Calculated Stability and Structure of Nickel Ferrite Crystal Surfaces in

Article pubs.acs.org/JPCC

Calculated Stability and Structure of Nickel Ferrite Crystal Surfaces in Hydrothermal Environments Christopher J. O’Brien,* Zs. Rák, and Donald W. Brenner Department of Materials Science and Engineering, North Carolina State University, Raleigh, North Carolina 27695, United States ABSTRACT: A comprehensive theoretical investigation of nickel ferrite (NiFe2O4) surfaces is undertaken to understand the structure and stability of nanocrystallites that would be present under conditions of hydrothermal synthesis (HTS). In particular, the focus is on conditions characteristic to an operating pressurized water nuclear reactor (PWR). Solid− liquid equilibrium is assumed between bulk nickel ferrite and the aqueous environment saturated with respect to nickel ferrite. A theoretical framework is developed in which the surface energies are evaluated in terms of concentrations of aqueous metal cations, pH, temperature, and pressure. The energies of the bare and water terminated surfaces are calculated and discussed. Surfaces that have more metal cations exposed are found to be more stable. Water adsorption on the nickel ferrite surfaces is an exothermic process, with the magnitude of exothermicity decreasing as a function of temperature. At temperatures relevant to operating PWRs, the energy gain due to water adsorption is negligible. The most stable surfaces are along the (111) planes and are predicted to have negative surface energies. This indicates that, in an operating PWR, nickel ferrite tends to increase its surface area, giving rise to a highly porous thermodynamic ground state. This provides an explanation for the porous nature of the nickel ferrite deposits observed to form on PWR fuel rods.

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INTRODUCTION There is an increasing interest in nickel ferrite (NiFe2O4) nanoparticles due to their potential application in technologies that include gas sensing, magnetic data storage, catalysis, and solid-oxide fuel cells. Nickel ferrite nanoparticles are synthesized through different physical and chemical methods; examples of the latter include hydrothermal reaction, and sol−gel combustion and coprecipitation.1−7 Nickel ferrite and other metal oxides also appear in deposits on heat transfer surfaces (i.e., fouling) in many industrial processes. For example, nickel ferrite is a primary component of porous oxide deposits that form on nuclear fuel rods during reactor operation.8−12 This includes both boiling water and pressurized water reactors (PWRs). Understanding and controlling the synthesis and deposition of nickel ferrite clusters and films start with a thorough understanding of their surface structure and energy, including the effect of aqueous environments. While first principles methods such as density functional theory (DFT) can be used to calculate relatively accurate surface energies, AB2O4 spinels provide challenges that can complicate the calculations. First, these compounds can have normal spinel structures in which the divalent and trivalent ions occupy tetrahedral sites and octahedral sites, respectively, or inverse structures where half of the trivalent cations occupy tetrahedral sites and the remaining divalent and trivalent ions are randomly distributed over octahedral sites. Second, low-index, bulk-terminated surfaces of AB2O4 spinels are Tasker type III (polar) surfaces13 that contain inherent dipoles normal to the surface that lead to divergent surface energies. Reconstructions, surface chemisorption, and charge rearrangement can remove the dipole, © 2014 American Chemical Society

although the choices of structures that both eliminate dipoles and produce low surface energies are not always clear. Third, DFT calculations are done at 0 K, while understanding processes such as hydrothermal synthesis (HTS) and formation of porous oxide deposits on PWR fuel rods requires surface structures and free energies in high temperature and pressure aqueous environments. Finally, nonstoichiometric surfaces that satisfy dipole and energy considerations require reference chemical potentials that depend on system conditions to calculate surface free energies. Analytic interatomic potentials have been used to characterize surfaces for Fe3O4,14−16 ZnCr2O4,17 and MgAl2O4,18,19 and similar calculations using first principles methods have been carried out for Co3O420,21 and Fe3O4.21−27 For these systems, all of which have a bulk spinel crystal structure, the calculations examined relative surface energies for various reconstructions, including surface vacancies and extra atoms. Compared to systems with the normal spinel structure, relatively little computational work has been done on surfaces of crystals with inverse spinel structures. Recently, Kumar et al.28 used DFT to calculate adsorption energies and dissociation barriers for water on (111) surfaces of NiFe2O4 with different Ni and O terminations. This surface was chosen based on experimental studies of nickel ferrite particles created by HTS that show a strong preference for (111) facets.1,2 Based on this study they conclude that water is more reactive toward the NiFe2O4 surface than the comparable surface of Fe3O4, a result that Received: January 9, 2014 Revised: February 13, 2014 Published: February 14, 2014 5414

dx.doi.org/10.1021/jp5002308 | J. Phys. Chem. C 2014, 118, 5414−5423

The Journal of Physical Chemistry C

Article

has important implications for catalytic and corrosion properties of nickel ferrite. The possibility of nonstoichiometric surfaces makes comparison of surface energies very difficult, as surface energy will depend on the species and phase chosen as the thermodynamic reservoir of the components. For example, Pentcheva et al.27 calculated a surface phase diagram for Fe3O4 that gives the stable surface structure as a function of temperature and oxygen pressure, while Guo and Barnard29 used calculated surface energies to construct a size-dependent phase diagram for Fe−O systems with different structures and stoichiometries that depends on the degree of water saturation. In this paper we use DFT to calculate energies of (111), (110), and (100) NiFe2O4 surfaces with different structures and terminations, and we then use a DFT-referenced thermodynamics scheme to reference the structures to aqueous conditions at temperatures, pressures, and species concentrations that represent those in pressured water nuclear reactors and used in HTS restricted to subcritical water conditions.30 Data is also given in the paper that can be used to determine surface energies for other conditions. Consistent with experiment, the calculations predict stable (111) facets under all conditions examined. In addition, under these conditions some of the surface energies are negative, similar to the results from recent calculations reported by Łodziana et al. for alumina.31

Equations 2 and 3 are solved using the ECPs derived from eq 1 and free energies of formation from the SUPCRT database.32,33 Further details are in ref 30. Details of the DFT calculations of the slab energies as described below are the same as those used to determine the effective chemical potentials in ref 30. The generalized gradient approximation (GGA) with effective rotationally invariant onsite Coulomb interactions as formulated by Dudarev34 was used in the Vienna ab-initio simulation package (VASP).35−38 In this simplified implementation, only the single parameter Ueff = U − J is used to describe the Coulombic repulsion. The values of Ueff utilized in the present investigation are 4.5 and 6 eV for Fe and Ni, respectively.29,39 The value of Ueff(Fe) = 4.5 eV has been chosen because it reproduces the experimental band gaps of hematite and goethite and also gives acceptable lattice parameters for iron oxides,29 while Ueff(Ni) = 6 eV reproduces the oxidation energy of Ni.39 Electrons within the ionic core were modeled with the projector augmented wave pseudopotentials40,41 included in the VASP. Electronic convergence was assumed when the energy difference between consecutive cycles was less than (1.5 × 10−7)n, where n is the number of ions in the system, and the geometry was optimized until the Hellmann−Feyman forces on each nucleus were less than 1.5 × 10−2 eV/Å. Surface energy calculations reported here were carried out using slab geometries, where periodicity was maintained in directions parallel to the surface and 15 Å of vacuum separated surface planes. Slab calculations for the (100), (011), (101), and (110) surfaces, illustrated in Figures 3−6, were performed with the outer three atomic layers allowed to move, while the remaining three internal layers were fixed. The exception is the (011)S surface (Figure 3c), where only the central layer was fixed. Calculations of the (111) surfaces (Figure 2) were conducted with the outer five layers allowed to relax, with the remaining layers being fixed. The total number of layers in the slab is illustrated in Figure 2. Partial occupancies of orbitals were set using 0.1 eV wide Gaussian smearing for all surface geometry optimizations. The use of symmetry was disabled for all surface calculations. The k-point mesh was reduced to a single k-point in the direction normal to the surface with the kgrid in the remaining directions scaled from that used for the bulk calculations. Surface supercells for the (001) surface were 8.41 × 8.41 Å2 with a k-point mesh of 5 × 5 × 1. The (110), (101), and (011) surface supercells were 8.41 × 5.94 Å with a k-point mesh of 5 × 7 × 1. All of the (111) surfaces were 5.94 × 5.94 Å2 with axes forming a 60° angle employing a Γcentered 7 × 7 × 1 k-point mesh. As mentioned above, nickel ferrite has an inverse spinel structure, with an equal number of Ni2+ and Fe3+ ions distributed over the octahedral sites, while the tetrahedral sites are occupied solely by Fe3+ ions. In the present study, the inverse spinel structure of nickel ferrite was modeled using the approach of Fritsch and Ederer,42 where the Ni2+ ions are distributed on the octahedral sites to maintain cubic symmetry but are reduced from the Fd3̅m space group (#227) to Imma (#74). With this ionic distribution, cuts along the (110), (011), and (101) planes produce different surfaces. The spin configuration that minimizes the energy for bulk nickel ferrite was also used for the surface calculations. In this configuration the spins of the octahedral Fe3+ and Ni2+ ions are aligned ferromagnetically, while the spins of tetrahedral and octahedral ions are aligned antiferromagnetically.42

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THEORETICAL METHODS The DFT-informed thermodynamics scheme has been described in detail elsewhere,30 and only a brief discussion is included here. In this scheme, a set of temperature-dependent effective chemical potentials (ECP) for each element in a group of oxides is determined by solving a set of linear equations of the form ° Δf G M (T ) = E M iOjHk (0 K) − iμM° (T ) − iOj Hk

−

j ° μ (T ) 2 O2

k ° μ (T ) 2 H2

(1)

where the EMiOjHk denote the energies of the metal oxides calculated from DFT, the μ°’s are the ECPs, and Δf G°MiOjHk(T) are temperature-dependent free energies of formation taken from experiment. The superscript naught on the latter denotes that the element is referenced to a thermodynamic standard state pressure of 1 bar (gas for O2 and H2, solid for oxides and metals). Data for the nine metal oxides (Fe2O3, FeO(OH), Fe3O4, NiO, ZnO, Co3O4, NiFe2O4, CoFe2O4, and ZnFe2O4) were used in eq 1 to set up an overdetermined system of linear equations that was solved to determine the ECPs for Fe, O, H, Ni, Zn, and Co. ECPs are determined for water, solvated Fe2+, Ni2+, and aqueous H2 by solving the following equations: (Δf G H2O(T , P))l = (μH O(T , P))l − 2

1 ° μ (T ) − μH° (T ) 2 2 O2 (2)

and (Δf G Mz+(T , P))aq = (μ M° z+ (T , P))aq − μM° (T ) z + μH° (T ) − z(μH° + (T , P))aq 2 2

(3) 5415



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so that solving eq 12 for μ°H2 and subtracting it from eq 11 yields the desired relationship:

Nickel ferrite is a challenging system in which to calculate surface free energies because the surface terminations may be nonstoichiometric, which can create a complicated relationship to their environment. The general form of the surface energy for nickel ferrite is given by

(ΔμH )aq = ΔμH 2

Because known concentrations are used below, it is more convenient to express eq 5 in terms of the chemical potential of ions and aqueous species. This can be accomplished by substituting eq 12 into 9 and solving for the chemical potential of the metal in the solid phase, so that z μM = (μ Mz+ )aq + zμH+ − ((μH )aq − (Δf G H2)aq ) 2 (14) 2

1 γ (T , P ) = (Gslab − nNiμ Ni (T , P) − nFeμFe (T , P) 2A − n H2μH (T , P) − nO2μO (T , P)) 2

2

(4)

where Gslab is the free energy of the slab, μi (i = Ni, Fe, O, and H) are the chemical potentials of the constituents i, and ni are the corresponding number of atoms in the slab. In eq 4 the chemical potentials μi are not the ECPs (μ°i ) discussed in the previous section but rather are floating chemical potentials that are determined from particular chemical equilibrium conditions. If NiFe2O4 is assumed to be in equilibrium with solid Ni, Fe, and water, the chemical potentials of H2 and O2 can be eliminated from eq 4, and the surface energy can be expressed as γ (T , P ) =

γ (T , P ) =

where the surface excess quantities, Γi, are given by nH2 ΓFe = nFe − nO2 + 2 ΓNi = nNi −

nO2 2

nO2 2

+

[(Δf G H2)aq − (μH )aq + 2μH + ]} 2

μ Mz+ = μ M° z+ + RT ln[xM]

(6)

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(8)

RESULTS AND DISCUSSION The surfaces examined in this work are defined by the cuts labeled with a letter along the low index planes (001), (110), (011), (101), and (111) as illustrated in Figure 1. Side views of surface projections for structures containing only Fe, O and Ni are illustrated in Figures 2−6. The calculated DFT+U energies along with the number of atoms that build up the various slabs are given in Tables 1 and 2. When combined with the effective chemical potentials reported in ref 30, the data in Tables 1 and 2 are all the information needed to calculate the surface energies. To investigate the stability of hydroxyl-terminated surfaces, multiple configurations were explored, including attaching OH groups onto all surface metal atoms and H to all surface oxygen atoms, attaching H onto surface O atoms, and attaching OH groups to surface metals alone. The number and type of adsorbate species that are attached to both sides of the slab are indicated in Tables 2 and 4, as well as in Figures 7 and 8. The (110), (101), (011), and (111) surfaces were found to have stable hydroxyl-terminated structures. Data for these surfaces are presented in Table 2, including the number and type of atoms in the slab and the DFT+U total energy required for calculating the surface energies. Despite testing multiple configurations, no (100) planes could be found with a stable terminated surface. The technique for calculating surface energy is general but can be difficult to interpret without fixing some of the chemical potentials (species concentrations), pressure, or temperature. For application to understanding corrosion in PWRs, the

While most of the previous surface energy calculations on spinel compounds assume solid−solid or solid−gas equilibrium conditions, the present study deals with solid−liquid equilibrium between the NiFe2O4 and the surrounding saturated aqueous solution. Because the solution is composed of ionic species such as Ni2+, Fe2+, Fe3+, and H+, it is convenient to write eq 5 in terms of the chemical potentials of aqueous ions. The free energy of formation of a cation with charge z is defined by42 M + z(H+)aq = (Mz+)aq + z/2(H2)g, with the free energy calculated via the corresponding equilibrium equation z (μ Mz+ )aq = μM − μH + zμH + (9) 2 2 which must also hold in the reference state z (Δf G Mz+)aq = (μ M° z+ )aq − μM° + μH° − zμH° + 2 2

(10)

Because the systems of interest here will not have access to gaseous H2, only to the solvated molecule, it is convenient to relate the chemical potential of the two phases as (μH )aq = μH + ΔG hyd 2

(11)

2

where ΔGhyd is the hydration energy of the H2 molecule. In the reference state (Δf G H2)aq = (μH° )aq − μH° 2

2

(16)

Therefore, in order to evaluate the surface energies using eq 15, the standard chemical potentials and the concentrations of constituent ions in the aqueous solution, shown in eq 16, are needed.

(7)

4

(15)

The chemical potentials in eq 15 can be related to readily observable quantities such as concentrations of solvated species (including pH) assuming an ideal solution and using molal concentration, [xM], as

nH2

−

2

2 4

nH2 4

1 {[Gslab − ΓFeμFe2+ − ΓNiμ Ni 2+ 2A − ΓNiFe2O4g NiFe O − n H2μH O] − [ΓFe + ΓNi]·

(5)

2

ΓNiFe2O4 =

Substituting eq 14 into eq 5, assuming divalent cations (z = 2), the surface energy can be expressed in terms of the chemical potential of aqueous species as

1 [Gslab − ΓFeμFe − ΓNiμ Ni − ΓNiFe2O4g NiFe O 2 4 2A − n H 2μ H O ]

(13)

2

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Figure 1. Views parallel to the nickel ferrite surfaces investigated herein showing the planes where surfaces are generated. The planes investigated are the (a) (111), (b) (110), (c) (011), (d) (101), and (e) (001). Surfaces are identified by both a plane index and a letter (A−E), which indicates the surface cut. The Ni2+ (gray), Fe2+/Fe3+ (brown), and O2− (red) spheres represent the position of the ions.

Figure 2. Denuded (111) surfaces of nickel ferrite with surfaces defined in Figure 1a: (a) A; (b) A′; (c) stoichiometric; (d) B; (e) C; (f) D; and (g) E.

surface energies are reported at conditions representative of PWR coolant at standard operating conditions. The concentrations of metal ions and aqueous hydrogen in the coolant (in molal units at 473 K) are taken from ref 43: [Fe2+] = 4.17 × 10−13, [Ni2+] = 1.66 × 10−14, and [H2]aq = 1.10 × 10−3. A concentration value for Fe3+ was not reported and should not make a significant contribution, as the coolant is kept in a reducing environment by the addition of (H2)aq. Coolant pressure is assumed to be a constant 155 bar, and pH is maintained at 7.2. The chemical potential of water is determined by the temperature and pressure alone, allowing for further reduction of the degrees of freedom. The calculated surface energies, under PWR and HTS conditions, with and without hydroxyl termination are provided in Tables 3 and 4, respectively. The stoichiometric surface energies are independent of the concentration of metal ion precursors, and the denuded stoichiometric surface energies are independent of temperature. This is because the surface energy was defined in terms of equilibrium between the bulk nickel ferrite, water, and aqueous ions. The origin of the temperature dependence in terminated stoichiometric surfaces is due to the chemical potential of the adsorbed water. Many of the surfaces considered are predicted to have a negative surface energy under PWR and HTS conditions, the lowest energies of which are plotted in Figures 7 and 8, respectively, as a function of temperature. Negative surface energies have been previously calculated and arise from chemical effects;31,44 in a multicomponent system, this is due to having to choose the component(s) that become the surface

Figure 3. Denuded (011) surfaces (a) A and (b) B, as defined in Figure 1c. Part c is the stoichiometric surface.

Figure 4. Denuded (110) surfaces of nickel ferrite with surfaces defined in Figure 1b: (a) A; (b) B.

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Table 1. Data Needed for Evaluating Various Denuded Nickel Ferrite Surfaces, Including Total DFT+U Energy and the Number and Type of Atomic Species, with Stoichiometric Surfaces also Provided and Identified as the S Surface Cut plane

surface cut (Figure 1)

E(0 K) [eV]

Ni

Fe

O2

H2

(111)

A A′ S B C D E A B A B S A B D A B S

−320.77 −321.70 −304.75 −273.96 −274.01 −264.72 −256.39 −398.50 −381.76 −377.45 −401.57 −336.66 −381.66 −398.75 −398.36 −416.92 −449.37 −433.79

7 7 7 7 7 7 5 9 9 10 8 8 9 9 9 10 10 10

14 14 14 14 14 12 12 17 19 16 20 16 17 19 19 18 22 20

16 16 14 12 12 12 12 18 18 18 18 16 18 18 18 20 20 20

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Figure 5. Denuded (101) surfaces of nickel ferrite with surfaces defined in Figure 1(d): (a) A, (b) B, and (c) D. (110)

excess quantities. The chemical interactions between the excess components and the stoichiometric surface can produce negative surface energies. In the present study the surface excess quantities are expressed in terms of water molecules and metal cations. The negative surface energy is an indication that, under PWR/HTS conditions, nickel ferrite tends to increase its surface area, giving rise to the highly porous morphology observed in the case of CRUD scrapes from PWR fuel rods.11 The (111) Surfaces. Along the (111) orientation, as shown in Figure 1a, six nonequivalent surfaces can be created: two with close-packed oxygen terminations (A and A′) and four with octahedral (B and D) and tetrahedral (C and E) cations exposed. The structures of the two oxygen-terminated surfaces are similar: both the (111)A and (111)A′ terminations consist of oxygen monolayers where the former covers a multilayer of tetrahedral and octahedral cations (Figure 2a) and the latter is situated on the top of an octahedral metal layer (Figure 2b). The similarity between the two O-terminated surfaces is also reflected by the relatively close values of the calculated surface energies (Table 3). In a 1 × 1 unit cell, the (111)B termination (Figure 2d) exposes three octahedral metal ions over four O atoms. The (111)E surface (Figure 2g) consists of 1/4 ML of tetrahedral cations covering a close-packed O monolayer. The (111)D and (111)C are obtained by adding a 1/4 ML of octahedral cations followed by another 1/4 ML of tetrahedral cations on top of the (111)E surface. Thus, the (111)C and D terminations expose metallic multilayers, consisting of octahedral and tetrahedral Ni2+/Fe3+ ions (Figure 2e and f). According to the present calculations, the most stable denuded surfaces, under PWR/HTS conditions, are (111)B and (111)C (Table 3). This is somewhat unexpected because surfaces that cut between the O monolayer and metallic layers (A, A′, B, and C) generate more broken bonds compared with cuts across the metallic multilayer (D and E). This unusual behavior can be explained by the fact that the simulations are carried out under O-poor conditions (the system is not in contact with a reservoir of molecular O); therefore, metal-terminated surfaces that expose less oxygen are the most stable.

(011)

(101)

(100)

Table 2. Data Needed for Evaluating Various Terminated Nickel Ferrite Surfaces, Including Total DFT+U Energy and the Number and Type of Atomic Species, with the Type and Number of Adsorbed Species on Both Sides on the Slab Models also Indicated plane


E(0 K) [eV]

Ni

Fe

O2

H2

(111)

A−2H2 A′−2H2 B−6H2O C−6H2O D−2H2O E−2H2O A−4H2O + 2H2 A−8H2O B−8OH B−8H2O A−4H2O + 2H2 A−8H2O B−2H2 B−8H2O

−368.62 −368.03 −365.53 −365.69 −295.17 −287.28 −464.24 −502.60 −481.42 −521.22 −460.39 −496.94 −428.91 −521.14

7 7 7 7 7 5 9 9 9 9 10 10 9 9

14 14 14 14 12 12 17 17 19 19 16 16 19 19

16 16 15 15 13 13 20 22 22 22 20 22 18 22

4 4 6 6 2 2 6 8 4 8 6 8 4 8

(110)

(011) (101)

The (111) surfaces can be further stabilized by capping the adsorption sites with H, OH, or dissociated H2O. In the case of the A and A′ surfaces, the adsorption of a single layer of H on top of the close-packed O layer is a highly exothermic process: under PWR/HTS conditions, the capped surfaces are ∼14−18

Figure 6. Denuded (001) surfaces of nickel ferrite with terminations defined in Figure 1e: (a) A; (b) B. Part c is the stoichiometric surface. 5418



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reveal that the adsorption of OH groups on the bare (111)C and B surfaces is energetically unfavorable. Guided by previous findings,25,26,28,45,46 the adsorption of dissociated water on the metal-terminated NiFe2O4 surfaces has also been investigated. This was modeled by placing the OH groups on top of the exposed metal ions and the corresponding H atoms above the O atoms in the subsurface monolayer. Based on the number of exposed cations, in a 1 × 1 unit cell, each of the (111) B and C surfaces accommodate three H2O molecules, while the (111) D and E surfaces adsorb one H2O molecule. In each case the adsorption process is slightly exothermic, suggesting that the systems gain energy by adsorbing dissociated water. As expected, this energy gain decreases with temperature such that above 600 K the bare surfaces become more stable. For instance, as listed in Tables 3 and 4, at 598.15 K the surface energy of (111)D is not changed by water adsorption, and at higher temperatures (not listed in Tables 3 and 4), the denuded surface is energetically favorable. The (110) Surfaces. In the case of denuded (011), (110), and (101) surfaces, as shown in Figures 3, 4, and 5, respectively, two types of terminations are possible: one that exposes only octahedral cations and O ions (denoted A in Figure 1b−d) and one that exposes both tetrahedral and octahedral cations and O ions (denoted B in Figure 1b−d). The number of broken bonds that are generated by creating the two types of surfaces are the same; therefore, intuitively, one would expect similar surface energies for the A- and B-type cuts. However, as listed in Table 3, the B-type surface, which exposes twice as many metal cations as the A-type surface (8 vs 4 in a 1 × 1 cell), is much more stable. This is in accordance with the observation that under PWR/HTS conditions the metal rich surfaces are more stable. In the case of a perfectly random distribution of the nickel cations over the octahedral sites, as is the case in the real material, the (011), (110), and (101) surfaces are equivalent; therefore, it would be inadequate to carry out a comparison between the energetics of these surfaces to determine the surface stability. Nevertheless, such a comparison can provide an estimate about the effect of cation distribution on surface energetics. It is assumed that the energies of the A- and B-type surfaces can be approximated by the average energies calculated for the three equivalent terminations and the effect of the cation distribution can be evaluated from the standard deviation of the energies from the average. The values and the standard deviations for the equivalent (110) surfaces are listed in Table 3, and they show very little temperature dependence. The relatively low standard deviations indicate that the (110), (011), and (101) surface energies tend to be close to the average, which means that the cation distribution has only a minor effect on the energetics of the NiFe2O4 surfaces. The process of dissociated water adsorption on the A-type (110) surfaces, under PWR/HTS conditions, is exothermic at low temperatures, and it becomes endothermic at temperatures above ∼420/320. In the case of the B-type terminations, water adsorption remains energetically favorable even at temperatures higher than 600 K. The (001) Surfaces. Similarly to the (110) surfaces, by cleaving a perfect NiFe2O4 crystal along the (001) planes, two types of terminations can be generated: one that exposes octahedral cations and O ions (denoted A in Figure 1e) and one that exposes tetrahedral Fe3+ ions (denoted B in Figure 1e). The calculated energies of the bare (001) surfaces, listed in Table 3, indicate that the B-type surface is more stable, despite

Figure 7. Energies of the (a) denuded and (b) hydroxylated surfaces of nickel ferrite as a function of temperature under conditions typical of PWRs. In the case of the hydroxylated surfaces (right panel), the type and number of adsorbents that are attached to both sides on the slabs are also indicated.

Figure 8. Energies of the (a) denuded and (b) hydroxylated surfaces of nickel ferrite as a function of temperature under conditions typical to HTS. In the case of the hydroxylated surfaces (right panel), the type and number of adsorbents that are attached to both sides on the slabs are also indicated.

J/m2 more stable than the bare surfaces. The H-terminated (111)A and A′ surfaces can also be regarded as (111)C and B terminated with OH groups. The values listed in Tables 3 and 4 5419



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Table 3. Energies of Denuded Nickel Ferrite Surfaces at Conditions Representative of PWR Coolant and HTS Conditionsa γ [J/m2] 298.15 K

598.15 K

plane


PWR

HTS

PWR

HTS

PWR

HTS

(111)

A A′ S B C D E A B S A B S

15.22 14.98 1.71 −7.94 −7.95 −2.01 5.29 4.10 ± 0.11 −0.80 ± 0.03 3.06 4.66 −2.05 1.23

19.33 19.08 1.71 −12.05 −12.06 −2.08 5.35 5.74 ± 0.80 −2.34 ± 0.56 3.06 6.41 −3.80 1.23

15.08 14.83 1.71 −7.80 −7.81 −1.96 5.24 4.08 ± 0.11 −0.78 ± 0.03 3.06 4.63 −2.01 1.23

19.40 19.15 1.71 −12.12 −12.13 −2.07 5.34 5.77 ± 0.82 −2.37 ± 0.61 3.06 6.45 −3.83 1.23

15.03 14.79 1.71 −7.75 −7.76 −1.94 5.22 4.07 ± 0.11 −0.77 ± 0.03 3.06 4.61 −2.00 1.23

19.47 19.22 1.71 −12.19 −12.20 −2.07 5.35 5.80 ± 0.83 −2.40 ± 0.62 3.06 6.48 −3.86 1.23

(110) (011) (100)

a

498.15 K

The average and standard deviation of the equivalent (110) surfaces are provided separately from the stoichiometric (011) surface.

Table 4. Energies of Terminated Nickel Ferrite Surfaces at Conditions Representative of PWR Coolant and HTS Conditionsa γ [J/m2] 298.15 K

498.15 K

598.15 K

plane


PWR

HTS

PWR

HTS

PWR

HTS

(111)

A − 2H2 A′ − 2H2 B − 6H2O C − 6H2O D − 2H2O E − 2H2O A −8H2O B −8H2O A − 4H2O + 2H2 B − 2H2 A − 4H2O + 2H2 B − 8OH

0.53 0.69 −8.60 −8.64 −2.22 4.97 3.90 ± 0.29 −1.43 ± 0.01 −0.21 −7.00 −0.34 6.27

0.53 0.69 −12.70 −12.74 −2.28 5.04 5.74 ± 1.14 −2.67 ± 0.01 0.96 −10.75 −0.36 7.55

0.80 0.95 −8.06 −8.10 −2.03 5.05 4.20 ± 0.29 −1.08 ± 0.01 0.05 −6.72 −0.08 6.37

0.80 0.95 −12.38 −12.42 −2.14 5.16 6.10 ± 1.16 −2.37 ± 0.01 1.24 −10.66 −0.11 7.73

0.93 1.09 −7.80 −7.84 −1.94 5.10 4.36 ± 0.30 −0.90 ± 0.01 0.19 −6.60 −0.06 6.44

0.93 1.09 −12.24 −12.28 −2.07 5.23 6.30 ± 1.17 −2.22 ± 0.01 1.39 −10.63 0.02 7.84

(110) (011) (101) (110) a

The average and standard deviation of the equivalent (110) surfaces covered with an integer number of water molecules are provided separately from the partially terminated surfaces, which are provided for each plane separately. The type and number of adsorbents that are attached to both sides of the slabs are also indicated.

calculated adsorption energies at 0 K and 298.15 K are listed in Table 5, and they can be compared to the those reported by Kumar et al.,28 for the (111)E surface (0.25 ML Fetet1 in Kumar’s notation) and the (111)D surface (analogous to Kumar’s 0.50 ML Feoct2−tet1 surface). Although the distribution

the fact that it contains fewer metal cations than the A-type surface. This, apparently, contradicts the assumption that under PWR/HTS conditions the surfaces that are rich in metals are more stable. The stability of the (001) B-type surface, however, can be explained by the observation that, after atomic relaxation, the outermost tetrahedral Fe ions are contracted toward the slab such that together with the subsurface octahedral cations and oxygens they form a single atomic layer. This way, the number of exposed surface metal cations becomes greater than in the case of the A-type surface. This is illustrated in Figure 6b, where the relaxed B-type surface is shown. As mentioned earlier, despite testing multiple initial configurations, no water molecule could be adsorbed on the (001) surfaces. Water Adsorption Energies. For surfaces that are covered by an integer number of water molecules, the water adsorption energies were also calculated. This quantity is defined as Eads = Eslab + nH O − Eslab − nμH O 2

2

Table 5. Water Adsorption Energies on Various Nickel Ferrite Surfaces with Respect to the Energy of an Isolated Water Molecule at 0 K, Where EH2O(0 K) = −14.21 eV, or a Water Molecule in Liquid at 298.15 K (mH2O(298 K))

(111)S (111)B (111)C (111)D (111)E (110), (101), (011)A − average (110), (101), (011)B − average

(17)

where n is the number of water molecules adsorbed on the surface and μH2O is the chemical potential of water. The 5420

Eads from EH2O(0 K) [kJ/mol]

Eads from μH2O(298 K) [kJ/mol]

−168.76 −102.27 −100.66 −97.42 −118.44 −69.07 −98.09

−108.55 −42.07 −40.45 −37.21 −58.23 −8.86 −37.88



Article

of nickel ferrite particles, as the decomposition into ions is thermodynamically favorable at conditions representative of PWR coolant. Based on solid−liquid equilibrium conditions, the surface stability of the denuded and hydroxylated NiFe2O4 surfaces has been investigated. As a general trend, under PWR/ HTS, the surfaces that expose a larger number of metal cations (Ni2+, Fe3+) are more stable. The process of water adsorption is exothermic for each of the surfaces investigated, with exothermicity decreasing as a function of temperature. Around 600 K, a temperature that is relevant to PWRs, the energies of denuded and hydroxylated surfaces are comparable, suggesting that nickel ferrite particles might be present with surfaces stripped of water molecules. The most stable nickel ferrite surface is the (111)B and C, with calculated negative surface energies. This indicates that, under conditions characteristic to PWR coolant, nickel ferrite crystals tend to expand their surface area, giving rise to a highly porous thermodynamic ground state. This provides an explanation for the porous nature of the nickel ferrite deposits found on PWR fuel rods. The remarkable stability of the (111)B and C surfaces has important implications for the formation of nickel ferrite particles and implicitly for the formation of CRUD in PWRs. Even though the bulk nickel ferrite is unstable in the coolant at PWR operating conditions, the energy cost of creating a volume of nickel ferrite is counterbalanced by the energy gain in creating the new interface between the bulk nickel ferrite and its aqueous environment. Although the present study does not address the problem of boron trapping on fuel rod surfaces, the formalism developed here can be utilized to describe and understand the reactivity of NFO surfaces toward boric acid, under PWR coolant conditions. The boron deposition on NiO and ZrO2 surfaces (both present in CRUD) has been recently described by Kumar et al.45 using first-principles methods and assuming solid−gas equilibrium between the surfaces and the environment. Their work represents a first step toward understanding the energetics and kinetics of the boron incorporation mechanism into CRUD. The method described in this paper combines firstprinciples results with experimental data and assumes solid− liquid equilibrium between the bulk material and its environment, allowing for the integration of variables, such as temperature, pressure, concentration, and pH, into the computational framework. Thus, the results provide insights into materials processes under PWR coolant conditions and can be used for adjusting the coolant chemistry in nuclear reactors.

of Ni ions over the octahedral sites differs between studies, the 0 K adsorption energies of a single water molecule on the (111)E surface are in good agreement; the value of −1.11 eV, reported by Kumar et al.,28 compares well to the −1.23 eV calculated in the present study. The adsorption energies on the (111)D surface are slightly different in the two studies; Kumar et al. report −2.30 eV for the adsorption of one water molecule, while in the present investigation the calculated value is −1.01 eV (Table 3). Because the (111)D slab, represented in Figure 6f, contains an octahedral adsorption site, the discrepancy is likely due to the differing arrangement of Fe and Ni cations over the octahedral positions. According to the values listed in Table 5, at 0 K, the presence of water on the NiFe2O4 surfaces decreases the energy of the surface in each case examined. At 298.15 K the adsorption becomes energetically less favorable but still exothermic for each surface. At around 600 K, the energy gain due to water adsorption on the most stable surfaces (i.e., (111)B and C), becomes insignificant, suggesting that in PWRs these surfaces might appear without hydroxyl termination and stripped of water molecules. Hydrothermal Synthesis. There are a number of experimental studies that characterize nanocrystalline nickel ferrite particles obtained via HTS.1−7 Under hydrothermal conditions, nickel ferrite typically forms from Fe3+ and Ni2+, where the precursor for Fe3+ is FeO(OH), Fe(NO)3, or FeCl3 and the mole ratio of Fe/Ni is 2:1. To simulate these conditions, the molal concentrations of Fe3+ and Ni2+ are assumed to be 2 × 10−3 and 1 × 10−3 mol/kg, respectively. Because it has been observed that the crystallization of nickel ferrite is greatly enhanced with the increasing alkalinity of solution,1 the value of the pH in the simulation is set to 11. The calculated energies of the bare and hydroxylated NiFe2O4 surfaces, under HTS conditions, are listed in Tables 3 and 4, respectively, and the values are plotted as a function of temperature in Figure 8. The comparison between the surface energies calculated under PWR and HTS conditions reveals a general trend: surfaces that have more metal cations exposed become more stable, while the metal-poor surfaces become less stable under HTS conditions. This is because under HTS conditions the concentration of aqueous metal cations is significantly higher than that under PWR conditions. Similarly to the PWR setup, the lowest surface energies belong to the (111)B and C terminations, and they become even more stable after adsorption of H2O molecules. The energetics of the water adsorption is also similar to that under PWR conditions: at lower temperatures all surfaces are stabilized by water adsorption, but the exothermicity of the process decreases rapidly with temperature (see Tables 3 and 4), such that above 600 K most surfaces become denuded. This behavior is consistent with the experimentally observed desorption of H2O molecules from nickel ferrite surfaces at ∼400 K, measured by IR powder spectroscopy.1 The fact that the (111) surfaces are the most stable suggests that nickel ferrite particles that are formed during HTS are octahedrons, enclosed by the (111) planes. This is consistent with published results of HTS that produce octahedral particles displaying (111) surfaces.1,2

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

This research was supported by the Consortium for Advanced Simulation of Light Water Reactors (http://www.casl.gov), an Energy Innovation Hub (http://www.energy.gov/hubs) for Modeling and Simulation of Nuclear Reactors under U.S. Department of Energy Contract No. DE-AC05-00OR22725.

CONCLUSIONS This study is unique in considering the stability of nickel ferrite surfaces with reference to aqueous cations. This approach is necessary to obtain a fundamental understanding of the stability 5421



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Article

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Calculated Stability and Structure of Nickel Ferrite Crystal Surfaces in

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