Native Defects in α-Mo2C: Insights from First-Principles

Oct 9, 2014 - molybdenum carbide phases in their low Miller-index surfaces using PBE/plane wave calculations. Their results showed that α-Mo2C has th...
0 downloads 0 Views 6MB Size
Download PDF
Article pubs.acs.org/JPCC

Native Defects in α‑Mo2C: Insights from First-Principles Calculations Cláudio de Oliveira,† Dennis R. Salahub,‡ Heitor A. de Abreu,§ and Hélio A. Duarte*,§ †

Department of Natural Science, Universidade Federal de São João Del Rei, 36301-160 São João Del Rei, Minas Gerais, Brazil Department of Chemistry, CMS (Centre for Molecular Simulation), IQST (Institute for Quantum Science and Technology) and ISEEE (Institute for Sustainable Energy, Environment and Economy), University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 § Grupo de Pesquisa em Química Inorgânica Teórica − GPQIT, Department of Chemistry − ICEx, Universidade Federal de Minas Gerais, 31.270-901 Belo Horizonte, Minas Gerais, Brazil ‡

S Supporting Information *

ABSTRACT: Molybdenum carbide is a promising material for replacing the hydrogenation catalysts used in industry. It has been synthesized using different sources of molybdenum oxides and carbon, such as molybdenum heptamolybdate, molybdenum oxides, glucose, alkanes, and toluene. Nonstoichiometric material is produced, normally forming carbon vacancies or molybdenum vacancies, depending on the synthetic route. An approach for calculating the Helmholtz free energy of vacancy formation has been proposed, taking as reference the carbon and molybdenum atomic energies in the solid. The Helmholtz free energy for the carbon and molybdenum vacancies for different temperatures has been calculated. The results show that at 650 K, 9.6% of molybdenum vacancies and 10.2% of carbon vacancies exist. For temperatures below 610 K, the concentration of molybdenum vacancies is larger than that of carbon vacancies. The electron localization function (ELF) has been calculated for both defective materials, showing that the carbon vacancy presented trapped electrons forming a network that extends throughout the crystal with nearly homogeneous density. For the molybdenum vacancies, the electron density is completely depleted, leading to an electron-deficient site. The Lewis acid and base sites formed upon the presence of vacancies must be of real importance for understanding the catalytic properties of such materials. The presence of both vacancies is predicted to exist at a temperature of about 620 K.

1. INTRODUCTION

crystallizes in space group P3m1 with lattice parameters of a = b = 3.002 Å and c=4.724 Å.12 However, it is important to mention that the Joint Committee on Powder Diffraction Standards (JCPDS) data files13 defined a different convention denoting α-Mo2C and β-Mo2C phases as hexagonal and orthorhombic structures, respectively. Here we will follow the Christensen convention, which is widely used in the catalysis community. The structural and electronic properties of orthorhombic and hexagonal Mo2C were investigated by Liu et al.14 using planewave density functional theory (DFT). They showed that the

Molybdenum carbide has been intensely investigated due to its structural and chemical stability, corrosion resistance, and catalytic activity toward a plethora of technologically important reactions.1−5 However, the complete understanding of the relation between the physical chemical properties, structure, and reactivity of the molybdenum carbides is not yet at hand,6−10 and this lack of comprehension has restricted its use in commercial applications. Following Christensen,11 molybdenum carbide in a 2:1 stoichiometry presents two crystalline forms: an orthorhombic α-Mo2C and a hexagonal β-Mo2C phases. The α-Mo2C phase has an orthorhombic crystal structure (space group: D14 2h, Pbcn) with lattice parameters a, b, and c equal to 4.732, 6.037, and 5.204 Å, respectively. β-Mo2C is hexagonal-close-packed, and it © 2014 American Chemical Society

Received: August 6, 2014 Revised: September 24, 2014 Published: October 9, 2014 25517

dx.doi.org/10.1021/jp507947b | J. Phys. Chem. C 2014, 118, 25517−25524

The Journal of Physical Chemistry C

Article

described by Vanderbilt ultrasoft pseudopotentials24 considering the valence configuration: Mo 4d55s15p0 (the pseudopotential was generated with a scalar-relativistic calculation) and C 2s22p2 (the pseudopotential was generated with a nonrelativistic calculation). The valence states were expanded in plane waves with a kinetic energy cutoff of 30 and 300 Ry for the charge density cutoff. The integration over the Brillouin zone was performed with the Monkhorst−Pack scheme25 using a 2 × 2 × 2 uniform shifted k-point mesh for structure optimization and 4 × 4 × 4 for the electronic structure calculation. A Marzari-Vanderbilt26 smearing width of 0.01 Ry was applied for Brillouin zone integrations. All the k-point grids used in this work are able to calculate the total energy with an error lower than 10−3 Ry in comparison with larger grids. For the isolated atoms a simple cubic unitary cell of 10 Å was used to perform spin-polarized calculations at gamma point with 2.0 (Bohr mag/cell) and 6.0 (Bohr mag/cell) for carbon and molybdenum atoms, respectively. This is consistent to the ground state electron configuration of [He]2s22p2 for carbon and [Kr]5s14d5 for molybdenum. The phonons were calculated using density-functional perturbation theory from the ground-state calculations. For the phonon calculations, the Mo4C8 primitive cell was used. The kinetic energy cutoff was kept to 30 Ry, and the charge density cutoff was set to 240 Ry. The 4 × 4 × 4 uniform shifted k-point mesh was used with the threshold for self-consistency tightened to 10−19 Ry. Further processing was performed to calculate the interatomic force constants. The phonon band structure and density of states are shown in Figure S1 of the Supporting Information. The phonon entropy and vibrational internal energy calculations were based on the quasi-harmonic approximation (QHA). All eigenvalues were real and positive, assuring that the correct convergence and a minimum in the potential energy surface were achieved. In order to investigate the native defects, a large Mo64C32 unit cell was used as shown in Figure 1. The stoichiometry changes for each atom removed by x = 0.0156 in Mo2(1−x)C and x = 0.0312 in Mo2C(1−x) for Mo and C defects, respectively.

orthorhombic (α-Mo2C) structure is more stable. They argued that the lack of a band gap at the Fermi level, the presence of a pseudogap below the Fermi level, and the band structure in the energy region from −8 to −4 eV with strong hybridization between the carbon p and molybdenum d states indicate that covalent and metallic bonds coexist in the system. Wang et al.7 also showed that the hybridization effect between Mo 3d and C 2p states is much stronger in α-Mo2C than in the hexagonal βMo2C structure. Abderrahim et al.15 investigated a series of carbides, including α-Mo2C by first-principles methods and Bader’s quantum theory of atoms in molecules (QTAIM). The QTAIM analysis also indicates that molybdenum carbide presents a strong covalent character. Recently, dos Santos Politi et al.16 investigated the molybdenum carbide phases in their low Miller-index surfaces using PBE/plane wave calculations. Their results showed that α-Mo2C has the stronger metallic character among the different phases investigated. The nonpolar (011) surface of α-Mo2C presents a high stability and a relatively low work function of only 3.4 eV. Reddy et al.17 investigated experimentally the structural stability of α/β-Mo2C during thermochemical processing. They used MoO3 with sucrose as a carbon source. They observed that varying the Mo:C ratio permitted the control of the formation of α- or β-Mo2C. They observed for a 1:1 Mo:C ratio that α-Mo2C is obtained with 5.2 wt % of carbon content which corresponds to a nonstoichiometric structure of Mo2C0.876. Pereira-Almao and co-workers18−20 used a different route to synthesize α-Mo2C, and they carefully analyzed the X-ray diffraction pattern to conclude that nonstoichiometric α-Mo2C with either molybdenum or carbon vacancies have been formed. In conclusion, this material can have carbon vacancies, molybdenum vacancies, or both depending on the carbon source used, the molybdenum precursor, and the temperature of carburization.17−21 The presence of the native defects and their effect on the electronic, mechanical, and structural properties of α-Mo2C is of crucial importance for understanding its reactivity toward catalysis. In the present work, an approach based on firstprinciples calculations is proposed to estimate the concentration of the defects with respect to the temperature. The presence of a Mo2+ or a C4− vacancy leads to different effects in the electronic structure of the material. For instance, the absence of molybdenum must lead to the increase of the oxidation number of the surrounding carbon atoms to compensate the charge. On the other hand, the absence of carbon atoms must lead to the decrease of the oxidation number of the surrounding molybdenum atoms. Experimental data shows that the concentration of the vacancies can be as large as 10% and, hence, it is expected to modify the mechanical, electronic and structural properties of the material. In the present work, the effect of the vacancies on the structural, mechanical, and chemical stability of the material is investigated and the concentration of the defects is estimated with respect to the temperature. The aim of this work is to fulfill the lack of information about the native defects of the αMo2C synthesized at different temperatures.

Figure 1. (a) Unit cell and (b) supercell (Mo64C32) used for defect calculations of Mo and C in the Mo2C structure.

3. RESULTS AND DISCUSSION 3.1. Bulk α-Mo2C and the Native Defects. Table 1 shows a comparison of the present results and the available experimental and calculated lattice constants, bulk modulus, and volumes for α-Mo2C. The results are in very good agreement with the previous calculated values using plane wave/DFT calculations with the PBE XC functional.7,14,16 The differences between the theoretical approaches can be

2. THEORETICAL APPROACH All the calculations were performed using DFT with the exchange and correlation functional (XC) proposed by Perdew and Wang (PW91)22 as implemented within the QuantumESPRESSO package, PWscf.23 The core electrons were 25518

dx.doi.org/10.1021/jp507947b | J. Phys. Chem. C 2014, 118, 25517−25524

The Journal of Physical Chemistry C

Article

similar and the possibility of the presence of both vacancies cannot be excluded. The cohesive energy of the defective α-Mo2C is given by eq 2.

Table 1. Calculated and Experimental Lattice Constants (Å), Bulk Modulus (GPa) and Volumes (Å3) of α-Mo2C this work PBEa PW91b PBEc PBEd experimentale experimentalf a

a

b

c

volume

bulk modulus

4.798 4.738 4.793 4.707 4.741 4.724 4.732

6.127 6.038 6.030 6.040 6.070 6.004 6.037

5.314 5.210 5.211 5.247 5.227 5.199 5.199

156.22 149.09

309.1

E cohesive =

302.35 150.19 147.46

attributed to the model and protocol used to perform the calculations. The simulation of the concentration of native defects in the bulk is a difficult task. As the defects are formed during the synthesis at a relatively high temperature (above 400 K), it is expected that thermodynamic processes are involved and the defects formed are arranged in the most stable configuration. However, from the third atom removed, the number of possible configurations to be assessed increases exponentially. We decided to use the following strategy: (1) calculations were performed for the removal of an atom from each nonequivalent site. (2) The most stable structure was taken to calculate the thermodynamic properties. (3) Calculations were performed for the removal of the second atom from the remaining nonequivalent sites, and the most stable structure was used to calculate the thermodynamic properties. (4) So on, until a defect level of 30% was achieved. This strategy is reasonable since we have tested all possible configurations for the removal of the first three atoms in the Mo64C32 unit cell. For the first carbon atom removal, 4 nonequivalent sites exist, for the second, about 15 nonequivalent sites, and for the third removal, about 59 nonequivalent sites. We observed that the most stable configurations of the deleted atoms always have the removed atoms of the previous configurations, following the strategy proposed above. We assumed that this tendency is maintained with the increase of defects. Figures S2 and S3 of the Supporting Information show these results graphically for the carbon and molybdenum defects. Although a large number of configurations for the vacancies were spawned, it is limited to the Mo64C32 unit cell used in the present work. The vacancy formation energy was calculated for the most stable configurations according to eq 1.

0 E Mo 2C

NMo + NC

(2)

The EMo2C is the total energy of the bulk or defective solid, and Eisolated are the total energy of the isolated atoms Eisolated Mo C calculated using a unit cell of the same size. NMo and NC are the number of molybdenum and carbon atoms in the respective unit cell. Figure 2 presents the formation energy with respect to

315.5

Ref 14. bRef 15. cRef 7. dRef 16. eRef 12. fRef 11.

Evac =

isolated EMo2C − NMoEMo − NCECisolated

Figure 2. Cohesive energy of the solid with respect to the concentration of vacancies.

the vacancy concentration. The two lines are quite straight up to x = 0.14 and x = 0.20 for Mo and C defective α-Mo2C, respectively, demonstrating that the formation of a vacancy does not depend on the presence of other vacancies. The carbon atom is bound more weakly to the solid than the molybdenum, and then it is more affected by the vacancies surrounding it. The influence of the defects on the equilibrium volume and bulk modulus are shown in Figures 3 and 4. The cell structure is relaxed with the presence of vacancies, and the volume

0 n isolated EMo − E Mo − nEatom 2C 2C

n

(1) n E Mo 2C

corresponds to the defect-free structure, corresponds to the structure with n defects, and Eisolated atom is the total energy either of the carbon or molybdenum atoms. Tables S1 and S2 of the Supporting Information show Evac for the different concentration of vacancies. The carbon vacancy formation energies vary from 8.268 to 8.496 eV/vac with the increase of the vacancy concentration. For the molybdenum vacancy, the values vary from 8.448 to 8.360 eV/vac. For tungsten and titanium carbides, the carbon vacancy formation energy (about 2.21 eV/vac) is smaller than for the metal vacancy (about 5.7 eV/vac), explaining the experimental observation of the presence of carbon vacancies.27,28 However, for the case of α-Mo2C, vacancy formations are predicted to be

Figure 3. Equilibrium volume (Å3) with respect to the concentration of carbon or molybdenum vacancies. 25519

dx.doi.org/10.1021/jp507947b | J. Phys. Chem. C 2014, 118, 25517−25524

The Journal of Physical Chemistry C

Article

3.2. Helmholtz Free Energy of Formation for the Carbon and Molybdenum Native Defects. The Helmholtz free energy of formation (ΔF) is defined according to eq 3. ΔF(Nvac) = ΔU (Nvac) − T ΔS(Nvac)

(3)

ΔU(Nvac) and ΔS(Nvac) are the internal energy and entropy involved in the formation of the defective solid with Nvac vacancy. T is the absolute temperature. ΔU(Nvac) can be described according to eq 4. ΔU (Nvac) = ΔE elec(Nvac) + ΔEvib(Nvac)

(4)

ΔE (Nvac) and ΔE (Nvac) are the electronic and the vibrational internal energy contributions to the internal energy, respectively. The electronic energy contribution to the internal energy formation is given by eq 5. elec

Figure 4. Bulk modulus (GPa) with respect to the concentration of carbon and molybdenum vacancies.

vib

ele ele ΔE ele(Nvac) = Etotal (Nvac) + NvacECele − Etotal (0) for carbon vacancy ele ele ele ΔE ele(Nvac) = Etotal (Nvac) + NvacEMo − Etotal (0) for molybdenum vacancy

changes at most by 4% with the increase of the concentration of vacancies up to 0.25. The main characteristics of the structure are preserved with such a large concentration of vacancies. This is due to the large covalent character of the bonding in αMo2C. Concerning the bulk modulus, it is much affected by the molybdenum vacancies, decreasing it about 20% with a 0.25 concentration of vacancies. For carbon vacancies, the change is much less, about 6% smaller with a 0.25 concentration of vacancies. The density of states (DOS) does not change appreciably with the increase of the concentration of vacancies. The DOS became structureless near the Fermi level with the increase of the vacancies as one can see in Figure S4 of the Supporting Information. However, the lack of a band gap near the Fermi level indicates that the metallic nature of α-Mo2C is preserved. As it was discussed by Liu et al.,14 the structure between −8 and −4 eV is due to strong hybridization between the carbon p and molybdenum d states, and the appearance of a pseudogap below the Fermi energy indicates that the bonds present large covalent character. The structure between −8 and −4 eV is lost with the increase of the molybdenum vacancies, increasing its metallic character. However, this loss of structure in the DOS is not observed for the carbon vacancies.

Eele Mo

(5)

Eele C

Normally, and are the energies per atom for solid molybdenum and for diamond or graphite. However, this turned out to be very computationally demanding, since we also need to perform the phonon calculations in order to evaluate the thermal contribution. In the present study, the goal is to calculate the Helmholtz free energy to understand how the concentration of vacancies changes with the temperature. Keeping in mind this goal, any reasonable reference can be used to perform these calculations. ele We decided to take Eele Mo and EC as the atomic energy contribution to the α-Mo2C bulk calculated according to eq 6. ECele = ele EMo =

C ) Etotal(0) − Etotal(N vac C N vac Mo Etotal(0) − Etotal(Nvac ) Mo Nvac

(6)

As we have observed in Figure 2, this contribution is approximately constant and can be estimated from a linear

ele Figure 5. (a) Eele Mo and (b) EC with respect to the vacancies.

25520

dx.doi.org/10.1021/jp507947b | J. Phys. Chem. C 2014, 118, 25517−25524

The Journal of Physical Chemistry C

Article

regression of the values calculated for different Nvac, as shown in Figure 5. The extrapolated value is −11.447057 (−155.75 eV) Ry and −29.421161 Ry (400.30 eV) for carbon and molybdenum, respectively, which has been used as the reference in the calculations. Concerning the vibrational internal energy contribution to the internal energy, as we have used the atomic reference, we can assume the equipartition of the vibrational contribution and estimate it by eq 7. vib ΔEvib(Nvac) = NvacEatom

agreement with the experimental observations made by PereiraAlmao and coauthors19,20 based on X-ray diffraction analysis. Figure 7 shows the component contributions to the Helmholtz free energy for the molybdenum vacancies at 650

(7)

is estimated from the ideal α-Mo2C bulk calculations for both carbon and molybdenum atoms. Concerning ΔS(Nvac), it can be separated into two contributions as suggested by Lucas et al.,29 according to eq 8. Evib atom

ΔS(Nvac) = ΔS vib(Nvac) + ΔS conf (Nvac)

(8)

For ΔS (Nvac), the additivity of the entropy is used according to eq 9. vib

vib ΔS vib(Nvac) = NvacSatom

Figure 7. Helmholtz free energy and partial energy contributions for the native molybdenum defects at 650 K.

(9)

Svib atom is also estimated from the ideal α-Mo2C bulk calculations for both carbon and molybdenum atoms. The defective α-Mo2C presents different vacancy configuration leading to a configurational entropy contribution. This contribution is given by eq 10.

K. At this temperature, the favored concentration of native molybdenum vacancies is predicted to be 0.096. This value is also in good agreement with the experimental values that are predicted to be about 10%. The variation of the Helmholtz free energy with respect to the temperature and vacancies is shown in Figures 8 and 9 for the carbon and molybdenum vacancies, respectively.

ΔS conf (Nvac) = Nsitek b[x ln(x) + (1 − x) ln(1 − x)] (10)

where Nsite is the number of sites available to the vacancy in the unit cell model used, kb is the Boltzmann constant, and x is the fraction of vacancy given by eq 11.

x=

Nvac Nsite

(11) 30

Rogal et al. reported recently an extensive review of ways to include the role of point defects in calculations of the free energy of compounds. The Helmholtz free energy and partial energy components for the carbon native defects at 650 K are shown in Figure 6. A minimum in ΔF is shown at x = 0.102. This is in very good

Figure 8. Variation of the Helmholtz free energy with respect to the carbon vacancies for different temperature. Red points are the minimum for each curve.

3.3. Electronic Structure of Nonstoichiometric αMo2C. It has been shown that the vacancies caused relatively small change in the structure. The equilibrium volume changes less than 2% for 10.2% carbon vacancies and 2.6% for 9.6% molybdenum vacancies. This is evidence that the structure has a large covalent character. The removal of carbon atoms leads to smaller change in the bulk modulus of about 2.2% with 10.2% vacancies. However, the removal of molybdenum atoms leads to a decrease of the bulk modulus of about 7% for 9.6% vacancies. The compressibility of the α-Mo2C increases with the molybdenum vacancies. Distortions of the orthorhombic

Figure 6. Helmholtz free energy and partial energy contributions for the native carbon defects at 650 K. 25521

dx.doi.org/10.1021/jp507947b | J. Phys. Chem. C 2014, 118, 25517−25524

The Journal of Physical Chemistry C

Article

interpreted as a complete localization of the electron density and 1/2 as a uniform electron gas. The ELF was calculated in the different planes of the crystal. Figure 10 shows the ELF for the carbon vacancy. The stoichiometric α-Mo2C is consistent with the Mo2+ and C4− formal oxidation states. The presence of the carbon vacancies has to be compensated with the decrease of the molybdenum oxidation number. Surrounding the carbon vacancies, there is an excess of electrons which form a network extending throughout the crystal and having nearly constant electron density (the homogeneous gas limit characterized by ELF = 0.5) (Figure 10). Actually, the electron-rich molybdenum atoms surrounding the carbon vacancy fill this space with electrons. This is consistent with the increase of the density of states at the Fermi level, as shown in Figure S4 of the Supporting Information. The molybdenum vacancy is characterized by a complete lack of electron density as shown in Figure 11. The molybdenum vacancy has to be compensated by the increase of the oxidation state of the carbon which becomes electron deficient. This explains the fact that the ELF is nearly zero in the molybdenum vacancy. This characteristic must have important consequences for catalysis since the Lewis acid sites (electron deficient) favor heterogeneous catalysts. In conclusion, the carbon vacancy has trapped electrons forming interconnected regions having a uniform electron gaslike density as evidenced by the ELF, and the molybdenum vacancy leads to a completely depleted electron density consistent with a Lewis acid site. Figure 12 shows the temperature versus the percentage of vacancies. It shows that for temperatures below 610 K, the percentage of molybdenum vacancies is larger than carbon vacancies in good agreement with the recent experimental work.19 For higher temperatures, the carbon vacancies are predominant as observed for most of the syntheses which are made at higher temperatures. One could argue whether the existence of the two types of vacancies is possible. In fact, the two vacancies must exist concomitantly in different ratios as the plot of Figure 12 suggests. In fact, the stoichiometry suggested in the experimental papers is related to the ratio of Mo/C in the solid and, it is not related to the stoichiometry in the unit

Figure 9. Variation of the Helmholtz free energy with respect to the molybdenum vacancies for different temperature. Red points are the minimum for each curve.

structure with respect to the stoichiometric α-Mo2C are less than 1% for up to 20% vacancies. The electron localization function (ELF)31,32 described by eq 12 permits the analysis of the topology of the nonspin-polarized electron density of the system. −1 ⎡ ⎛ D( r ⃗) ⎞2 ⎤ ELF( r ⃗) = ⎢1 + ⎜ 0 ⎟ ⎥ ⎢⎣ ⎝ D ( r ⃗) ⎠ ⎥⎦

(12)

where D0(r)⃗ is the kinetic energy of a uniform electron gas given by 3 D0( r ⃗) = (6π 2)2/3 ρ5/3 ( r ⃗) (13) 5 and D(r)⃗ is the corresponding function given by D( r ⃗) = τ( r ⃗) −

1 ⎛ (∇ρ( r ⃗)2 ) ⎞ ⎜ ⎟ 4 ⎝ ρ( r ⃗) ⎠

(14)

The quantities τ(r)⃗ and ρ(r)⃗ are the kinetic energy and the total electron density, respectively. The value of ELF(r)⃗ equal to 1 is

Figure 10. Electron localization function for the carbon vacancy. The red ball is the position of the carbon vacancy. 25522

dx.doi.org/10.1021/jp507947b | J. Phys. Chem. C 2014, 118, 25517−25524

The Journal of Physical Chemistry C

Article

Figure 11. Electron localization function for the molybdenum vacancy. The purple ball is the position of the molybdenum vacancy.

ray diffraction was simulated for the Mo58C26 unit cell. Surprisingly, the diffractogram became very similar to the ideal structure, showing that the symmetry of α-Mo2C is preserved.

4. FINAL REMARKS Molybdenum carbide is a conducting metal-like solid. It has been reported in the literature that the synthesis of α-Mo2C using different sources of molybdenum (generally molybdenum oxides or heptamolybdate) and carbon (glucose, toluene, and alkanes) can be tuned to produce either molybdenum or carbon vacancies.18,19 This is a nonstoichiometric material that has the potential to replace the hydrogenation catalysts in industry. An alternative approach for calculating the Helmholtz free energy for vacancy formation has been proposed. At a temperature of about 650 K, the vacancy is estimated to be about 9.6% molybdenum vacancy and 10.2% carbon vacancy, consistent with the available experimental results.19,20 The vacancy formed for both defects have been analyzed by the ELF. The molybdenum vacancy leads to depleted electron density in the hollow, and the carbon vacancy presents trapped electrons consistent with an electron gas. The molybdenum and carbon sites are consistent with the Lewis acid and base, respectively. Therefore, the presence of the molybdenum and carbon native defects must be very important for understanding the surface reactivity of the α-Mo2C toward catalysis. In conclusion, our calculations indicate that the hydrothermal control of the type and concentration of vacancies formed is realistic. However, the synthesis and the complete characterization of this type of material remain to be achieved.

Figure 12. Temperature as a function of the vacancies.

cell. In fact, different ratios of Mo/C can be related to the same stoichiometry. If one takes the Mo64C32 as a reference, one could have Mo64C29 (xMo = 0.0; xC = 0.093), Mo62C28 (xMo = 0.031; xC = 0.096), Mo60C27 (xMo = 0.063; xC = 0.100), or Mo58C26 (xMo = 0.0938; xC = 0.103). For all of these configurations, the vacancy of the carbon with respect to the Mo2C(1−x) is about 10%. The estimate of the stoichiometry is based on the X-ray diffraction analysis, which assumes that the equilibrium volume of the cell does not change with the vacancies formation. In fact, we showed that upon 10% vacancy formation the equilibrium volume decreases less than 10%, hence, validating this assumption. Actually, one has to keep in mind that removing two Mo atoms, one degree of freedom for the carbon vacancy is decreased in the unit cell. Therefore, all the approaches used for calculating the Helmotz free energy of formation and the configuration entropy, eq 10, remain valid. The X-ray diffraction pattern for the ideal structure, molybdenum vacancy, carbon vacancy, and for the structure with both vacancies are shown in Figure S5 of the Supporting Information. For both defective structures, the peaks are displaced to higher angles. For the molybdenum defective structure, the crystallinity is enhanced. For comparison, the X-

■

ASSOCIATED CONTENT

* Supporting Information S

Figures of phonon band structure, phonon density of states, electron density of states, simulated X-ray diffractogram, contributions to the free energy of carbon and molybdenum vacancy formation as a function of temperature; tables with the optimized lattice constants, volumes of super cell, Bulk modulus, cohesive and vacancy formation energies, values of the different contributions to the Helmholtz free energy according to the temperature and vacancy concentration. This 25523

dx.doi.org/10.1021/jp507947b | J. Phys. Chem. C 2014, 118, 25517−25524

The Journal of Physical Chemistry C

Article

(16) dos Santos Politi, J. R.; Vines, F.; Rodriguez, J. A.; Illas, F. Atomic and Electronic Structure of Molybdenum Carbide Phases: Bulk and Low Miller-Index Surfaces. Phys. Chem. Chem. Phys. 2013, 15, 12617−12625. (17) Reddy, K. M.; Rao, T. N.; Revathi, J.; Joardar, J. Structural Stability of Alpha/Beta-Mo2C During Thermochemical Processing. J. Alloys Compd. 2010, 494, 386−391. (18) Guzman, H. J.; Xu, W.; Stacchiola, D.; Vitale, G.; Scott, C. E.; Rodriguez, J. A.; Pereira-Almao, P. In Situ Time-Resolved X-Ray Diffraction Study of the Synthesis of Mo2c with Different Carburization Agents. Can. J. Chem. 2013, 91, 573−582. (19) Vitale, G.; Frauwallner, M. L.; Hernandez, E.; Scott, C. E.; Pereira-Almao, P. Low Temperature Synthesis of Cubic Molybdenum Carbide Catalysts Via Pressure Induced Crystallographic Orientation of MoO3 Precursor. Appl. Catal., A 2011, 400, 221−229. (20) Vitale, G.; Frauwallner, M. L.; Scott, C. E.; Pereira-Almao, P. Preparation and Characterization of Low-Temperature Nano-Crystalline Cubic Molybdenum Carbides and Insights on Their Structures. Appl. Catal., A 2011, 408, 178−186. (21) Griboval-Constant, A.; Giraudon, J. M.; Leclercq, G.; Leclercq, L. Catalytic Behaviour of Cobalt or Ruthenium Supported Molybdenum Carbide Catalysts for Ft Reaction. Appl. Catal., A 2004, 260, 35−45. (22) Perdew, J. P.; Wang, Y. Accurate and Simple Analytic Representation of the Electron-Gas Correlation-Energy. Phys. Rev. B 1992, 45, 13244−13249. (23) Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G. L.; Cococcioni, M.; Dabo, I.; et al. Quantum Espresso: A Modular and Open-Source Software Project for Quantum Simulations of Materials. J. Phys.: Condens. Matter 2009, 21, 395502−395521. (24) Laasonen, K.; Pasquarello, A.; Car, R.; Lee, C.; Vanderblit, D. Car-Parrinello Molecular-Dynamics with Vanderbilt Ultrasoft Pseudopotentials. Phys. Rev. B 1993, 47, 10142−10153. (25) Monkhorst, H. J.; Pack, J. D. Special Points for Brillouin-Zone Integrations. Phys. Rev. B 1976, 13, 5188−5192. (26) Marzari, N.; Vanderbilt, D.; De Vita, A.; Payne, M. Thermal Contraction and Disordering of the Al(110) Surface. Phys. Rev. Lett. 1999, 82, 3296−3299. (27) Kong, X. S.; You, Y. W.; Xia, J. H.; Liu, C. S.; Fang, Q. F.; Luo, G. N.; Huang, Q. Y. First Principles Study of Intrinsic Defects in Hexagonal Tungsten Carbide. J. Nucl. Mater. 2010, 406, 323−329. (28) Zhao, S. J.; Xue, J. M.; Wang, Y. G.; Huang, Q. First-Principles Investigation of the Intrinsic Defects in Ti3SiC2. J. Phys. Chem. Solids 2014, 75, 384−390. (29) Lucas, M. S.; Delaire, O.; Winterrose, M. L.; Swan-Wood, T.; Kdresch, M.; Halevy, I.; Fultz, B.; Hu, J.; Lerche, M.; Hu, M. Y.; et al. Effects of Vacancies on Phonon Entropy of B2 Feal. Phys. Rev. B 2009, 80, 214303−21413. (30) Rogal, J.; Divinski, S. V.; Finnis, M. W.; Glensk, A.; Neugebauer, J.; Perepezko, J. H.; Schuwalow, S.; Sluiter, M. H. F.; Sundman, B. Perspectives on Point Defect Thermodynamics. Phys. Status Solidi B 2014, 251, 97−129. (31) Becke, A. D.; Edgecombe, K. E. A Simple Measure of Electron Localization in Atomic and Molecular-Systems. J. Chem. Phys. 1990, 92, 5397−5403. (32) Savin, A.; Jepsen, O.; Flad, J.; Andersen, O. K.; Preuss, H.; Vonschnering, H. G. Electron Localization in Solid-State Structures of the Elements - the Diamond Structure. Angew. Chem., Int. Ed. 1992, 31, 187−188.

material is available free of charge via the Internet at http:// pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +553134095748 Fax: +55-313409-5700. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We would like to thank Prof. Pedro Pereira-Almao, Dr. Hector Guzman, and Dr. Gerardo Vitale for fruitful discussions during the development of this work. We thank the Inter-American Materials Research Collaboration (CIAM-2008) for supporting us during the work missions. The Brazilian agencies CNPq, FAPEMIG, and CAPES and the Canadian agency NSERC are gratefully acknowledged. This work has also been supported by the Brazilian National Institute of Science and Technology for Mineral Resources, Water and Biodiversity (INCT-ACQUA).

■

REFERENCES

(1) Alexander, A. M.; Hargreaves, J. S. J. Alternative Catalytic Materials: Carbides, Nitrides, Phosphides and Amorphous Boron Alloys. Chem. Soc. Rev. 2010, 39, 4388−4401. (2) Furimsky, E. Metal Carbides and Nitrides as Potential Catalysts for Hydroprocessing. Appl. Catal., A 2003, 240, 1−28. (3) Kim, H. G.; Lee, K. H.; Lee, J. S. Carbon Monoxide Hydrogenation over Molybdenum Carbide Catalysts. Res. Chem. Intermed. 2000, 26, 427−443. (4) Navarro, R. M.; Pena, M. A.; Fierro, J. L. G. Hydrogen Production Reactions from Carbon Feedstocks: Fossils Fuels and Biomass. Chem. Rev. (Washington, DC, U. S.) 2007, 107, 3952−3991. (5) Zaman, S.; Smith, K. J. A Review of Molybdenum Catalysts for Synthesis Gas Conversion to Alcohols: Catalysts, Mechanisms and Kinetics. Catal. Rev. 2012, 54, 41−132. (6) Rutkowska-Zbik, D.; Grybos, R.; Tokarz-Sobieraj, R. DFT Investigation of Molybdenum (Oxo)Carbide Formation from MoO3. Struct. Chem. 2012, 23, 1417−1424. (7) Wang, X. R.; Yan, M. F.; Chen, H. T. First-Principle Calculations of Hardness and Melting Point of Mo2C. J. Mater. Sci. Technol. 2009, 25, 419−422. (8) Lo, R. L.; Fukui, K.; Otani, S.; Oyama, S. T.; Iwasawa, Y. CTerminated Reconstruction and C-Chain Structure on Mo2c(0001) Surface Studied by Low Energy Electron Diffraction and Scanning Tunneling Microscopy. Jpn. J. Appl. Phys. 1999, 38, 3813−3815. (9) Lo, R. L.; Fukui, K.; Otani, S.; Iwasawa, Y. High Resolution Images of Mo2C(0001)-(Root 3 X Root 3)R30 Degrees Structure by Scanning Tunneling Microscopy. Surf. Sci. 1999, 440, L857−L862. (10) St Clair, T. P.; Oyama, S. T.; Cox, D. F.; Otani, S.; Ishizawa, Y.; Lo, R. L.; Fukui, K.; Iwasawa, Y. Surface Characterization of AlphaMo2c (0001). Surf. Sci. 1999, 426, 187−198. (11) Christensen, A. N. Neutron-Diffraction Investigation on a Crystal of Alpha-Mo2C. Acta Chem. Scand., Ser. A 1977, 31, 509−511. (12) Parthe, E.; Sadagopa, V. Structure of Dimolybdenum Carbide by Neutron Diffraction Techniques. Acta Crystallogr. 1963, 16, 202−205. (13) Powder Diffraction File; JCPDS Intenational Center for Diffraction Data: Pennsylvania, 2004. (14) Liu, H.; Zhu, J.; Lai, Z.; Zhao, R.; He, D. A First-Principles Study on Structural and Electronic Properties of Mo2C. Scr. Mater. 2009, 60, 949−952. (15) Abderrahim, F. Z.; Faraoun, H. I.; Ouahrani, T. Structure, Bonding and Stability of Semi-Carbides M2C and Sub-Carbides M4C (M=V, Cr, Nb, Mo, Ta, W): A First Principles Investigation. Phys. B 2012, 407, 3833−3838. 25524

dx.doi.org/10.1021/jp507947b | J. Phys. Chem. C 2014, 118, 25517−25524