First-Principles Analysis of Cation Diffusion in ... - ACS Publications

Subscriber access provided by KUNGL TEKNISKA HOGSKOLAN

Article

A First Principles Analysis of Cation Diffusion in Mixed Metal Ferrite Spinels Christopher L Muhich, Victoria J. Aston, Ryan M Trottier, Alan W. Weimer, and Charles B. Musgrave Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.5b03911 • Publication Date (Web): 03 Dec 2015 Downloaded from http://pubs.acs.org on December 8, 2015

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Chemistry of Materials is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Chemistry of Materials

A First Principles Analysis of Cation Diffusion in Mixed Metal Ferrite Spinels Christopher L. Muhich1,†, Victoria J. Aston1 , Ryan M. Trottier 1, Alan W. Weimer1* and Charles B. Musgrave1,2* 1

Department of Chemical and Biological Engineering, University of Colorado, Boulder, Colorado 80309-5096 2

Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309

†

Present address: Department of Mechanical and Process Engineering, ETH, 8092 Zürich, Switzerland *

email: [email protected], [email protected]

*To whom correspondence should be addressed

Abstract Ferrite spinels are metal oxides used in a wide variety of applications, many of which are controlled by the diffusion of metal cations through the metal oxide lattice. In this work, we used density functional theory (DFT) to examine the diffusion of Fe, Co and Ni cations through the Fe3O4, CoFe2O4, and NiFe2O4 ferrite spinels. We apply DFT and crystal field theory to uncover the principles that govern cation diffusion in ferrite spinels. We found that a migrating cation hops from its initial octahedral site to a neighboring octahedral vacancy via a tetrahedral metastable intermediate separated from octahedral sites by a trigonal planar transition state (TS). The cations hop with relative activation energies of Co ≈< Fe < Ni; the ordering of the diffusion barriers is controlled by the crystal field splitting of the diffusing cation; specifically, the orbital splitting and number of electrons which must be promoted into the higher energy t2g orbitals of the tetrahedral metastable intermediate as the cations move along the minimum energy pathway of hopping. Additionally, for each diffusing cation, the barriers are inversely proportional to the spinel lattice parameter, leading to relative barriers for cation diffusion of Fe3O4 < CoFe2O4 < NiFe2O4. This results from the shorter cation-O bonds at the TS for spinels with smaller lattices, which inherently possess shorter bond lengths, and consequently higher system energies at their more constricted TS geometries.

1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1. Introduction Ferrite metal oxides are a class of commonly encountered materials, many of which possess desirable properties for a wide range of applications. However, they are also often the undesired products of the deleterious oxidation of useful materials. Broadly, ferrites are Fe containing metal oxides which adopt a range of stable configurations. In this work, we only considered the common Fe3O4 spinel and its derivatives, CoFe2O4 and NiFe2O4. These materials are abundant and relatively inexpensive. This, coupled with their many desirable properties, makes ferrites useful for a wide variety of applications such as: electronics, where they make up the magnetic layers of hard-drives; catalysis,1-3 batteries,4 and oxygen carriers in chemical looping and solar thermal water splitting processes.5-11 However, as we noted above, the presence and formation of ferrites can also be detrimental, most prominently as corrosion products from the oxidation of iron and iron alloys where oxidation degrades the mechanical, electrical, aesthetic, and other advantageous properties of the original material. In this contribution we examine cation diffusion in ferrites using the ferrite oxidation step of chemical looping as an example to illustrate the fundamental principles of cation diffusion in metal oxides. The principles we investigate will be transferrable to other processes involving cation diffusion in ferrites and other metal oxides. Chemical looping and solar thermal water splitting serve as examples for processes controlled by cation diffusion because recent studies have demonstrated significant differences in the oxidation rates of ferrites of different compositions. When oxidized using water, reduced Fe3O4 oxidizes faster than reduced CoFe2O4, which oxidizes faster than reduced NiFe2O4,12 and CoFe2O4 oxidizes faster than NiFe2O4 when oxidized by O2.13 Additionally, Lu et al. reported that cations diffuse through CoFe2O4 faster than through Fe3O4.14 This demonstrates that cation diffusion affects the oxidation rates of ferrite


Page 2 of 35

Page 3 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


spinels and that composition can be used to tune the rates of processes based on ferrite spinel oxidation. Chemical looping is a two-step cycle in which a metal oxide acts as an oxygen carrier to isolate individual reduction and oxidation

reactions,

and

is

illustrated

schematically in Figure 1. This is usually performed for the purpose of simplified

Figure 1: A schematic of the chemical looping process. The right side shows the reduction of MOx by reducing gases such as CH4, H2 or CO to produce the reduced oxide MO y. The left side shows the oxidation of reduced MO y by oxidizing gases such as O2, H2O, and CO2 to re-oxidized MOy to MOx.

product separation or purification where the production of the gaseous products of oxidation and reduction are temporally, or spatially disassociated so that they can be collected separately as they are produced.15 In ferrite based chemical looping the ferrite is exposed to a reducing gas such as H2, CO, or a hydrocarbon to generate H2O, and/or CO2 and a reduced material, i.e. a metallic iron alloy, although the reaction is sometimes terminated prior to complete reduction.1618

This partial reduction is shown on the right side of Figure 1. The reduced solid is then easily

separated either spatially or temporally from the gaseous H2O, and/or CO2 byproducts. If a mixture of H2O and CO2 is produced, the component gases are easily separated from one another by condensation of H2O, enabling the CO2 to be sequestered, stored for use as a pure chemical gas, or released. The partially reduced ferrite, or metallic iron alloy produced if reduction continues to completion, is then exposed to an oxidizing gas that is reduced to a useful product, such as heat, H2, or CO. This reaction also oxidizes the ferrite to regenerate the starting metal oxide, enabling cycling of this process. This is shown on the left side of Figure 1. The oxidizing gases are commonly air, which results in significant heat generation; H2O, which generates H2 with no side products; or CO2, which generates CO with no side products. The heat released



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 35

from air oxidation is useful for power generation, which enables simple CO2 collection from the reduction step for later sequestration and eliminates noxious NOx production because the N2 in air is not exposed to the radical oxygen species which are produced by direct methane combustion in air.19-21 Oxidizing the reduced material with H2O or CO2 is useful for steam or dry methane reforming, respectively, where methane is the reducing gas.22,

23

Once again, this

enables simplified product separation of the byproduct CO2 from the desired H2 or CO gases, while simultaneously eliminating the need for expensive catalysts. Additionally, water gas shift and reverse water gas shift reactions are accomplished by using streams of H2, CO or mixtures thereof as the reducing gas, enabling the generation of H2 and CO in desirable ratios for FischerTropsch synthesis, other gas to liquid pathways, or for direct use.24, 25 In H2 generating chemical looping with Co and Ni ferrites, the oxidation step is substantially slower than the reduction step.12 Therefore, accelerating the rate of oxidation can result in significant improvements in the overall cycle efficiency and process economics. Iron alloys oxidize through a multi-banded shrinking core reaction, where the external surface of the alloy first oxidizes to MxFe1-xO, then MxFe2-xO4 and finally, if sufficient driving force exists, to MxFe2-xO3, where M is an iron alloying element.26-30 The partially reduced intermediate structures thus consist of stratified layers oxidized to different extents with the degree of oxidation decreasing with depth into the bulk of the material. The rate of iron oxidation is limited by the diffusion of cations through the ferrite spinel (MxFe2-xO4) layer.29 Additionally, ferrites exhibit relative oxidation rates of Fe3O4 > CoFe2O4 > NiFe2O4 when oxidized by steam and relative oxidation rates of CoFe2O4 > NiFe2O4 when oxidized by O2.12, 13 However, it is not clear a priori why the relative oxidation rates order as Fe3O4 > CoFe2O4 > NiFe2O4. By understanding what governs the relative rates of oxidation, not only can new chemical looping ferrites be


Page 5 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


designed which accelerate the oxidation step but, by extension, modifications to ferrites in other applications can also be made to either increase or decrease the MxFe2-xO4 oxidation rates as desired. The ferrite oxidation reactions are cation diffusion-limited under oxygen rich environments, as present in both O2 and H2O based chemical looping oxidation steps.26-29, 31, 32 This occurs because the equilibrium concentration of cation vacancies is considerably higher than oxygen (anionic) vacancies in ferrites under oxidative conditions.14,

33, 34

However, few

studies have examined the atomic level processes by which cations diffuse through mixed metal ferrites and, therefore, the mechanisms and related phenomena that govern the rates of oxidation of these materials are largely not understood. To the best of our knowledge, only Hendy et al. have used first principles calculations to investigate the hopping of cations through the ferrite lattice and their study was limited to the migration of Fe cations in Fe3O4.35,

36

As many

processes involve more complex ferrite materials, an understanding of the migration of cations in these materials, including the effect of composition and the identity of the diffusant on cation diffusion, requires an evaluation of the diffusion of different cations in these more complex structures. Furthermore, while crystal field theory has been used to explain relative cationic diffusion preferences,37-40 it has not been used to analyze cation diffusion in ferrite spinels. Mixed metal ferrites (MFe2O4, M=Ni, Co, and Fe for the ferrites examined in this work) take on the spinel crystal structure, which is characterized by an AB2O4 stoichiometry where the A cation has an oxidation state of +2 and the B cations have oxidation states of +3. The ordering of the A and B cations on the lattice sites of the spinel structure ranges from “normal” to “inverse”. In completely normal spinels, the A2+ cations exclusively occupy tetrahedral sites, while the B3+ cations exclusively occupy octahedral sites, as shown in Figure 2a. In completely



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

inverse spinels, the octahedral sites are randomly occupied by A2+ and B3+ cations with half the octahedral sites occupied by A2+ cations and half occupied by B3+ cations, while the tetrahedral sites are exclusively occupied by B3+ cations, as shown in Figure 2b. The extent of inversion is measured by the inversion parameter X, which ranges from 0 (completely normal) to 1 (completely inverse). The degree of tetrahedral/octahedral site

Figure 2: The primitive unit cell representations of the completely a) normal and b) inverse spinels. The large silver, large blue and smaller red spheres represent B3+, A2+ and O2ions, respectively. The dotted-dashed lines indicate the coordination of the metal centers in that plane.

preference for the A and B ions, and therefore, the degree of inversion, has been widely

studied and depends on the size of the ion, its oxidation state and crystal field effects. In general, the elemental preference for octahedral site occupation is: Cr2+ > Ni2+ > Mn3+ > Al3+ > Fe2+ > Co2+ > Fe3+ > Mn2+.41 These relative preferences for occupying the octahedral sites suggest that the majority of mixed metal spinel ferrites are inverse spinels, including the Fe3O4, CoFe2O4 and NiFe2O4 spinels we focus on in this study. This agrees with experimental observations and computational predictions that have shown that Fe3O4, CoFe2O4 and NiFe2O4 are indeed inverse spinels.42-44 In this work, we used Density Functional Theory (DFT) based atomistic modeling to calculate the hopping pathways and approximate relative rates of diffusion of Fe, Co and Ni in Fe3O4, CoFe2O4 and NiFe2O4. The results of this study will facilitate the design of ferrite based chemical looping materials and by extension, other processes governed by the rate of cation


Page 6 of 35

Page 7 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


diffusion in ferrites, including those where facile diffusion is either detrimental or beneficial to the system, by identifying the material properties that significantly influence the rates of cation diffusion.

2. Methods We performed plane wave periodic boundary condition DFT calculations using the Vienna Ab initio Simulation Package (VASP).45, 46 Calculations employed the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) exchange-correlation functional47 coupled with projector augmented wave (PAW) pseudopotentials.48 PAWs described the oxygen 2s and 2p and iron, cobalt and nickel 4s and 3d electrons explicitly. All calculations utilized a 500 eV plane-wave cut-off energy based on a convergence study over the range of 450 to 550 eV. We conducted calculations using a Γ-point centered 222 Monkhorst-Pack k-point mesh because our k-point convergence study determined that the more computationally expensive 444 kpoint mesh resulted in an energy only 0.01 eV/supercell lower than that calculated using the 222 k-point expansion. Because DFT using GGA exchange-correlation functionals overestimates lattice constants and the lattice constants of mixed metal ferrites significantly influence the activation energies of cation hopping, as discussed in Section 4.2, we calculated lattice constants using the hybrid HSE06 functional.49,

50

The resulting lattice constants agree

closely with experiment, as shown in Table 1. We used the PBE+U method for all other calculations, including transition state searches, because of its considerably lower computational resource requirements. The spinel supercell model we employed is composed of eight (222) primitive unit cells and includes 112 atoms, as shown in Figure 2.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

A Hubbard51 onsite correction term Ueff of 3, 2 and 5 eV for the Fe, Co and Ni d-electrons, respectively, was included in the DFT functional because the GGA functional does not adequately describe the strong electron correlation between d electrons of the same metal center for these systems. These values of Ueff have been shown to accurately predict the electronic structures of ferrite spinels.44 The Hubbard correction attempts to describe strong electron correlation between electrons localized within the d-shell (or f-shell for atoms with multiple electrons occupying the f-shell of the atom) more accurately by imposing an energy penalty for partial electron occupation of the d-orbitals on a given atom. Bader charge analysis was conducted using software from the Henkelman group.52, 53 Minimum energy pathways for cation hopping were determined using the nudged elastic band and dimer methods as implemented in VASP by the Henkelman Group.54, 55 First, nudged elastic band calculations were conducted to determine the individual single steps and locations of stable intermediates along the migration path. Then, for each step, the highest energy geometry from the NEB calculation was used as the starting geometry for the dimer method transition state (TS) search. The dimer method was used to determine the TS geometry because it explicitly locates the saddle point along a potential energy surface rather than finding the energies of evenly spaced points along the migration path. Additionally, because the dimer method only optimizes one atomic configuration, it is less computationally expensive than NEB calculations which require minimization of the energies of multiple geometries.56 All geometries were optimized until the forces were less than 0.02 eV/Å. We will only discuss the results of the dimer method because the NEB results were only used to generate approximate structures to be further refined by the dimer method. The rate of vacancy mediated diffusion of cation i is the product of the concentration of vacancies and the hopping rate of cation i in ferrite F:57 8 ACS Paragon Plus Environment

Page 8 of 35

Page 9 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−∆𝐸𝑓 𝑣𝑎𝑐

𝐷𝑖 ≈ [𝑉𝐹 ] 𝑟𝑖 = [𝑀]𝑒

(

𝑅𝑇

)

−𝐸𝑎,𝑖

𝑣𝑖 𝑒

(

𝑅𝑇

)

(1)

where Di is the diffusion constant of cation i, [VF] is the concentration of cationic vacancies in ferrite F, ri is the rate of hopping of cation i into vacancies, [M] is the concentration of cationic sites, ΔEf vac is the formation energy of a cationic vacancy, νi is the attempt frequency for cation i, and Ea,i is the activation energy for hopping of cation i. We note that vacancies in different sites will have different formation energies and so in practice the vacancy concentration would be calculated using the site concentrations and vacancy formation energies for each type of site in each material. Because DFT+U calculations result in erroneous energies of the metallic forms of d- and f-block metals, it fails to accurately predict the formation energies of cationic vacancies in metal oxides. Therefore, we calculate the approximate relative rates of cationic hopping through the ferrite spinels using ratios of the Arrhenius equations for the various hops and their associated activation energies:

𝑟𝑖,𝑗 =

−𝐸𝑎 𝑖) 𝑅𝑇 −𝐸𝑎 𝑗

𝑒𝑥𝑝( 𝑒𝑥𝑝(

𝑅𝑇

(2)

)

where ri,j is the approximate relative rate of cations i and j hopping between sites, and Ea is the associated activation energy. This estimate assumes that the pre-exponential factors for the two hopping events are relatively similar, which should be valid given the similarity of the masses of the hopping cations, bond strengths, surrounding material and diffusion pathways. 3.0 Results Here, we report the activation energies for the vacancy mediated diffusion of cations through MFe2O4 where M=Fe, Co and Ni. We have calculated the lattice constants of these three ferrite spinels using the HSE06 method and find that they agree closely with experiment, as 9 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 35

shown in Table 1. We also considered several atomic and spin configurations to calculate the hopping activation energies including the fully normal and inverse spinel atomic configurations, and ferromagnetic and anti-ferromagnetic spin configurations. As shown in Table 1, CoFe2O4 and NiFe2O4 are inverse spinels; as such these geometric configurations were used for the reaction path calculations. The lowest energy spin configurations were used to calculate the reaction energies.

Table 1: Structural parameters of the mixed metal ferrites Energy preference for spin orderinga

Lattice parameter Ferrite

Energy preference for inverse structure

Calculated

Experimental

Ferromagnetic

AntiFerromagnetic: Tet/Octb

AntiFerromagnetic: Layersc

Fe3O4

8.453 Å

8.396 Å58

1.08 eV

0.16 eV

0.00 eV

n/ad

CoFe2O4

8.387 Å

8.391 Å59

1.48 eV

0.00 eV

0.20 eV

0.30 eV

NiFe2O4

8.348 Å

8.339 Å59

1.47 eV

0.00 eV

0.32 eV

1.5 eV

aRelative

energies were calculated per formula unit, and are the energy differences between the lowest energy configuration and the configuration being compared. The energy of the lowest energy spin configuration is shown in bold font for each ferrite. bThe tetrahedral sites are spin up while the octahedral sites are spin down. cEach layer of cations separated by a plane of oxygen atoms is either spin up or down with the neighboring cation layer having the opposite spin. dBecause the difference between inverse and normal Fe O is only in the distribution of the oxidation states of the Fe cations 3 4 rather than the location of different elements, we could not force a normal structure as the system collapses into the inverse structure.

We investigated all three cations migrating through Fe3O4, but we only examined the diffusion of Co and Fe in CoFe2O4 and Ni and Fe in NiFe2O4, i.e. we only investigated the migration of explicitly present cations for the cases of CoFe2O4 and NiFe2O4. We refer to the migrating cation, i.e. the diffusant, as the active cation. We first briefly examined the vacancy site preference (tetrahedral vs. octahedral) before calculating the activation energies because, as described below, a sufficiently strong site preference exists for cation vacancies to occupy the


Page 11 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


octahedral sites whereas the equilibrium concentration of tetrahedral site vacancies is predicted to be extremely low. Complete projected density of states plots (PDOS) can be found in Figures SI1-SI7 of the supporting information (SI).

3.1 Site preferences of cation vacancies Our calculations predict that in ferrite spinels, cationic vacancies preferentially occupy octahedral sites over tetrahedral sites, as shown by the relative energies of vacancy site occupation in Fe3O4 we report in Table 2. This preference arises because the M-O bonds of the tetrahedral site cations are stronger on average than those occupying octahedral sites. Fe3+ cations occupy the tetrahedral sites and form strong Fe-O bonds. However, octahedral sites are occupied by a combination of Fe2+ and Fe3+ cations and Fe2+ cations form weaker Fe-O bonds than Fe3+ cations. Consequently, the average Fe-O bond strength of Fe cations occupying octahedral sites is weaker and thus the predicted formation energies for octahedral cation vacancies is lower than those of tetrahedral vacancies; formation energies of octahedral site vacancies is at least 1.01 eV lower than that of tetrahedral site vacancies, as shown in Table 2. This results in higher octahedral cation vacancy concentrations than tetrahedral vacancies where even at the relatively high temperature of 800°C used in chemical looping only ~1 in 55,000 vacancies occupy tetrahedral sites at equilibrium. Not only does thermodynamics impose a substantial driving force for vacancies to remain on octahedral sites, but kinetics also limit vacancy exchange between octahedral and tetrahedral sites. The activation barrier for a vacancy to hop from an octahedral site to a tetrahedral site must be at least the reaction energy of 1.01 eV for the case of a vacancy moving onto an Fe tetrahedral site. Thus, this barrier is at least 0.31 eV higher than the 0.70 eV barrier for octahedral-octahedral site hopping for the case of a vacancy moving onto an Fe site. Therefore, a simple estimate based on a Boltzmann equilibrium 11 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 35

distribution suggests that octahedral-octahedral site transitions are at least 28 times faster than octahedral-tetrahedral transitions at 800°C. This is consistent with experimental observations that suggest that cation diffusion through ferrites is dominated by octahedral-octahedral hops.60 Thus, we limit our study to the migration of cations between octahedral sites and do not explicitly calculate the barrier or reaction path for octahedral to tetrahedral, or tetrahedral to tetrahedral site diffusion.

Table 2: Vacancy formation energy difference (EF,Tet-EF,Oct) and activation energies for cation hopping in Fe3O4

aThe

Active cation

Vacancy formation energy difference (eV)a

Octahedral-octahedral hop activation energy (eV)

Fe

1.01

0.70

Co

2.22

0.76

Ni

3.97

1.21

energy required to exchange a vacancy on an octahedral site with a cation (Fe, Co or Ni) on a tetrahedral site.

3.2 Hopping activation energy in ferrites In this section, we report our calculated activation energies for octahedral vacancy mediated diffusion of active cations through Fe3O4, CoFe2O4 and NiFe2O4. We discuss the ferrites in the order given above and each active cation in the following order: Fe, then Co, then Ni. For CoFe2O4 and NiFe2O4 we only examine Fe and either Co or Ni diffusion, respectively. We begin by describing the migration of Fe through Fe3O4 and use this as a model reaction to describe subsequent cases in terms of their differences from that of Fe in Fe3O4.


Page 13 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.2.1 Diffusion through Fe3O4 We first give a brief description of the Fe3O4 cell before discussing the migration of individual cations. The pure ferrite spinel has the largest lattice spacing of those studied here with a lattice parameter of 8.45 Å. In the inverse spinel Fe3O4 octahedral sites are randomly occupied by Fe2+ and Fe3+ cations and distinguishing their oxidation state based on the structure alone is not possible; however, a Bader charge analysis allows us to determine the cation oxidation states. Of the 47 cations in the Fe3O4 supercell with one Fe vacancy, 33 have been oxidized by ~1.80 e, while 14 have been oxidized by ~1.66 e, as

Figure 3: Reaction pathway for octahedral-octahedral migration in Fe3O4. The top panel shows the reaction path energies for Fe, Co and Ni cations while the bottom set of panels shows the geometries of the migrating cation at unique positions along the reaction path. The three representations show the initial position of the cation in the octahedral site (left), the cation in the trigonal transition state geometry (middle) and the cation in the tetrahedral meta-stable state geometry (right). The reaction coordinate labels of the bottom panel correspond with those of the diffusion potential energy plot. The large blue, large gold and small red spheres represent the migrating cation, Fe cations, and O anions, respectively. The dotted circles and green arrows indicate cationic vacancies and the direction of the migrating cation along the reaction path, respectively.

estimated by the Bader analysis. The numbers of cations with Bader charges of 1.80 e and 1.66 e are equal to the expected numbers of cations with +3 and +2 oxidation states, suggesting that these cations can be assigned formal oxidation states of +3 and +2, respectively. This assessment agrees with the fact that the fully occupied supercell is comprised of thirty-two +3 cations and sixteen +2 cations and that, when an Fe is removed from Fe3O4, two of the +2 cations should oxidize to +3 oxidization states to maintain the O formal charge of 2. The Bader charge analysis also shows that the Fe cations nearest the Fe vacancy are oxidized by 1.80 e, suggesting that the electron density redistributes to produce Fe3+ ions nearest the Fe vacancy while all of the



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 35

Fe2+ cations are located further from the vacancy. It is also worth noting that the O anions neighboring the Fe vacancy are oxidized by ~0.1 e relative to the other O anions.

3.2.1.1 Fe diffusion in Fe3O4

Iron cations move from octahedral sites to neighboring octahedral vacancies by first hopping to an intermediate meta-stable tetrahedral interstitial site. This involves an activation energy of 0.70 eV and a reaction energy to form the tetrahedral intermediate of 0.65 eV, as shown in Figure 3. At the TS, the Fe cation is coordinated to three O atoms in a trigonal planar configuration. From the metastable tetrahedral site intermediate, hopping to the initial vacancy or the one formed by the hop to the tetrahedral intermediate are equivalent and the reaction potential energy surface is symmetric, with activation energies of 0.05 eV from the tetrahedral intermediate to the two neighboring octahedral vacancies. At temperatures below 800°C, the difference in these activation barriers of 0.65 eV indicates that the Fe cation hops out of the metastable state faster than it hops to the metastable state by at least three orders of magnitude. Thus, the time spent in the metastable intermediate state is insignificant, and we therefore do not discuss the rate of leaving the metastable state in the remainder of this work. The Fe octahedral site neighboring an octahedral vacancy is distorted with O-Fe bond lengths of 1.94, 1.99, 2.04, 2.04, 2.13 and 2.21 Å. At the TS, the Fe cation resides in a distorted trigonal planar configuration bound to three O atoms with two O-Fe bond lengths of 1.89 Å while one is elongated to ~2.02 Å. At the trigonal planar TS not only have three O-Fe bonds been broken, but the remaining three O-Fe bond lengths have shortened, causing ionic repulsion of the electrons localized on the diffusing Fe and its neighboring O atoms. The meta-stable tetrahedral site is also severely distorted. The Fe cation of the tetrahedral intermediate is


Page 15 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


coordinated to two O atoms with bond lengths of ~1.89 Å, which is the Fe-O bond length for tetrahedral Fe in Fe3O4, while the other two Fe-O bonds are elongated to 2.10 and 2.13 Å. This distortion stems from the interstitial nature of the site and the presence of the two neighboring octahedral vacancies. These structures are shown in the bottom panel of Figure 3. Throughout the remaining discussion, the initial, transition and meta-stable interstitial states will be referred to as the octahedral, trigonal and tetrahedral sites, respectively. As the Fe cation hops from the initial octahedral site, the surrounding electron density redistributes such that the migrating cation is partially reduced. As described above, Fe3+ ions neighbor Fe vacancies, so the migrating Fe cation begins its hop in the +3 formal oxidation state. However, the hop of Fe from the octahedral site to the tetrahedral interstitial involves a 0.3 e reduction of the cation according to the Bader charge analysis which estimates that its charge changes from +1.8 e to +1.5 e. Thus, the Fe cation in the tetrahedral intermediate is the most reduced cation in the system by ~ 0.1 e

Figure 4: The total and PDOS for the migration of Fe through Fe3O4 where panels a, b and c show the DOS of the octahedral, trigonal transition state and tetrahedral meta-stable state, respectively. The vertical dotted-dashed line shows the Fermi energy of the system. The total DOS is scaled to show both the total and PDOS on the same graph. The occupation of the dorbitals of the hopping Fe cation is indicated in d. Here, the dashed arrow indicates the electron that the migrating Fe3+ cation gains as it hops from the octahedral site.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 35

and it therefore possesses a formal charge of +2. The majority of the charge transfer occurs as the Fe moves from the octahedral site to the transition state, which is oxidized by only 0.01 e relative to the tetrahedral site. The majority (~68%) of the charge gained by the migrating Fe is contributed by other Fe ions in the system, with the remainder contributed by the O atoms of the supercell. As the Fe cation hops from an octahedral site into a neighboring Fe vacancy, the dorbitals rearrange, as shown in the PDOS depicted in Figure 4. A schematic of the ordering and occupation of the orbitals is shown in Figure 4d. The complete PDOS for the migrating Fe is shown in Figure SI 1. For Fe3O4 with an octahedral vacancy, the filled states of the valence band associated with the migrating Fe cation in the octahedral site are primarily composed of t2g and eg orbitals, where the anti-bonding t2g orbitals occupied by spin up electrons are lower in energy than the anti-bonding eg orbitals, as expected for an octahedral site. A spin down electron also occupies the octahedral t2g orbital of the migrating Fe cation; this state is higher in energy than its spin up t2g counterparts because unlike the spin up electrons, it has few exchange interactions with nearby like-spin electrons. As the Fe cation hops, the crystal field evolves along the reaction coordinate to change the relative energies of the t2g and eg orbitals of the hopping Fe cation as it moves from the octahedral state to the tetrahedral state. When the active Fe cation occupies the tetrahedral site, the t2g and eg orbitals reorder such that the t2g orbitals now lie above the eg states. Similar to the spin down electron occupying the t2g octahedral orbital, significant electron density is associated with the spin down eg orbital and its lower number of exchange interactions also makes it highenergy. At the trigonal site, no distinct separation of the orbitals into any clear grouping consistent with linear, trigonal planar or other traditional crystal field theory symmetry is


Page 17 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


present.

This is shown for the eg and t2g states in

Figure 4b and for the total projection in SI Figure SI 1. Instead, all of the individual d orbitals from across the depth of the valence band contribute to the density of states of the TS. This suggests that the trigonal planar TS is distorted enough towards the linear geometry that little crystal field interactions exist to break the dorbital degeneracy normally associated with either trigonal planar or linear geometries. Therefore, this geometry and associated electronic structure represents a transition point, where the respective increases and decreases in the energy of the t2g and eg orbitals are equal, and the d orbitals are roughly degenerate.

3.2.1.2 Co diffusion in Fe3O4

Analogous to the case of Fe migration, the cobalt cation migrates from one octahedral site to another through a tetrahedral metastable site with an activation barrier of 0.77 eV and energy to form the tetrahedral intermediate of 0.63 eV. Once again, the TS is characterized by the Co atom being bound to three O atoms in a distorted trigonal planar arrangement with Co-O bond lengths of 1.87, 1.87 and 1.95 Å. The

Figure 5: The total and PDOS for the migration of Co through Fe3O4 with panels a, b and c showing the octahedral, trigonal transition state and tetrahedral meta-stable state, respectively. The vertical dotted-dashed line indicates the Fermi energy of the system. The total DOS has been scaled to show both the total and PDOS on the same graph. Panel d indicates the occupation of the dorbitals of the hopping Co cation.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 35

energy of the Co cation in the metastable tetrahedral intermediate (0.63 eV) is slightly lower than that of the diffusing Fe cation (0.65 eV), while the activation energy for Co migration (0.77 eV) is slightly higher than that of Fe migration (0.70 eV). However, within the typical errors associated with the electronic structure methods employed, the Fe and Co cations have equal reaction and activation energies. While the migration path of the Co atom closely mimics that of Fe migration, the oxidation state behavior of the migrating Co differs. Unlike Fe atoms, which are in the Fe3+ oxidation state when neighboring a cationic vacancy, Co atoms at these sites remain Co2+ cations based on the Bader calculated oxidation state of +1.40, which is the same as the Co2+ cations in CoFe2O4. This result is not unexpected as Fe preferentially oxidizes over Co. Although the Co cation has a +2 formal oxidation state, it reduces slightly as it migrates; it reduces by 0.09 e at the trigonal state and 0.05 e at the tetrahedral site relative to the octahedral site. This is significantly less than the negative charge gained by a migrating Fe atom. We attribute the slight change in charge to the fact that the Co cation is already in the +2 oxidation state and that Co very rarely adopts a +1 oxidation state. As in the case of the migrating Fe cation, the changing crystal field along the reaction coordinate of Co cation hopping reorders the t2g and eg orbitals of the hopping Co cation as it moves from the octahedral state to the tetrahedral state, as shown in Figure 5. Additionally, the density of states at the TS is also characterized by a roughly equal contribution of the d-orbitals from across the valence band. This suggests that the electronic behavior of the Co cation is similar to that of the diffusing Fe atom at the TS.


Page 19 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.2.1.3 Ni diffusion in Fe3O4 Ni cations migrate through Fe3O4 along an analogous path to those of Fe and Co described above with an octahedral to interstitial tetrahedral reaction energy of 0.92 eV and an activation energy of 1.21 eV. As the Ni ion proceeds through the trigonal planar TS, the Ni-O bonds are more distorted from the trigonal site relative to the Fe and Co cases; two of the Ni-O bonds shorten to 1.84 Å, while one lengthens to 1.97 Å. Unlike Fe and Co cation migration, which have relatively similar reaction and activation energies, Ni diffusion has a considerably higher activation barrier and reaction energy. Therefore, Fe and Co cations hop to neighboring octahedral vacancies ~450 and 200 times faster than the Ni cation at 800 °C. The oxidation state of the migrating Ni cation is more similar to Co than Fe. Bader charge analysis predicts that the Ni cation in Fe3O4 has a Bader charge of +1.32. By comparison to the Bader charge of Ni atoms in NiFe2O4, we assign the migrating Ni atom a formal oxidation state of +2 using similar logic to that described in Section 3.2.1.2. As the Ni cation hops it reduces by less than 0.01 electrons. This is expected

Figure 6: The total and PDOS for the migration of Ni through Fe3O4. Panels a, b and c show the octahedral, trigonal transition state and tetrahedral meta-stable state, respectively. The vertical dotteddashed line marks the Fermi energy of the system. The total DOS is scaled to show both the total and PDOS on the same graph. Panel d indicates the occupation of the d-orbitals of the hopping Ni cation.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 35

because Ni, like Co, is not prone to adopting the +1 oxidation state. The lower extent of reduction of the Ni cation as it hops relative to the Co cation arises because Ni (Bader charge of +1.32) is already more reduced than Co (Bader charge of +1.40) and, therefore, less driving force exists to reduce it as it moves along the entrance channel of the hopping potential energy surface towards the TS and tetrahedral metastable site. Crystal field theory and the analogous analysis for Fe and Co migration described above suggest that the Ni cation in the octahedral and tetrahedral configurations exhibits a clear separation between the eg and t2g states, as shown in Figure 6. However, unlike the Fe and Co cations, a significant down spin electron density localizes on the Ni atom. In the octahedral configuration, this spin down density consists primarily of the t2g orbitals as the three spin down electrons normally associated with a Ni2+ ion occupy the three lower energy t2g orbitals. In the tetrahedral configuration, spin down electron density contributes to both the lower energy eg orbitals and the higher energy t2g orbitals. This is consistent with the promotion of one electron into a higher energy t2g orbital of the tetrahedral site as illustrated schematically in Figure 6d.

3.2.2 Co and Fe Diffusion in CoFe2O4 In CoFe2O4, the Co and Fe cations are nominally in the +2 and +3 formal oxidation states, respectively, and have Bader charges of roughly +1.40 and +1.85. The CoFe2O4 lattice is slightly smaller than the Fe3O4 lattice which have lattice parameters of 8.39 Å and 8.45 Å, respectively. As in the case of Fe3O4, the Fe cations neighboring the Fe vacancy are slightly more oxidized (by about 0.1 e) than the Fe cations further removed from the Fe vacancy. We first discuss Fe migration, then Co migration in CoFe2O4. Because cation hopping behavior in CoFe2O4 is similar to that of Fe3O4, we will only highlight the differences between them.


Page 21 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.2.2.1 Fe Diffusion in CoFe2O4

The Fe cation diffuses by moving through the octahedral–trigonal–tetrahedral path described above with an activation energy of 0.96 eV and an energy to form the tetrahedral intermediate of 0.72 eV. As it hops, the cation reduces, going from a Bader charge of +1.92 to charges of +1.82 and +1.83 at the TS and meta-stable states, respectively. This suggests that although the diffusing Fe cation reduces, it does not adopt the +2 oxidation state as it does in the Fe3O4 case. The O-Fe bond lengths again shorten from their distorted octahedral lengths (1.90, 1.91, 1.92, 2.23, 2.38 and 2.40 Å) to their lengths in the trigonal state (1.81, 1.85 and 1.90 Å). At the TS, the Fe-O bond lengths are roughly 0.07 Å shorter in CoFe2O4 than in Fe3O4. As above, the t2g and eg orbitals split considerably as shown in the PDOS of Figure 7. While the orbital ordering remains consistent with that of octahedral and tetrahedral configurations, the orbitals are substantially more separated in this case. We note that, unlike the Fe3O4 case, the electrons of the hopping cation flip their spin to transition from

Figure 7: The total and PDOS for the migration of Fe through CoFe2O4. The octahedral, trigonal transition state and tetrahedral meta-stable state are shown in panels a, b and c, respectively. The vertical dotted-dashed line indicates the Fermi energy of the system. The total DOS has been scaled to fit both the total and PDOS on the same graph. Panel d shows the occupation of the dorbitals of the hopping Fe cation.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 35

predominately spin up at the octahedral site (µ = 4.12, where µ = number of spin up – spin down electrons) to predominately spin down at the transition and metastable states (µ= -4.03 and -4.00, respectively). This suggests that as the Fe cation progresses along the entrance channel towards the TS a spin flip minimizes the energy of the TS and meta-stable positions. Due to the high-spin nature of the material and the large spinorbit coupling of heavy atoms, this transition should be facile. As in the previous cases, no obvious breaking of the d-orbital degeneracy at the TS occurs.

3.2.2.2 Co Diffusion in CoFe2O4

The Co cation hops by moving along an analogous reaction path to that described above with a barrier of 0.83 eV and energy to form the tetrahedral intermediate of 0.67 eV. At the TS, the Co-O bond lengths are again shorter relative to those of its counterpart in Fe3O4, having bond lengths of only 1.84, 1.86, and 1.94 Å compared to 1.87, 1.87 and 1.95 Å. Similar to the case of Co in Fe3O4, minimal reduction of the hopping Co occurs; Co gains only 0.10 electron as it moves from the octahedral site

Figure 8: The total and PDOS for the migration of Co through CoFe2O4. Panels a, b and c show the octahedral, trigonal transition state and tetrahedral meta-stable state, respectively. The vertical dotteddashed line marks the system Fermi energy. The total DOS is scaled to fit both the total and PDOS on the same graph. Panel d shows the occupation of the dorbitals of the hopping Co cation.


Page 23 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(+1.45 Bader charge) through the trigonal planar TS (+1.35 Bader charge) and 0.05 e as it moves to the metastable site (+1.40 Bader charge). The distribution of electronic states associated with the migrating Co cation in CoFe2O4 closely matches that of the migrating Co in Fe3O4, as shown by the PDOS of Figure 8. This indicates that, in contrast to the hopping Fe ion, diffusing Co ions behave similarly between the two materials. Hence, we do not discuss Co hopping in CoFe2O4 further. 3.2.3 Cation Diffusion in NiFe2O4

In NiFe2O4, Ni takes on a +2 formal oxidation state while Fe adopts a +3 formal oxidation state, with Bader predicted charges of roughly +1.35 and +1.85, respectively. Once again, the Fe cations neighboring the Fe vacancy are slightly reduced (~0.1 e) relative to the Fe cations further removed from the vacancy. The similarity of the Fe Bader charge to that of the Fe cations in CoFe2O4 and the result that Ni is slightly reduced compared to the Co of CoFe2O4 suggests that the Ni-O bonds are slightly more covalent than the Co-O bonds, as expected. The NiFe2O4 spinel has the shortest lattice parameter of the ferrites considered here with a lattice parameter of 8.34 Å compared to the 8.39 and 8.45 Å lattice parameters of CoFe2O4 and Fe3O4, respectively. The contraction of the lattice as Fe is substituted with Co or Ni atoms is consistent with the relative sizes of the three ions. Once again, we briefly discuss Fe and then Ni cation migration and only highlight the differences from the previous cases.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3.2.3.1 Fe Diffusion in NiFe2O4

Fe cations hop via the diffusion mechanism described above with an activation energy of 1.00 eV and an energy to form the tetrahedral intermediate of 0.80 eV. These are the highest reaction and activation energies for Fe migration of the three ferrites studied herein. Unsurprisingly, given the short lattice parameters, the Fe-O bond lengths at the TS are the shortest of the three ferrites, being only 1.80, 1.82 and 1.94 Å. As in the case of Fe diffusion in CoFe2O4, Fe reduces slightly from +1.92 to +1.82 and +1.83 at the TS and metastable sites, respectively; but this is insufficient to suggest that Fe3+ reduces to Fe2+ as it hops. The Fe PDOS, shown in Figure 9, is qualitatively similar to that of Fe in CoFe2O4 and is characterized by a large separation of the t2g and eg states. However, unlike in CoFe2O4, the electrons associated with the hopping Fe cation remain spin up along the entire reaction path. Figure 9: The total and PDOS for the migration of Fe through NiFe2O4. Panels a, b and c show the octahedral, trigonal transition state and tetrahedral meta-stable state, respectively. The vertical dotteddashed line shows the system Fermi energy. The total DOS is scaled to show both the total and PDOS on the same graph. Panel d indicates the occupation of the dorbitals of the hopping Fe cation.


Page 24 of 35

Page 25 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.2.3.2 Ni Diffusion in NiFe2O4

Ni migration in NiFe2O4 has the highest activation energy (1.40 eV) and energy to form the tetrahedral intermediate (1.32 eV) of any of the cations and ferrites considered in this work. While the Ni-O bond lengths shorten from 1.97, 2.01, 2.04, 2.04, 2.06, and 2.06 Å to 1.84, 1.90 and 1.95 Å at the TS, the Ni-O bond lengths are not much shorter at the TS than those of the TS of Fe3O4 with Ni-O bond lengths of 1.88, 1.88 and 1.96 Å. Only the shortest Ni-O bond length is significantly shorter. As in the case of Ni hopping in Fe3O4, the Ni cation reduction along the hopping path is insignificant, gaining less than 0.06 electrons. The PDOS of Figure 10 shows that a Ni cation diffusing through NiFe2O4 exhibits the same t2g and eg ordering behavior in the octahedral, trigonal planar and tetrahedral positions we described above for Ni hopping in Fe3O4 in Section 3.2.1.3.

4.0 Discussion Based on the results described above, we suggest that the reaction and activation energies of

Figure 10: The total and PDOS for the migration of Ni through NiFe2O4. Panels a, b and c show the octahedral, trigonal transition state and tetrahedral meta-stable state, respectively. The vertical dotteddashed line marks the system Fermi energy. The total DOS has been scaled to fit both the total and PDOS on the same graph. Panel d shows the occupation of the dorbitals of the hopping Ni cation.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 35

cation migration in ferrites are governed by the occupation and energies of the t2g and eg states of the active cation, and the O-cation bond lengths at the TS. We will briefly discuss each of these effects and then the implications for the relative migration rates of Fe, Co and Ni cations through ferrites.

4.1 Hopping energies of Fe, Co and Ni Cations The

cations

studied

herein

have

relative activation energies for hopping of Co2+ ≈ Fe2+ < Fe3+ < Ni2+ within a given ferrite, as shown in Figure 11 and Table 3. This suggests that the identity and oxidation state of the diffusing cation affects the diffusion rate. First, we discuss the effect of cation identity, and then the effect of the oxidation

state

on

cation

hopping.

As

described above and shown in Figure 11a, as the cations move from the octahedral site to the tetrahedral site the relative energies of the

Figure 11: a) The filling of the t2g and eg orbitals for the Fe, Co, and Ni cations and b) the activation energies for the cations in the various ferrites studied herein.

t2g and eg orbitals switch. In the octahedral position, the three low energy t2g orbitals are available for occupation, while in the tetrahedral configuration only the two eg orbitals are low energy. For the metal centers migrating as +2 cations, the Fe, Co and Ni respectively have 6, 7, and 8 valence electrons, which must be accommodated in the five d-orbitals of the cation. The relative rates of cation hopping can be understood to arise from the number of electrons which must be promoted into higher energy orbitals of the transition and tetrahedral intermediate states. In the cases of 26 ACS Paragon Plus Environment

Page 27 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Fe2+ and Co2+, only one electron must be promoted into a higher energy orbital during the octahedral to tetrahedral transition, while two electrons must be promoted in the case of Ni2+, as illustrated in Figure 11a. The larger number of high energy electrons associated with the Ni2+ cation indicates that the Ni2+ transition and tetrahedral states will be higher in energy than those of Co2+ and Fe2+. This is reflected in the higher activation and meta-stable state energies for Ni2+ migration. Additionally, because Co2+ and Fe2+ require the same number of electronic promotions, their TS and tetrahedral intermediate energies are roughly the same. This analysis, based on the calculated splitting of the t2g and eg orbitals, is in agreement with previous work which invoked crystal field theory to explain relative cationic diffusion preferences.37-40 The effect of the oxidation state on the activation energy is most apparent by comparing the activation and tetrahedral intermediate energies of Co and Fe in Fe3O4 and CoFe2O4. In Fe3O4 both Co and Fe diffuse as +2 cations and have similar activation energies of 0.77 and 0.70 eV, respectively. In contrast, in the CoFe2O4 ferrite Co and Fe cations respectively diffuse as +2 and +3 cations and have less similar activation energies of 0.83 and 0.96 eV. We attribute this bifurcation in behavior to the different oxidation states of the Fe cation in Fe3O4 and CoFe2O4. The difference in activation energies for Fe2+ and Fe3+ cation diffusion stems from the relative strengths of the three Fe-O bonds that dissociate as Fe hops from its octahedral position to the TS; Fe3+-O bonds are stronger than Fe2+-O bonds, leading to a larger barrier for Fe3+ hopping in CoFe2O4. Despite the energy penalty associated with dissociating the stronger M-O bonds of +3 cations, the energy penalty for promoting Ni2+’s additional valence electron into the higher energy t2g orbitals of the transition and tetrahedral intermediate states is even larger because Ni2+’s activation energy for hopping is greater than that of Fe3+. This suggests that cations with a given oxidation state, including those not considered here, that minimize electron promotion to



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 35

high energy t2g orbitals, i.e. those left of Fe on the periodic table, have lower activation energies for diffusion in ferrites than those studied here. Conversely, cations with more than six delectrons require promotion of additional electrons into the high energy t2g orbitals during hopping, and will consequently possess higher barriers to hopping.

4.2 Cation hopping energies in different ferrites For a given cation, the hopping barriers in different ferrites increase with decreasing lattice parameter, giving an ordering of the activation energies of: Fe3O4 < CoFe2O4 < NiFe2O4. This is shown graphically in Figure 11b. This results from the smaller lattices inherently having shorter cation-O bond lengths, which further contracts the already compressed nature of the trigonal planar TS structure. The shorter cation-O bond lengths increase electrostatic repulsion and steric hindrance and, therefore, increase the activation energies for hopping.

4.3 Relative migration rates of the cations Our calculated approximate relative cation hopping rates are reported in Table 3 for cation hopping at 800°C. The lattice size significantly affects the relative hopping rates of cations, where smaller lattices result in substantially higher activation energies and slower hopping rates, as shown in Figure 11b and Table 3. This ordering is consistent with the relative rates of oxidation of Fe3O4, CoFe2O4, and NiFe2O4 observed in chemical looping of these ferrites.12, 13 Additionally, the prediction that Fe hops through Fe3O4 slightly faster than Co but that Co hops through CoFe2O4 faster than Fe is consistent with experimental cation tracer diffusion measurements by Lu et al.14 Therefore, if faster cation diffusion is desired, as in the case of the oxidation step of chemical looping, the material could be doped either with larger cations that expand the ferrite lattice, such as Mn, Cr or In; or cations with fewer d electrons,


Page 29 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


such as Mn, Cr, or V. Conversely, if cation diffusion is undesirable, the material could be doped with cations that retard hopping by either compressing the ferrite lattice, such as Al or Zn; or with cations with more d-electrons, such as Cu or Zn.

Table 3: Relative rates (and activation barriers) of cation hopping in mixed metal ferrites at 800°C

Diffusing cation Fe Co Ni

Fe3O4 4,400 (0.70 eV) 2,080 (0.77 eV) 10 (1.21 eV)

CoFe2O4 190 (0.96 eV) 930 (0.83 eV) n/a

NiFe2O4 125 (1.00 eV) n/a 1 (1.40 eV)

5.0 Conclusions We have investigated cation diffusion in mixed-metal ferrite spinels using density functional theory and employed crystal field theory to explain the results. We find that the relative cation hopping barriers order as: Co2+ ≈ Fe2+ < Fe3+ < Ni2+ and that smaller ferrite lattices result in higher cation hopping barriers, such that the same migrating ion diffuses faster in one material over another with the following relation: cation diffusion in Fe3O4 is faster than in CoFe2O4 and cation diffusion in CoFe2O4 is faster than in NiFe2O4 given cationic vacancy concentrations that do not differ significantly. The relative rates of cation hopping between the cations is dictated by two factors: 1) the number of electrons which must be promoted to high energy t2g orbitals of the cation in the TS and tetrahedral meta-stable state as the cation hops, and 2) the oxidation state of the diffusing cation. The hopping barrier increases with the number of electrons that must be promoted to the t2g orbitals of the transition and tetrahedral intermediate states and with the oxidation state of the cation. Of these two effects, electronic promotion has a larger effect. The inverse relationship between lattice spacing and diffusion barrier stems from the extent of compression of the M-O bonds at the TS, where smaller lattices exhibit greater compression and hence higher activation energies. Our results are consistent with experimental 29 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 35

oxidation rates of mixed metal ferrites undergoing chemical looping. By understanding the effects of lattice spacing, oxidation state and especially the required number of electron promotions on the rate of cation diffusion, new materials can be designed which either accelerate or impede cation migration depending on the application. Accelerated cation diffusion could enable faster reactions for processes such as chemical looping or longer materials lifetimes for ferrite materials prone to oxidation by slowing cation migration.

6.0 Supporting Information The Supporting Information is available free of charge on the ACS Publications website. It includes the individual d-orbital PDOS plots for the octahedral, trigonal planar, and tetrahedral sites for each cationic hopping and ferrite combination studied.

7.0 Acknowledgements This paper presents results from an NSF project (award number CBET-1433521) competitivelyselected under the solicitation “NSF 14-15: NSF/DOE Partnership on Advanced Frontiers in Renewable Hydrogen Fuel Production via Solar Water Splitting Technologies”, which was cosponsored by the National Science Foundation, Division of Chemical, Bioengineering, Environmental, and Transport Systems (CBET), and the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Fuel Cell Technologies Office. The authors also gratefully acknowledge NSF and DOE for support of this research via NSF grant CBET0966201 and DOE grant DE-EE0006671. CLM would like to thank the Department of Education for support through a Renewable and Sustainable Energy Graduate Assistance in Areas of National Need (GAANN) Fellowship. This work utilized the Janus supercomputer, which is supported by the NSF (CNS-0821794) and the University of Colorado Boulder. The Janus


Page 31 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


supercomputer is a joint effort of the University of Colorado Boulder, the University of Colorado Denver and the National Center for Atmospheric Research.

8.0 References 1. Tinkle, M.; Dumesic, J., Isotopic Exchange Measurements of the Rates of Adsorption/Desorption and Interconversion of CO and CO2 over Chromia-Promoted Magnetite: Implications for Water-Gas Shift. J. Cat. 1987, 103, 65-78. 2. Kotani, M.; Koike, T.; Yamaguchi, K.; Mizuno, N., Ruthenium Hydroxide on Magnetite as a Magnetically Separable Heterogeneous Catalyst for Liquid-Phase Oxidation and Reduction. Green Chem. 2006, 8, 735-741. 3. Huff, G. A.; Satterfield, C. N., Intrinsic Kinetics of the Fischer-Tropsch Synthesis on a Reduced Fused-Magnetite Catalyst. Ind. Engin. Chem. Process Design and Development 1984, 23, 696-705. 4. Jang, B.; Park, M.; Chae, O. B.; Park, S.; Kim, Y.; Oh, S. M.; Piao, Y.; Hyeon, T., Direct Synthesis of Self-Assembled Ferrite/Carbon Hybrid Nanosheets for High Performance LithiumIon Battery Anodes. JACS 2012, 134, 15010-15015. 5. Allendorf, M. D.; Diver, R. B.; Siegel, N. P.; Miller, J. E., Two-Step Water Splitting Using Mixed-Metal Ferrites: Thermodynamic Analysis and Characterization of Synthesized Materials. Energy & Fuels 2008, 22, 4115-4124. 6. Svoboda, K.; Siewiorek, A.; Baxter, D.; Rogut, J.; Pohořelý, M., Thermodynamic Possibilities and Constraints for Pure Hydrogen Production by a Nickel and Cobalt-Based Chemical Looping Process at Lower Temperatures. Energy Conversion and Manag. 2008, 49, 221-231. 7. Fletcher, E. A.; Moen, R. L., Hydrogen- and Oxygen from Water. Science 1977, 197, 1050-1056. 8. Kuo, Y.-L.; Hsu, W.-M.; Chiu, P.-C.; Tseng, Y.-H.; Ku, Y., Assessment of Redox Behavior of Nickel Ferrite as Oxygen Carriers for Chemical Looping Process. Ceramics Inter. 2013, 39, 5459-5465. 9. Muhich, C. L.; Ehrhart, B. D.; Al-Shankiti, I.; Ward, B. J.; Musgrave, C. B.; Weimer, A. W., A Review and Perspective of Efficient Hydrogen Generation Via Solar Thermal Water Splitting. Wiley Interdisciplinary Reviews: Energy and Environment 2015. 10. Muhich, C. L.; Evanko, B. W.; Weston, K. C.; Lichty, P.; Liang, X.; Martinek, J.; Musgrave, C. B.; Weimer, A. W., Efficient Generation of H2 by Splitting Water with an Isothermal Redox Cycle. Science 2013, 341, 540-542. 11. Muhich, C. L.; Ehrhart, B. D.; Witte, V. A.; Miller, S. L.; Coker, E. N.; Musgrave, C. B.; Weimer, A. W., Predicting the Solar Thermochemical Water Splitting Ability and Reaction Mechanism of Metal Oxides: A Case Study of the Hercynite Family of Water Splitting Cycles. Energy & Environ. Science 2015, 8, 3687-3699. 12. Aston, V. J.; Evanko, B. W.; Weimer, A. W., Investigation of Novel Mixed Metal Ferrites for Pure H2 and CO2 Production Using Chemical Looping. Inter. J. Hydrogen Energy 2013, 38, 9085-9096. 31 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 35

13. Fan, Y.; Siriwardane, R., Novel New Oxygen Carriers for Chemical Looping Combustion of Solid Fuels. Energy & Fuels 2014, 28, 2248-2257. 14. Fu-Hsing, L.; Tinkler, S.; Diekmann, R., Point Defects and Cation Tracer Diffusion in (CoxFe1−X)3−ΔO4 Spinels. Solid State Ionics 1993, 62, 39-52. 15. Hossain, M. M.; de Lasa, H. I., Chemical-Looping Combustion (CLC) for Inherent Separations— A Review. Chem. Eng. Science 2008, 63, 4433-4451. 16. Mattisson, T.; Lyngfelt, A.; Cho, P., The Use of Iron Oxide as an Oxygen Carrier in Chemical-Looping Combustion of Methane with Inherent Separation of CO2. Fuel 2001, 80, 1953-1962. 17. Scheffe, J. R.; Allendorf, M. D.; Coker, E. N.; Jacobs, B. W.; McDaniel, A. H.; Weimer, A. W., Hydrogen Production Via Chemical Looping Redox Cycles Using Atomic Layer Deposition-Synthesized Iron Oxide and Cobalt Ferrites. Chem. Mat. 2011, 23, 2030-2038. 18. Cocchi, S.; Mari, M.; Cavani, F.; Millet, J.M. M., Chemical and Physical Behavior of CoFe2O4 in Steam–Iron Process with Methanol. App. Cat. B: Environ. 2014, 152–153, 250-261. 19. Yang, H.; Xu, Z.; Fan, M.; Gupta, R.; Slimane, R. B.; Bland, A. E.; Wright, I., Progress in Carbon Dioxide Separation and Capture: A Review. J. Environ. Sci. 2008, 20, 14-27. 20. Adanez, J.; Abad, A.; Garcia-Labiano, F.; Gayan, P.; de Diego, L. F., Progress in Chemical-Looping Combustion and Reforming Technologies. Progress in Energy and Combustion Science 2012, 38, 215-282. 21. Li, F.; Kim, H. R.; Sridhar, D.; Wang, F.; Zeng, L.; Chen, J.; Fan, L. S., Syngas Chemical Looping Gasification Process: Oxygen Carrier Particle Selection and Performance. Energy & Fuels 2009, 23, 4182-4189. 22. Solunke, R. D.; Veser, G., Hydrogen Production Via Chemical Looping Steam Reforming in a Periodically Operated Fixed-Bed Reactor. Ind. Engin. Chem. Research 2010, 49, 11037-11044. 23. Bhavsar, S.; Najera, M.; Veser, G., Chemical Looping Dry Reforming as Novel, Intensified Process for CO2 Activation. Chem. Engin. Tech. 2012, 35, 1281-1290. 24. Gupta, P.; Velazquez-Vargas, L. G.; Fan, L.-S., Syngas Redox (SGR) Process to Produce Hydrogen from Coal Derived Syngas. Energy & Fuels 2007, 21, 2900-2908. 25. Nestl, S.; Voitic, G.; Lammer, M.; Marius, B.; Wagner, J.; Hacker, V., The Production of Pure Pressurised Hydrogen by the Reformer-Steam Iron Process in a Fixed Bed Reactor System. J. Power Sources 2015, 280, 57-65. 26. Wagner, C., Beitrag Zur Theorie Des Anlaufvorgangs. Z. Phys. Chem. B 1933, 21, 25-41. 27. Slattery, J. C.; Peng, K.-Y.; Gadalla, A. M.; Gadalla, N., Analysis of Iron Oxidation at High Temperatures. Ind. Engin. Chem. Research 1995, 34, 3405-3410. 28. Yuan, L.; Wang, Y.; Cai, R.; Jiang, Q.; Wang, J.; Li, B.; Sharma, A.; Zhou, G., The Origin of Hematite Nanowire Growth During the Thermal Oxidation of Iron. Mat. Sci. and Engin.: B 2012, 177, 327-336. 29. Chen, R. Y.; Yeun, W. Y. D., Review of the High-Temperature Oxidation of Iron and Carbon Steels in Air or Oxygen. Oxidation of Metals 2003, 59, 433-468. 30. Bohn, C. D.; Müller, C. R.; Cleeton, J. P.; Hayhurst, A. N.; Davidson, J. F.; Scott, S. A.; Dennis, J. S., Production of Very Pure Hydrogen with Simultaneous Capture of Carbon Dioxide Using the Redox Reactions of Iron Oxides in Packed Beds. Ind. Engin. Chem. Research 2008, 47, 7623-7630. 31. Feitknecht, W.; Gallagher, K. J., Mechanisms for the Oxidation of Fe3O4. Nature 1970, 228, 548-549. 32 ACS Paragon Plus Environment

Page 33 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


32. Li, F.; Sun, Z.; Luo, S.; Fan, L.-S., Ionic Diffusion in the Oxidation of Iron-Effect of Support and Its Implications to Chemical Looping Applications. Energy & Environ. Sci. 2011, 4, 876-880. 33. Dieckmann, R.; Schmalzried, H., Defects and Cation Diffusion in Magnetite (II). Berichte der Bunsengesellschaft für physikalische Chemie 1977, 81, 414-419. 34. Lu, F.-H.; Dieckmann, R., Point Defects and Cation Tracer Diffusion in (Co, Fe, Mn)3−ΔO4 Spinels: I. Mixed Spinels (CoxFe2yMny)3−ΔO4. Solid State Ionics 1992, 53–56, Part 1, 290-302. 35. Hendy, S.; Walker, B.; Laycock, N.; Ryan, M., Ab Initio Studies of the Passive Film Formed on Iron. Phys. Rev. B 2003, 67, 085407. 36. Hendy, S.; Laycock, N.; Ryan, M., Atomistic Modeling of Cation Transport in the Passive Film on Iron and Implications for Models of Growth Kinetics. J. Electrochem. Soc. 2005, 152, B271-B276. 37. Cox, M. G. C.; McEnaney, B.; Scott, V. D., A Chemical Diffusion Model for Partitioning of Transition Elements in Oxide Scales on Alloys. Philosophical Magazine 1972, 26, 839-851. 38. Batra, A. P.; Hernandez, J. P.; Slifkin, L. M., Crystal-Field Effects and the Migration of Transition-Metal Ions in Agcl. Phys. Rev. Let. 1976, 36, 876-878. 39. Crispin, K. L.; Van Orman, J. A., Influence of the Crystal Field Effect on Chemical Transport in Earth's Mantle: Cr3+ and Ga3+ Diffusion in Periclase. Phys. Earth and Planetary Interiors 2010, 180, 159-171. 40. Saha, S.; Bengtson, A.; Crispin, K. L.; Van Orman, J. A.; Morgan, D., Effects of Spin Transition on Diffusion of Fe2+ in Ferropericlase in Earth's Lower Mantle. Phys. Rev. B 2011, 84, 184102. 41. Miller, A., Distribution of Cations in Spinels. J. App. Phys. 1959, 30, S24-S25. 42. Henderson, C.; Charnock, J.; Plant, D., Cation Occupancies in Mg, Co, Ni, Zn, Al Ferrite Spinels: A Multi-Element EXAFS Study. J. Phys. Cond. Mat. 2007, 19, 076214. 43. Carta, D.; Casula, M. F.; Falqui, A.; Loche, D.; Mountjoy, G.; Sangregorio, C.; Corrias, A., A Structural and Magnetic Investigation of the Inversion Degree in Ferrite Nanocrystals MFe2O4 (M = Mn, Co, Ni). J. Phys. Chem. C 2009, 113, 8606-8615. 44. Walsh, A.; Wei, S.-H.; Yan, Y.; Al-Jassim, M. M.; Turner, J. A.; Woodhouse, M.; Parkinson, B. A., Structural, Magnetic, and Electronic Properties of the Co-Fe-Al Oxide Spinel System: Density-Functional Theory Calculations. Phys. Rev. B 2007, 76, 165119. 45. Kresse, G.; Furthmüller, J., Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B 1996, 54, 11169. 46. Kresse, G.; Furthmüller, J., Efficiency of Ab-Initio Total Energy Calculations for Metals and Semiconductors Using a Plane-Wave Basis Set. Comp. Mat. Sci. 1996, 6, 15-50. 47. Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized Gradient Approximation Made Simple. Phys. Rev. Let. 1996, 77, 3865. 48. Kresse, G.; Joubert, D., From Ultrasoft Pseudopotentials to the Projector AugmentedWave Method. Phys. Rev. B 1999, 59, 1758-1775. 49. Heyd, J.; Scuseria, G. E.; Ernzerhof, M., Hybrid Functionals Based on a Screened Coulomb Potential. J. Chem. Phys. 2003, 118, 8207-8215. 50. Heyd, J.; Scuseria, G. E.; Ernzerhof, M., Erratum: `Hybrid Functionals Based on a Screened Coulomb Potential'' [J. Chem. Phys. 118, 8207 (2003)]. J. Chem. Phys. 2006, 124, 219906-1.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 35

51. Hubbard, J., Electron Correlations in Narrow Energy Bands. Proc. Royal Soc. London Series a-Math. Phys. Sci. 1963, 276, 238-257. 52. Henkelman, G.; Arnaldsson, A.; Jónsson, H., A Fast and Robust Algorithm for Bader Decomposition of Charge Density. Comp. Mat. Sci. 2006, 36, 354-360. 53. Sanville, E.; Kenny, S. D.; Smith, R.; Henkelman, G., Improved Grid-Based Algorithm for Bader Charge Allocation. J. Comp. Chem. 2007, 28, 899-908. 54. Henkelman, G.; Uberuaga, B. P.; Jonsson, H., A Climbing Image Nudged Elastic Band Method for Finding Saddle Points and Minimum Energy Paths. J. Chem. Phys. 2000, 113, 99019904. 55. Henkelman, G.; Jónsson, H., A Dimer Method for Finding Saddle Points on High Dimensional Potential Surfaces Using Only First Derivatives. J. Chem. Phys. 1999, 111, 70107022. 56. Heyden, A.; Bell, A. T.; Keil, F. J., Efficient Methods for Finding Transition States in Chemical Reactions: Comparison of Improved Dimer Method and Partitioned Rational Function Optimization Method. J. Chem. Phys. 2005, 123, 224101. 57. Pornprasertsuk, R.; Ramanarayanan, P.; Musgrave, C. B.; Prinz, F. B., Predicting Ionic Conductivity of Solid Oxide Fuel Cell Electrolyte from First Principles. J. App. Phys. 2005, 98, 103513. 58. Wechsler, B. A.; Lindsley, D. H.; Prewitt, C. T., Crystal Structure and Cation Distribution in Titanomagnetites (Fe3-xTixO4). Am. Mineralogist 1984, 69, 754-770. 59. Carta, D.; Casula, M.; Falqui, A.; Loche, D.; Mountjoy, G.; Sangregorio, C.; Corrias, A., A Structural and Magnetic Investigation of the Inversion Degree in Ferrite Nanocrystals MFe2O4 (M= Mn, Co, Ni). J. Phys. Chem. C 2009, 113, 8606-8615. 60. Peterson, N. L.; Chen, W. K.; Wolf, D., Correlation and Isotope Effects for Cation Diffusion in Magnetite. J. Phys. Chem. of Solids 1980, 41, 709-719.


Page 35 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TOC:


First-Principles Analysis of Cation Diffusion in ... - ACS Publications

Recommend Documents