(111) slab as a function of mC mC .
bulk bulk DH0f ðNbCÞ ¼ mbulk NbC mNb mC
(24)
It is obtained that DH0f ðNbCÞ ¼ 1:46eV. Integrating Eqs. (24) and (16), it can be derived that:
0 bulk mNb mbulk Nb ¼ DHf ðNbCÞ mC mC
(25)
Because mNb mbulk 0, mC mbulk 0, the following relationC Nb ship is obtained:
DHf0 ðNbCÞ mC mbulk 0 C
(26)
Integrating Eqs. (22), (23) and (26), the relationship between
sNbCð111Þ for the 9-layered NbC(111) slabs and mC mbulk are disC played Fig. 10b. From Fig. 10b, sNbCð111Þ ðNb terminatedÞ is smaller than sNbCð111Þ ðC terminatedÞ over the entire range of mC mbulk C , which means that the Nb-terminated slab is more stable from thermodynamics. Therefore, the follow-up interface models are conducted by the Nb-terminated NbC(111) slab and M7C3(0001) slab.
5.4. Properties of M7C3(0001)/NbC(111) interface 5.4.1. Interface models Based on the different atoms on the sub-surface of the M7C3(0001) slab, two kinds of M7C3(0001)/NbC(111) interface models are conducted, which are shown in Fig. 11a and c. The interfaces are marked by the dotted lines. In order to clearly display the stacking modes, Fig. 11b and d schematically show the sites of interfacial and sub-interfacial atoms from positive z-axis and the atoms are numbered separately(the equivalent atoms are numbered as the same). On account of the crystal structure complexity of M7C3 carbide, the electronic structures on ð1120Þ
and ð0110Þ planes are analyzed for comprehensive understanding, which are marked by the dotted lines in Fig. 11b and d. 5.4.2. Interfacial electronic structures Fig. 12 shows the electron densities and electron density differences on ð1120Þ plane and ð0110Þ plane for M7C3(0001)/ NbC(111) interface I, where the interface is marked by the dotted lines. From the electron density on ð1120Þ plane in Fig. 12a, the atomic spacing between Nb4 and C5 atoms is small, and strong bond between them is formed. It can be found that Nb1 and Cr1 atoms also form bond, while it is weaker than the Nb4-C5 bond. Besides, Nb1 atom and Fe atom on the Fe3Cr4C3 side also form bond. The electron density difference on ð1120Þ plane in Fig. 12b illustrates the localized interfacial charge redistribution for M7C3(0001)/ NbC(111) interface I. The population of Nb4-C5 bond is 0.31, which suggests the covalent characteristic of Nb4-C5 bond. While, the obvious charge depletion of Nb4 atom and charge accumulation of C5 atom further suggest the ionic characteristic of Nb4-C5 bond, so it is polar covalent/ionic bond. The charge of Nb1 atom and Cr1 atom transfers medially, suggesting the metallic characteristic of Nb1-Cr1 bond. Similarly, the Nb1-Fe bond is also metallic bond. From the electron density on ð0110Þ plane in Fig. 12c, the atomic spacing between Nb3 and C6 atoms, Nb3 and Cr3 atoms, Nb4 and C5 atoms as well as Nb4 and Cr3 atoms are small, and bonds between them are formed. The electron density difference on ð0110Þ plane in Fig. 12d displays obvious charge depletion of Nb3 atom and charge accumulation of C6 atom. Besides, the population of Nb3-C6 bond is 0.31, which illustrates its polar covalent/ionic characteristic. The charge of Nb3 atom and Cr3 atom transfers medially, suggesting the metallic characteristic of Nb3-Cr3 bond. Similarly, it also can be known that the Nb4-Cr3 bond is metallic bond. Fig. 13 shows the electron densities and electron density differences on ð1120Þ plane and ð0110Þ plane for M7C3(0001)/ NbC(111) interface II. From the electron density on ð1120Þ plane in
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Fig. 11. (a) M7C3(0001)/NbC(111) interface I and (b) schematic illustrations of the site of interfacial atoms before relaxation. (c) Fe3Cr4C3(0001)/NbC(111) interface II and (d) schematic illustrations of the site of interfacial atoms before relaxation.
Fig. 12. Electron density and electron density difference for M7C3(0001)/NbC(111) interface I. (a) Electron density on ð1120Þ plane; (b) electron density difference on ð1120Þ plane; (c) electron density on ð0110Þ plane; (d) electron density difference on ð0110Þ plane.
Fig. 13a, the atomic spacing between Nb1 and Cr1 atoms, Nb1 and Cr4 atoms as well as Nb1 and Fe1 atoms are small, and strong bonds between them are formed. C1 atom on NbC side and Cr4 on M7C3 side also form bond. The electron density difference on ð1120Þ plane in Fig. 13b also illustrates the localized interfacial charge redistribution for M7C3(0001)/NbC(111) interface II. The medially transferred charge illustrates the metallic characteristic of Nb1-Cr1 bond, Nb1-Cr4 bond and Nb1-Fe1 bond. The obvious charge depletion of Cr4 atom and charge accumulation of C1 atom suggests the ionic characteristic of C1-Cr4 bond. Together with the fact that the population of C1-Cr4 bond is 0, the C1-Cr4 bond is pure ionic bond.
From the electron density on ð0110Þ plane in Fig. 13c, bonds between Nb3 and Fe2 atoms, Nb3 and Cr3 atoms, Nb4 and Fe2 atoms as well as Nb4 and Cr3 atoms are formed. It can be known from the medially transferred charge in Fig. 13d that the Nb3-Fe2 bond, Nb3Cr3 bond, Nb4-Fe2 bond and Nb4-Cr2 bond are all metallic bonds. 5.4.3. Work of adhesions and interfacial energies The work of adhesion Wad is defined as the reversible work of separating an interface into two free surfaces. A large value of Wad implies that the binding strength of the interface is strong. The Wad of M7C3(0001)/NbC(111) interface is expressed as:
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Fig. 13. Electron density and electron density difference for M7C3(0001)/NbC(111) interface II. (a) Electron density on ð1120Þ plane; (b) electron density difference on ð1120Þ plane; (c) electron density on ð0110Þ plane; (d) electron density difference on ð0110Þ plane.
bulk Fig. 14. Interface energy of M7C3(0001)/NbC(111) interface I (a) and interface II (b) versus mCr mbulk Cr and mC mC .
Wad
1 ENbC þ EM7 C3 EM7 C3 =NbC ¼ A
(27)
where EM7 C3 and ENbC are the total energies for M7C3(0001) and NbC(111) slab after fully relaxation, namely the Eslab in Eqs. (11) and (15); EM7 C3 =NbC is the total energy for M7C3(0001)/NbC(111) interface model; A is the interfacial area. By putting the corresponding values into Eq. (27), Wad for the M7C3(0001)/NbC(111) interface model I and interface model II are 1.23 J/m2 and 2.24 J/m2, which means that bonding of interface II is stronger than that of interface I. The interfacial energy gM7 C3 =NbC of M7C3(0001)/NbC(111) interface is defined as the additional internal energy of increasing unit interfacial area, which is expressed as follow:
gM7 C3 =NbC ¼ sM7 C3 ð0001Þ þ sNbCð111Þ Wad
(28)
Integrating Eqs. (18), (22), (28) and Wad , gM7 C3 =NbC of the two
interface models as a function of mCr mbulk and mC mbulk can be Cr C drawn in Fig. 14, in which gSL of M7C3 carbide is marked by the yellow planes. From Fig. 14b, it can be seen that there exists a region where gM7 C3 =NbC of interface II is lower than gSL of M7C3 carbide. Therefore, there exists the possibility that NbC can act as the heterogeneous nucleus of M7C3 carbide. Cooperating with the experimental results that NbC precipitates previously to primary M7C3 carbide and exists in the primary M7C3 carbide, the refinement mechanism of NbC on primary M7C3 carbides is approved. 6. Conclusion (1) In order to verify the effects of NbC on primary M7C3 carbides, two hypereutectic Fe-Cr-C alloys with 0 Nb and 1.20 wt % Nb are manufactured. By experimental methods, the primary M7C3 carbides in the hypereutectic Fe-Cr-C alloy with 1.20 wt% Nb are obviously smaller than those in the hypereutectic Fe-Cr-C alloy with 0 Nb. Besides, NbC is found to
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precipitate previously to primary M7C3 carbide and exist in the primary M7C3 carbide. (2) By molecular dynamics simulations, it is acquired that the melting point of M7C3 carbide is Tmz(1625 ± 5)K. According to the classical nucleation theory, the solid-liquid interfacial energy of M7C3 carbide is calculated as gSL ¼ 3:312J=m2 . (3) By first-principles calculations, the interfacial electronic structures of the two kinds of M7C3(0001)/NbC(111) interface models with different atomic stacking modes are analyzed. M7C3(0001)/NbC(111) interface I is combined by polar covalent/ionic bonds and metallic bonds, while M7C3(0001)/ NbC(111) interface II is combined by ionic bonds and metallic bonds. (4) The works of adhesion for M7C3(0001)/NbC(111) interface I and II are 1.23 J/m2 and 2.24 J/m2 respectively, which means that the bonding of interface II is stronger than that of interface I. There exists a region where interfacial energy of interface II is lower than the solid-liquid interfacial energy of M7C3 carbide. Therefore, the refining mechanism of NbC on primary M7C3 carbide is approved theoretically, which is the heterogeneous nucleus theory.
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Acknowledgements The authors would like to express their gratitude for projects supported by the National Natural Science Foundation of China (No. 51471148), the Hebei province Basic Research Foundation of China (No. 16961008D) and the Innovation Fund for Graduate Students of Hebei Province (No. CXZZBS2017046).
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