Predicting Surface Energies and Particle Morphologies of Boehmite (Î³

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Predicting Surface Energies and Particle Morphologies of Boehmite (#-AlOOH) from Density Functional Theory Micah P Prange, Xin Zhang, Mark E. Bowden, Zhizhang Shen, Eugene S Ilton, and Sebastien Kerisit J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b00278 • Publication Date (Web): 24 Apr 2018 Downloaded from http://pubs.acs.org on April 24, 2018

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The Journal of Physical Chemistry

Predicting Surface Energies and Particle Morphologies of Boehmite (γAlOOH) from Density Functional Theory Micah P. Prange,*,† Xin Zhang,† Mark E. Bowden,‡ Zhizhang Shen,† Eugene S. Ilton,† and Sebastien N. Kerisit*,† †

Physical Sciences Division, Physical & Computational Sciences Directorate, Pacific Northwest

National Laboratory, Richland, Washington 99352, United States. ‡

Environmental Molecular Sciences Division, Earth & Biological Sciences Directorate, Pacific

Northwest National Laboratory, Richland, Washington 99352, United States.

March 28th 2018

ACS Paragon Plus Environment

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ABSTRACT: Particle morphology is an important property affecting the reactivity of solid phases. The aluminum oxyhydroxide boehmite (γ-AlOOH) is one example for which particle morphology has relevance to geochemistry, environmental management, and catalysis. Accurate predictions of the morphology of boehmite particles are currently missing, and the dependence of these predictions on the level of theory and complexity of the physics involved is not known. Density functional theory calculations were therefore performed for a wide range of exchangecorrelation (XC) functionals, including a hybrid functional, to calculate the energetics of the four main low-index surfaces of boehmite ((100), (010), (001), and (101)) in aqueous conditions and predict equilibrium and growth morphologies.

The results highlight the critical need for

converged plane-wave basis sets to obtain accurate surface energies as well as the strong influence the XC functional can have on surface energies and water adsorption energies. The predicted morphologies were compared to particle morphologies obtained from a variety of synthesis routes both carried out in this work and published in the literature. The comparison showed that, at basic pH, growth morphologies dominated at low temperatures and short time periods, whereas equilibrium morphologies were achieved as the temperature and/or aging time increased. This behavior is consistent with a slow stacking growth mode of boehmite along the [010] direction. Finally, calorimetry experiments are often used to derive surface energies, but excess water on hydrophilic surfaces can add significant uncertainty. An approach for correcting the surface energetics obtained by calorimetry using electronic structure calculations is presented.


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INTRODUCTION Aluminum oxyhydroxides are an important family of compounds with relevance to geochemistry, environmental management, and catalysis. Boehmite (γ-AlOOH) and gibbsite (γAl(OH)3) are ubiquitous minerals in soils and sediments1 and are critical components of the radioactive waste tanks at the Hanford Site, WA, U.S.A.2 Boehmite also has important industrial applications, primarily as a precursor for alumina-based catalysts such as γ-Al2O3.3 In every one of these examples, the morphology of the aluminum oxyhydroxide particles plays a central role in determining their impact on the environment, reactivity in aqueous media, or usefulness to a particular application. For example, because of the topotactic nature of the γ-AlOOH/γ-Al2O3 transformation, the morphology of boehmite precursor particles determines that of the alumina catalyst.4 In another example, the acid-base properties of boehmite particles can be manipulated by varying the proportions of basal to lateral surfaces,5 as the former only develops a surface charge at extreme pH values. Reliable predictions of particle morphologies are dependent on an accurate determination of surface energetics. Atomistic simulations are well-suited to fulfill this role. For boehmite and gibbsite, early work relied primarily on potential models6,7 while later evolving to electronic structure calculations.8-11 Boehmite particles can show remarkably varied morphologies depending on synthesis factors such as temperature, pH, and the presence of additives.5,12-16 However, particle morphologies predicted from electronic structure calculations have not been able to reproduce the observed morphologies.8 Therefore, gaps exist in our fundamental understanding of the influence of synthesis factors on boehmite morphology.


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A main potential issue is the ability to perform these calculations accurately. While the dependence of theoretical predictions on the level of theory or complexity of the physics involved has received close attention for bulk properties17-21 it is often overlooked for surface energetics. In particular, an important question is whether hybrid exchange-correlation (XC) functionals, which are becoming widely used in density functional theory (DFT) calculations but remain computationally expensive compared to semi-local functionals, are needed to obtain accurate morphologies. Another important issue is whether the observed morphologies are driven by thermodynamics (equilibrium morphology) or kinetics (growth morphology). Independent validation of the predicted surface energetics against calorimetric data can also be problematic because water is not easily removed from hydrophilic surfaces,1 which can introduce significant uncertainties. A combination of well-controlled morphologies and accurate quantification of surface energies from DFT would therefore offer a pathway to remediate this issue. To address these issues, we performed plane-wave DFT calculations of the four low-index surfaces observed experimentally for boehmite: (100), (010), (001), and (101).12,13,22,23 Convergence of the surface energetics with respect to the plane-wave cutoff energy was carefully checked. A wide-range of XC functionals was considered including a hybrid functional. The projector augmented-wave method and norm-conserving pseudopotential approach were compared to evaluate the influence of the approximation for treating core electrons and the effects of dispersion corrections were also investigated. Surface energies and attachment energies were used to predict equilibrium and growth morphologies, respectively, which, in turn, were compared to morphologies derived in this work and in the literature from both transmission electron microscopy (TEM) micrographs and X-ray diffraction (XRD) microstructural


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characterization of boehmite samples synthesized in a range of conditions. Finally, the energetics of water adsorption at boehmite surfaces were used to correct published calorimetry data1 for the presence of excess water. METHODS

Figure 1. Polyhedral model of the boehmite crystal structure along two viewing directions. Al, O, and H atoms are shown in blue, red, and white, respectively. The three Al−O bond distances (Al−O(1), Al−O(2), and Al−OH) are also shown. Structure of boehmite. Boehmite adopts an orthorhombic structure with cell parameters a = 3.693 Å, b = 12.221 Å, and c = 2.865 Å.24 Note that the A21am space group is used throughout


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this work. Because of proton disorder, an effective space group of higher symmetry, Amam, is often reported, together with a fractional occupancy of the hydrogen sites, in experimental crystallographic studies. The h and l indices are switched in the sometime-used Cmc21 and Cmcm corresponding space groups. In what follows, when referring to previous studies, the Miller indices were converted to the A21am space group, if the referred study used the Cmc21 or Cmcm space groups. The boehmite structure consists of layers of edge-sharing AlO4(OH)2 octahedra stacked along the [010] direction and linked via hydrogen bonds between hydroxide ions that lie parallel to the [100] direction (Figure 1). The hydroxide ions on either side of an AlOOH layer can point in the same or opposite direction along [100], which are referred to as ‘parallel’ or ‘alternating’ configurations, respectively. The alternating configuration, shown in Figure 1, was adopted throughout for consistency. Although the two configurations are degenerate in the bulk, they may affect surface relaxation for some of the surfaces, such as for the (101) surface, as described below. Computational methods. Plane-wave density functional theory calculations were performed with two main approaches: the projector augmented-wave25,26 (PAW) method and norm-conserving pseudopotentials (PP).27-31 The PAW calculations were carried out with the ab-initio totalenergy and molecular dynamics program VASP (Vienna Ab-initio Simulation Package) developed at the Fakultӓt für Physik of the Universitӓ Wien.32-34 The PP calculations were carried out with the pseudopotential plane-wave DFT module (NWPW) of the NWChem computational chemistry package.35,36 The calculations made use of a series of generalized gradient approximation (GGA) XC potentials, including that of Becke, Lee, Yang, and Parr (BLYP),37,38 Perdew, Burke, and Ernzerhof (PBE),39,40 the revised PBE functional (revPBE) of Zhang and Yang,41 the alternative revision of the PBE functional due to Hammer et al. (RPBE),42


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and the modified PBE functional for solids (PBEsol),43 as well as the local density approximation (LDA).44,45 The PAW calculations were performed with and without the Grimme dispersion corrections46 for all functionals. The PBE047 hybrid functional was also used throughout. Two sets of PAW potentials, one for the GGA and hybrid functional calculations and one for the LDA calculations, were obtained from the VASP database for aluminum (10), oxygen (2), and hydrogen (0), with the number of core electrons shown in parentheses. Softened Hamann pseudopotentials27,29 modified into a separable form suggested by Kleinman and Bylander28 were used for aluminum (10), oxygen (2) and hydrogen (0) in the PP calculations. These pseudopotentials were constructed using the following core radii: rcs = 1.241 au, rcp = 1.577 au, and rcd = 1.577 au for Al; rcs = rcp = rcd = 0.7 au for O; and rcs = rcp = 0.8 au for H.48,49 The convergence criterion for the electronic self-consistent calculation was 10−5 eV throughout. For the constant-pressure energy minimizations (ionic positions, cell volume, and cell shape are allowed to relax) of the boehmite bulk unit cell and constant-volume energy minimizations (only the ionic positions are allowed to relax) of boehmite slabs, convergence was reached when the force on any atom was less than 0.01 eV/Å. Brillouin zone integrations were performed using the tetrahedron method. Since all systems studied here have band gaps much larger than thermal energies, integral occupancies (0 or 2) were used.


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Table 1. Simulation parameters used in the surface energy calculations. Miller index

Bulk

(001)

(010)

(100)

(101)

Cell/slab size1

1×1×1

2×1×4

2×2×1

1×2×4

1×2×4

N(AlOOH)

4

32

16

32

32

N(H2O)2

0

16

8

16

16

k-point mesh1 12×3×12 7×3×1 5×7×1 3×7×1 3×3×1 2,3 Surface area n/a 83.6; 95.0 41.4; 44.4 64.4; 72.6 105.6; 120.3 (Å2) Slab n/a 11.3; 11.8 11.4; 12.6 14.7; 15.1 11.3; 11.8 thickness3,4 (Å) 1 Scaling factors and k-point meshes for surfaces are given with respect to the rotated unit cell where the third digit corresponds to the normal to the slab surface. 2 Both faces of the slab were hydrated. N(H2O) is the total number of water molecules whereas the surface area is for one face of the slab. Water coverage is therefore N(H2O)/(2×surface area). 3 Surface area and slab thickness depend on the XC functional. Minimum and maximum values are therefore shown. 4 Slab thickness was calculated as the scaled repeating unit normal to the slab surface. The vacuum gap was at least 15 Å in each case. As described below, an investigation of the effect of the plane-wave cutoff energy was performed, from which values of 750 eV (55 Ry) and 993 eV (73 Ry) were selected for the remaining PAW and PP calculations, respectively. The number of water molecules was selected to represent 1 monolayer (ML) based on the results of molecular dynamics simulations.50 Unless otherwise noted, the calculations used the parameters listed in


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Table 1. Surface energetics. The surface energy of crystal face hkl, γhkl, was defined as:

, AlOOH ice XI − × bulk − × bulk = 2

(1)

, where is the energy of a boehmite slab of crystal face hkl with n AlOOH stoichiometric

AlOOH units and m adsorbed water molecules, bulk is the bulk energy of AlOOH per stoichiometric

ice XI unit, bulk is the bulk energy of ice XI per H2O molecule, and A is the surface area. Because the

calculations performed in this work did not account for temperature, ice was chosen as the

reference phase for H2O rather than liquid water. Ice XI is the equilibrium structure of ice below ~70 K at ambient pressure and is the hydrogen-ordered form of ordinary ice. The lattice parameters of ice XI calculated with all the XC functionals considered in this work are tabulated in Table S1 and are compared with those obtained by Leadbetter at al.51 at 5 K from neutron powder diffraction data. 2 The water adsorption energy on crystal face hkl, ∆ , was defined as:

H O

H2 O ∆

=

, , − − × gp2

H O

(2)

where gp2 is the energy of a water molecule in the gas phase. H O

AlOOH The attachment energy at crystal face hkl, ∆ , was defined as:

AlOOH ∆

, AlOOH ice XI × bulk + × bulk − "# =

(3)


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, is the energy of a single repeating unit of crystal face hkl made of n AlOOH formula where s-hkl , units and with m water molecules adsorbed. s-hkl can therefore be obtained from an unrelaxed

dry slice (or “as cleaved”), relaxed dry slice, or relaxed slice with a water monolayer adsorbed on both faces, which is the choice used in this work. Calculations of pH effects. The model of Jolivet et al.52 was employed to quantify the effect of solution pH and ionic strength on surface energies. The full derivation can be found in the original article and is thus not reproduced in full here. Briefly, this model relies on the Gibbs adsorption isotherm, which relates the change in surface energy, ∆γ, to the surface density, Γi, and change in chemical potential, dµi, of adsorbed species i:

d = − ( )* d+* *,-

(4)

where k is the number of adsorbed species. In this work and as in the original article, the adsorbed species is considered to be H+, and its surface density is controlled by two surface acidbase couples (denoted with subscripts 1 and 2 throughout) with equilibrium constants K1 and K2:

S-OH21

δ +1

↔ S-OH δ1 +H +

S-OH δ2 +OH - ↔S-Oδ2#- +H2 O

(5a)

(5b)

where S represents a surface site and δ1 and δ2 are the formal charges of hydroxo sites (S−OH). If pK1 and pK2 are separated by at least 3 pH units, the cases where pH < PZC and pH > PZC (PZC = point of zero charge) can be treated separately and as being dominated by equilibriums 5a and 5b, respectively.


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In the case where pH > PZC and equilibrium 5b dominates, the Gibbs adsorption isotherm becomes:

d = )H d+H

(6)

where ΓH is a function of the surface charge density, σ0:

)H =

2345 − 3 6 7

(7)

where F is the Faraday constant and 345 is the maximum surface charge at pH

(8)

where R is the Boltzmann constant, T the temperature, and 34# is the minimum surface charge at

pH >> pK2 due to equilibrium 5b. Substituting Equations 7 and 8 into Equation 6 yields:

d = 2345 − 3 6d> +

89 5 d3 234 − 34# 6 23 − 34# 6 7

(9)

Jolivet et al. then used the Grahame equation, which relates the surface charge to the surface potential for an aqueous solution of ionic strength I:

σ = @8BCC 89sinh

7> 7> = Dsinh 289 289

(10)

where ε is the dielectric constant of water and ε0 is the permittivity of vacuum, to integrate Equation 9 with the condition γ = γ0 for σ0 = 0, which yields:


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∆ = − =

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89 3 3 E2D + 2345 sinh#- F G − 2HD4 + 34 + 2345 − 34# 6ln I1 − # JK 7 D 34

(11a)

A similar equation can be written for the case where pH < PZC and equilibrium 5a dominates:

∆ = − =

89 3 3 E2D + 23-# sinh#- F G − 2HD4 + 34 + 23-5 − 3-# 6ln I1 − 5 JK 7 D 3-

(11b)

At equilibrium the chemical potential of surface protons is equal to that of protons in solution; therefore, using the acid-base equilibrium in Equation 5a as an example:

L=

MS-OH4 O

N 5-

P

(12a)

;S-OH NO =;H + =exp2−7> ⁄896

3 = R- SS-OH NO T + 2R- + 16MS-OH4 O P N 5-

U = SS-OH NO T + MS-OH4 O

N 5-

3 = U V2R- + 16 −

1

P

(12b)

(12c)

W

L;H + =exp2−7> ⁄896

(12d)

And the electrostatic potential at the surface, ψ0, is thus determined self-consistently by satisfying both Equation 10 and Equation 12d. The pK values and formal charges reported by Jolivet et al.52 for the four boehmite surfaces, as obtained from the multisite complexation (MUSIC) model,53,54 were used in this work. Morphologies. Equilibrium and growth morphologies of boehmite, and their dependence on the XC functional, were calculated using the Wulff construction55 functionality implemented in VESTA.56 The normalized distance between the origin and the surface plane for each crystal


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face, X , is needed as input. The equilibrium morphology minimizes the surface energy for a given volume whereas the growth morphology results from the relative growth rates of the

different crystal faces. As was done in previous studies,6,57-59 the growth rate of a given crystal face was assumed to be proportional to its attachment energy. The reader is referred to published works58,59 and references therein for a discussion of this approach. Therefore, X was defined as

X =

8 8 *.

(13)

where 8 is either the surface energy γhkl, as defined by Equation 1, or the attachment energy AlOOH ∆ , as defined by Equation 3, of the hkl face in the calculation of the equilibrium and

growth morphologies, respectively, and 8 *. is the surface or attachment energy with the lowest magnitude.

Boehmite synthesis and characterization of particle morphologies. Boehmite samples were synthesized using a hydrothermal method. Al(OH)3 amorphous powders (Sigma-Aldrich) were dispersed into 0.2 M NaOH to reach an aluminum concentration of 0.25 M and a pH of approximately 13. 16 mL of the solution was transferred to a 20 mL Teflon container, which was sealed into a steel vessel and then heated for 48 hours in an electric oven at 120 °C and 200 °C for samples 1 and 2, respectively. The resulting white product was recovered by centrifuging and washing with deionized water three times. The solid samples thus obtained were dried at 80 °C overnight. The samples were characterized with XRD and TEM. For each sample, particle morphologies were determined in two ways: (1) by averaging particle size measurements of at least 25 particles from TEM micrographs; and (2) by whole-pattern fitting. The XRD patterns


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were collected using a Panalytical Bragg-Brentano instrument, and fitted using the fundamental parameters approach implemented in TOPAS v5 (Bruker AXS). The method described by Ectors et al.60 was used to model anisotropic crystallite sizes. Based on TEM observations, an elliptical cylinder model, which approximates hexagonal plates, was used for sample 1, and a cuboid model with the dimensions perpendicular to [001] constrained to be equal was used for sample 2.


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RESULTS AND DISCUSSION

Figure 2. Dependence of the boehmite lattice parameters and bulk energy per stoichiometric unit on the plane-wave cutoff energy for the PBE XC functional with the PAW and PP approaches. Obtaining accurate surface energies. A first series of simulations was performed with the PBE XC functional and both the PAW and PP approaches to determine the effect of the plane-wave cutoff energy on the lattice parameters and energy per stoichiometric unit (bulk energy) of


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boehmite (Figure 2). A plane-wave cutoff energy of 750 eV was necessary to obtain converged properties with the PAW approach, whereas a higher cutoff energy of approximately 1000 eV was necessary with the PP approach. Calculations of the surface energy of the boehmite (010) surface with a slab model (


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Table 1) demonstrated the importance of obtaining well-converged bulk energies (Figure 3). If the cutoff energy was below that quoted above, the surface energy diverged with increasing slab thickness because of a lack of full cancellation of errors between the bulk and slab models. Previous work on boehmite surfaces with plane-wave basis sets used cutoff energies of 300 eV (ultrasoft pseudopotentials)8 and 400 eV (PAW)11, i.e. below the values determined here as necessary for obtaining converged surface energies.

Figure 3. Surface energy of the boehmite (010) surface as a function of slab thickness as calculated with the PBE XC functional for a range of plane-wave cutoff energies and with the PAW and PP approaches. Effect of the exchange-correlation functional on bulk structure and surface energies. A series of energy minimizations was performed using the plane-wave cutoff energies determined in the previous section for a range of XC functionals with both the PAW and PP approaches and with and without dispersion corrections for the PAW calculations (Figure 4).


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Figure 4. Dependence of the lattice parameters of boehmite on the nature of the XC functional for both the PAW and PP approaches and with and without the Grimme dispersion corrections (PAW only). The horizontal lines represent the experimental lattice parameters. Also shown is the root mean-square deviation (RMSD) in each case. With deviations from experiment as high as 6%, the b parameter was more problematic than the a and c parameters, which, with one exception, showed deviations of at most 2.3%. The b parameter depends strongly on the strength of the hydrogen bonding between AlOOH layers, which GGA functionals usually have difficulties with. As expected, LDA underestimated the a and c lattice parameters while the GGA functionals overestimated these parameters in all but one


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case. Including dispersion corrections systematically decreased the lattice parameters but it did not systematically improve the root mean-square deviation (RMSD) between experiment and calculated parameters. Overall, PBE0, the only hybrid functional used in this comparison, performed best but the PBE functional without dispersion corrections showed the second lowest RMSD and none of the three PBE revisions considered here offered any overall improvement. Cleavage of the (100), (010), (001), and (101) surfaces exposed five-, six-, four-, and three-fold coordinated surface Al atoms, respectively (Figure 5). Surface Al atoms retained their coordination upon energy minimization in vacuum. One exception was the (101) surface, for which surface reconstruction is needed to reach the global minimum, as Mercuri et al.11 showed using a simulated annealing approach. In the reconstructed structure of Mercuri and co-workers, the originally three-fold coordinated surface Al atoms are displaced into the interlayer space between atomic layers parallel to the (101) plane and thereby increase their coordination number with oxygen to four.


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Figure 5. Polyhedra models showing the four surfaces of interest as cleaved (top) and after an energy minimization with one water monolayer adsorbed (bottom). Al, O, and H atoms are shown in blue, red, and white, respectively. In this work, the three-fold coordinated surface Al atoms were manually displaced to best reproduce the reconstructed structure of Mercuri and co-workers. Unlike for all the other surfaces, for which both faces of the slab were allowed to relax, only one face of the (101) slab was reconstructed and allowed to relax while the atoms of the other face were kept fixed at their bulk positions. This approach reflected that of Mercuri and co-workers, who fixed the positions of the two bottom layers of the (101) slab in their calculations. The energy of an as-cleaved (101) slab was used to isolate the surface energy of the reconstructed face only. Notably, Mercuri and co-workers used the parallel configuration of hydroxide ions with all hydroxide ions pointing toward the reconstructed face whereas our calculations used the alternate configuration with only half of the hydroxide ions pointing toward the reconstructed face and the other half


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toward the fixed face. The position and coordination of the surface Al atoms in the interlayer space of the former reproduced well the structure of Mercuri and co-workers, while the surface Al atoms in the interlayer space of the latter were able to relax further inward, due to the increased separation with the nearest interlayer protons, and thereby increase their coordination number to five (Figure S1). The increase in coordination number is likely to result in a more stable reconstructed surface than that reported by Mercuri et al. The adsorption of a water monolayer completed the coordination shell of all the surface Al atoms and thus reformed the AlO6 octahedra that were truncated upon cleavage of the surfaces (Figure 5). In all cases, the starting configuration was generated by adsorbing a water monolayer on the as-cleaved surface. The coordination shells of the surface Al atoms upon hydration are consistent with those obtained by Chiche et al.,61 except for the (001) surface, for which Chiche et al. reported half of the surface Al atoms to be in five-fold coordination. This is surprising given the very similar coordination environments of the two topmost Al atoms of the (001) surface unit cell and may stem from the fact that Chiche et al. used, as starting point for their energy minimizations, configurations from ab initio molecular dynamics simulations obtained at 350 K and with a water surface arrangement of lower symmetry. For the (101) surface and in agreement with Motta et al.10 and Chiche et al.,61 partial dissociative adsorption of the water monolayer was found to be energetically favored over molecular adsorption, whereby two of the three water molecules added per surface Al atom were dissociated so that the resulting hydroxide ions directly coordinated to the surface Al atoms and the protons recombined with surface bridging oxygens to form hydroxide ions (Figure 5).


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Figure 6. Surface energies of the four boehmite surfaces ((100), (010), (001), and (101) from top to bottom) in vacuum (left panels) and with a water monolayer adsorbed (right panels) as a function of the length of the bulk Al−O bond cleaved to generate the corresponding surface or the b parameter for the (010) surface (vertical dash lines represent the experimental values). In each plot, each data point is a different XC functional. The lines indicate linear correlations


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between the surface energy and structural parameters that resulted from varying the XC functional while keeping other approximations fixed. Surface energies of the four surfaces, relaxed either with or without a water monolayer adsorbed, were then calculated with all XC functionals. The surface energies are displayed in Figure 6 as a function of the length of one of the Al−O bonds cleaved to generate the corresponding surface. All Al−O bond distances are tabulated in Table S2. One exception is the (010) surface, which results in only hydrogen bonds being cleaved and for which the surfaces energies were therefore displayed as a function of the b parameter instead, as a proxy for hydrogen-bonding strength. The surface energy is linearly correlated to the Al−O bond distances for each of the three groups (PAW with Grimme corrections, PAW without Grimme corrections, and PP without Grimme corrections). For surfaces cleaved in vacuum, the gradients were steep and were often as high as 1600 mJ m–2 per 0.1 Å. In contrast, the gradients for surfaces with a water monolayer adsorbed were much gentler, with the highest value half that just quoted for the surfaces in vacuum. The diminished dependence on the XC functional of the hydrated surface energy relative to the vacuum surface energy is likely due to a cancellation of errors. For example, XC functionals that overestimate the Al−O bond strength in boehmite, and will thus overestimate the corresponding vacuum surface energy, will also overestimate the Al−(OH2) bond strength to a similar extent. Although this finding is encouraging in terms of obtaining accurate hydrated surface energies, it also means that the water adsorption energy, another important interfacial property, is also strongly correlated to Al−O bond distances and thus to the nature of the XC functional with gradients as high as 200 kJ mol−1 per 0.1 Å (Figure 7).


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Figure 7. Water adsorption energies obtained with all the XC functionals considered in this work for four boehmite surfaces ((100), (010), (001), and (101) from top to bottom) as a function of the length of the Al−O bond cleaved to generate the corresponding surface or the b parameter for the (010) surface.


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The slopes obtained from surface energies and water adsorption energies calculated with the PP approach were similar but often gentler than those obtained with the PAW approach. Adding Grimme corrections generally increased the surface energy and decreased the absolute adsorption energy for a given Al−O bond distance but resulted in similar slopes than without the corrections. PBE0 generally yielded lower surface energies and lower absolute adsorption energies than the GGA XC functionals for a given Al−O bond distance. Two surfaces – (001) and (101) − have more than one type of cleaved bond. Plots similar to Figure 6 and Figure 7 but for all bonds cleaved for each of these two surfaces are included in the Supporting Information (SI; Figures S2 to S5) and show essentially identical linear correlations. Equilibrium and growth morphologies. The calculated hydrated surface energies were used in Wulff constructions to determine the predicted equilibrium morphologies (see top row of Figure 8 for examples). The fractional surface areas of all four surfaces for all the XC functionals are tabulated in Table S3. The morphologies obtained with all the XC functionals were similar: none predicted the (001) surface to appear in the equilibrium morphologies; the fractional surface area of the (100) surface varied from 0 to 0.17; and the remaining surface area was close to equally distributed between the (010) and (101) surfaces.


25


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Figure 8. Examples of equilibrium and growth morphologies obtained from Wulff constructions based on surface and attachment energies calculated with the PAW approach and PBE0 and PBE (with or without Grimme corrections) XC functionals. “Reduced R100” corresponds to the case where the attachment energy of the (100) surface was artificially reduced by half (see text for


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explanation). The model of Jolivet et al.52 (see Methods section) was then employed to account for the effects of pH on the calculated surface energies, and thus the equilibrium morphologies, and to provide a more appropriate comparison with experimental morphologies. In this model, the surface energy decreases as the pH departs from the pK value of a given surface acid-base couple. This decrease in surface energy is steeper the higher the temperature and the higher the ionic strength, with the temperature effect being more pronounced than the ionic strength effect. Therefore, temperature (200 °C) and ionic strength (3 M) values in the upper range of those reported in the literature for boehmite synthesis were selected as an example of a limiting case. A plot of the pH-dependent PBE0 surface energies for pH values between 4 and 12 is shown in Figure S6 and the fractional areas thus obtained from equilibrium morphologies at pH 4, 10, and 12 are listed in Table S4. Examples of the equilibrium morphologies calculated with PBE and Grimme corrections at pH 4, 10, and 12 are shown in Figure 8 (middle row). The predicted equilibrium morphologies (pH = 12) are in good qualitative agreement with the morphology derived by Chiche et al.23 from powder XRD of a boehmite sample synthesized at pH = 11.5, for which the fractional surface areas were 0.01, 0.51, 0.01, and 0.47 for the (001), (010), (100), and (101) surfaces, respectively. The synthesis route consisted in aging at 100 °C for up to two months a precipitate formed from an aluminum nitrate aqueous solution with a fixed ionic strength set by 3 M NaNO3. Pardo et al.5 followed a similar synthesis route and derived apparent crystallite sizes in the three crystallographic directions by applying the Scherrer equation to their XRD data. Assuming an equilibrium morphology where only the (010) and (101) surfaces are expressed (as obtained for PBE0 for example, top row of Figure 8), the Scherrer sizes translate to fractional surface areas of approximately 0.43 and 0.57 for the (010)


27


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and (101) surfaces, respectively, for the samples aged more than 24 h at 200 °C. Similar fractional surface areas were calculated from TEM measurements of boehmite particles reported by Pardo et al.62 in a previous study. Boehmite particles synthesized by hydrothermal treatment by de Souza Santos et al.,14 who used either gibbsite or bayerite precursors, and by He et al.,15 who used a precipitate formed from an aluminum chloride solution, showed similar morphologies but the information provided in these studies was not sufficient to calculate fractional surface areas. Fractional surface areas calculated from the Scherrer sizes reported by Pardo et al.5 for their samples only aged for 24 h at 200 °C showed an increased proportion of the (010) surface (from 0.43 to 0.63). The same effect was observed when translating the crystallite sizes obtained from XRD measurements by Bokhimi et al.63,64 to fractional surface areas, again assuming a particle morphology where only the (010) and (101) surfaces are expressed. Bokhimi et al.,63,64 subjected boehmite seeds to hydrothermal treatment for 18 h at temperatures varying from room temperature to 240 °C, which resulted in fractional surface areas of the (010) surface between 0.59 and 0.70 by our calculations. These observations are consistent with the characterization of the two boehmite samples synthesized in this work via a hydrothermal approach (see Methods section). XRD patterns and typical TEM micrographs of both samples are shown in Figure 9, and particle/crystallite dimensions, based on the definitions presented in Figure S7, and resulting fractional surface areas are tabulated in Table S5. Both samples were obtained by reacting amorphous Al(OH)3 powder precursors for 48 h but sample 1, which was synthesized at 120 °C, was made of small and thin platelets with a pseudo-hexagonal morphology, similar to the predicted growth morphologies, with a fractional surface area of the (010) surface of 0.70 (from XRD) or 0.64 (from TEM). In contrast, sample 2, which was synthesized at a higher temperature


28

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(200 °C), showed larger and thicker particles with little to no contribution from the (100) surface, in agreement with the calculated equilibrium morphologies.

Figure 9. XRD patterns (top) of boehmite samples 1 (a) and 2 (b) and TEM micrographs of samples 1 (middle) and 2 (bottom). Samples 1 and 2 were synthesized using a hydrothermal method (48 h) at 120 °C and 200 °C, respectively.


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As discussed by Pardo et al.,5 these observations indicate that growth rates along the [010] direction are initially slow compared to the growth of lateral surfaces so that short times and/or low temperatures yield particles with a large fraction of basal surfaces. As the annealing time and/or the temperature increase, particle-particle growth through stacking of particles along the [010] direction allows the particle morphology to converge towards the predicted equilibrium morphology. This interpretation is supported by the growth morphologies obtained from the calculated attachment energies (see bottom row of Figure 8). Indeed, the calculated fractional surface areas of the (010) surface are 0.65 and 0.68 for PBE with and without Grimme corrections, respectively (PAW approach), in good quantitative agreement with those given above for boehmite particles synthesized in this work at the lower temperature, in the study of Pardo et al.5 at the shortest aging time, and in the work of Bokhimi et al.63,64 at the lowest temperatures. As discussed in the Introduction, boehmite is an important component of the radioactive waste storage tanks at the Hanford Site. In particular, boehmite is present in alkaline tanks containing waste from the reduction-oxidation (REDOX) Pu extraction process, as this process involved the addition of large amounts of Al(NO3)3.65 TEM micrographs of samples of REDOX tank waste sludge66 were analyzed to determine boehmite particle dimensions, as for samples 1 and 2 (Table S5). The boehmite particles were thin platelets with a large fraction of the total surface area due to the (010) surface (0.75), indicating that they had not reached their equilibrium morphology. This observation may seem surprising in the light of the discussion in the previous paragraph since the particles have been “aging” for decades and heating from radioactive decay caused some of the tanks to boil. Nonetheless, the TEM images showed a significant extent of stacking of boehmite particles along the [010] direction.65,66 One hypothesis to explain this observation is


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therefore that the particle-particle growth mode discussed above is much less efficient in the waste tanks due to the extremely complex chemical environment, and the presence of other solid phases, resulting in particle stacking rather than coalescence. Another hypothesis to explain the departure from equilibrium morphologies obtained in simpler environments free of additives is the change in surface energetics, and thus equilibrium morphologies, from preferential adsorption of ions at specific surfaces. For boehmite synthesis carried out at low pH, Chiche et al.23 found that, unlike for the particle morphologies they obtained at basic pH, the (100) surface was significantly expressed if the synthesis was carried out at an acidic pH of 4.5 (fractional surface areas of 0.04, 0.32, 0.35, and 0.29 for the (001), (010), (100), and (101) surfaces, respectively). This observation is consistent with Alemi et al.,67 who showed scanning electron micrographs of particles with needle-like morphology at pH 4, and Karouia et al.,4 who reported fractional surface areas of the (100) surface that varied from 0.25 to 0.35 at pH 4.5 depending on the hydrothermal treatment period. De Souza Santos et al.14 also reported morphologies that were stretched along the [001] direction upon addition of 0.1 M acetic acid and resulted in lath-like crystals. These experimental morphologies do not match the calculated low-pH equilibrium morphologies (middle row of Figure 8), neither do they match the calculated growth morphologies. The low-pH equilibrium morphologies predicted in this work differ significantly from those reported by Jolivet et al.52 because their predictions were based on the surface energies calculated by Raybaud et al.8 In addition to potential issues related to the use of a low plane-wave cutoff energy in that study and to differences that might arise from the use of a different parameterization of the GGA XC functional (PW9168,69) not considered in this work, as discussed above, the hydrated surface geometry considered by Raybaud et al.8 consisted in part of four-fold coordinated Al atoms


31


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compared to six-fold coordinated surface Al atoms in this work. As a result, the surface energy of the (101) was overestimated and this surface was not as strongly expressed as in the current equilibrium morphologies. A subsequent publication by the some of the same authors61 reconsidered the structure of the hydrated (101) surface and reported the same termination as considered in this work, i.e. with only six-fold coordinated Al atoms. If the attachment energy at the (100) surface is artificially reduced by half, the resulting growth morphologies (bottom row of Figure 8) logically express the (100) surface to a greater extent, but the fractional surface area of this surface (~0.3) becomes commensurate with the values derived from XRD by Chiche et al.23 and Karouia et al.4 and the relative proportions of the three surfaces are consistent with the TEM images of De Souza Santos et al.14. This exercise suggests that, as the pH is lowered, the morphology of boehmite particles reflects changes in growth rates rather than in surface energies and that the (100) surface is the only surface whose growth rate is significantly affected. Comparison of surface energetics with calorimetry data. Majzlan et al.1 determined the surface enthalpy of boehmite, using high-temperature oxide-melt calorimetry, from the dependence of the enthalpy of formation of boehmite powder samples on their surface area. The enthalpy of formation from the oxides, ∆H6, was calculated using the thermodynamic cycle shown in Figure 10.


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Figure 10. Thermodynamic cycle used by Majzlan et al.1 to determine the enthalpy of formation of boehmite from oxides ∆H6. The labels ‘cr’, ‘l’, and ‘g’ stand for crystal, liquid, and gas. Majzlan et al.1 measured ∆H3 (drop solution enthalpy of corundum) and ∆H1 (drop solution enthalpy of boehmite samples) from drop solution experiments with corundum and boehmite, respectively, and obtained ∆H2 (enthalpy change from liquid water at 298 K to gaseous water at 975 K) from thermodynamic tables. Excess water was present in the samples used in that study (x in Figure 10). Although Majzlan et al. were able to quantify the amount of excess water in the boehmite samples, they could not determine the enthalpy of water removal from those samples (∆H8) and they thus assumed it to be zero. In other words, the excess water was energetically equivalent to bulk water. The excess water was likely present as chemisorbed water molecules, particularly at the (101) surface, which has a large fractional surface area and for which the water adsorption energy is highly exothermic, as described above. Assuming that water was present as adsorbed water molecules and that, for sub-monolayer water coverage of the boehmite particle, water was distributed on the particles so as to maximize the energy gained through adsorption, the electronic structure calculations presented in this work can be used to determine ∆H8:


33


∆HZ = − ( 7* [* *,

H2 O

\∆*

H2 O

Page 34 of 49

ice XI − bulk ]

(14)

where 7* is the calculated fractional surface area of crystal face hkl and [*

H2 O

is the fraction of a

monolayer of water adsorbed at crystal face hkl and is defined as:

[2 = ^ H O

2 _excess − ∑c∆dH2O cfc∆dH2O c [*

H O

ghi

e

- ML _

7 × 0,

H2 O

7* × _*- ab

, 7 > 0 7 = 0

(15)

- ML 2 where _excess is the excess water per unit area measured by Majzlan et al.1 and _ is the H O

surface density of water molecules at 1 ML coverage on crystal face hkl. The boehmite crystal faces are ranked in order of decreasing absolute water adsorption energies (i.e., (101), (001), (100), and (010)) and the sum in Equation 15 is only performed over surfaces ranked higher than crystal face hkl. [2 is set to 1 if the result of Equation 15 is greater than 1. If the amount of H O

excess water was greater than that required to cover the entire particle with one monolayer, the remaining water was considered to be physisorbed atop the chemisorbed water monolayer and to be energetically equivalent to bulk water. A table listing the values of ∆H8 thus obtained for all XC functionals, the corresponding enthalpies of formation from the oxides and corrected experimental surface energies, and the calculated dry particle surface energies is given in the SI (Table S6). The particle surface energy, on which the comparison with experiment is based as powders were used, is defined as

n

dry

= ( 7* * *,

(16)


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where γi is the surface energy of crystal face hkl as determined from a vacuum calculation. Because the samples used by Majzlan et al.1 were not completely dry and, in any case, the particles are not expected to modify their morphologies upon drying, the data in Table S6 assumed the particle morphologies were those determined from the surface energies with 1 ML of water adsorbed (equilibrium morphologies with no pH effects). As illustrated in Figure 11, the gradient of a linear regression of the enthalpy of formation from the oxides as a function of the surface area yields the surface energy (Table 2). In their calorimetric measurements, Majzlan et al.1 used four boehmite powders with surface areas ranging approximately from 100 to 16000 m2 mol−1; the powder with the lowest surface area was obtained from a different vendor (referred to ALCAN) than that for the other three powders (referred to as PURAL). When translating the excess water measured by Majzlan et al. to the number of molecules per nm2, the PURAL powders had effective coverage values within 10 H2O nm−2, commensurate with a water monolayer, whereas that of the ALCAN powder differed by close to two orders of magnitude. It is thus likely that the excess water in that sample was not simply due to adsorbed water, as would be the case if the sample contained a non-negligible amount of gibbsite for example, and the ALCAN powder was therefore excluded from any correlation in this work.


35


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Figure 11. Enthalpy of formation from oxides (∆H6) as a function of surface area for the uncorrected experimental data (∆H8 = 0) and following correction based on the PBE XC functional (∆H8 = PBE), with and without Grimme corrections, and on equilibrium or growth morphologies. The gradient of the linear regression is the surface energy. Amounts of excess water are also shown in units of water molecules per nm2. For each XC functional, the corrected experimental surface energy can then be compared to that obtained from the calculated surface energies in vacuum for that same particle morphology (see Table S6 for all the XC functionals and Table 2 for a subset). For PBE for example, the corrected experimental surface energy (γ = 956 ± 120 mJ m−2, with the uncertainty reported by Majzlan et al.1) compares well with the calculated value at the equilibrium morphology (γp = 918 mJ m−2). Because the morphology of the PURAL boehmite samples is not known, the procedure was repeated assuming the particle morphologies obtained from the attachment energies (growth morphologies) for the PBE XC functional, with and without Grimme corrections and the results are presented in Table S7, Table 2, and Figure 11. Because the growth morphologies express the


36

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(010) surface more predominantly and because water adsorption is less exothermic at this surface than at the other three boehmite surfaces, the magnitude of the correction is lower than for the equilibrium morphologies. Nonetheless, the experimental surface energy obtained via correction with PBE is 747 mJ m−2, close to 30% greater than the uncorrected surface energy of 580 mJ m−2, highlighting the importance of the correction. Table 2.

Particle surface energies (mJ m−2) (experimental corrected and calculated) and

percentage differences (∆γ) between the corrected experimental and calculated particle surface energies. The experimental uncorrected particle surface energy is 580 mJ m−2.1

∆γ (%)

956

ndry

918

−4

equilibrium

943

1144

+21

No

equilibrium

989

949

−4

PBE

No

growth

747

664

−11

PBE

Yes

growth

834

950

+14

XC

Grimme

Morph.

γexp. corrected

PBE

No

equilibrium

PBE

Yes

PBE0

Overall, these results indicate that electronic structure calculations can be used to determine the energetics of excess water in calorimetry experiments and thus help apply critical corrections that would not otherwise be accessible. This approach has the potential to be quantitative if the calorimetry experiments can be combined with measurements that allow for a quantitative determination of particle morphology (e.g. TEM). CONCLUSIONS The structure and energetics of the four main low-index surfaces of boehmite were calculated using density functional theory with a wide range of exchange-correlation functionals. The energy of dry surfaces was very sensitive to small differences in the length of the cleaved Al−O


37


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bonds predicted by the different XC functionals. Although this effect was diminished for hydrated surfaces due to a cancellation of errors, it was shown to still affect the predicted equilibrium morphologies and to translate to large variations between the water adsorption energies predicted by the different XC functionals. The results of the calculations also strongly highlighted the critical need for converged plane-wave basis sets to obtain accurate surface energies. Experimental particle morphologies obtained at low temperatures and/or short hydrothermal treatments at basic pH were well reproduced by calculated growth morphologies while calculated equilibrium morphologies matched those observed at higher temperatures and/or longer reaction times. This finding could be explained by the slow stacking of boehmite particles along the [010] direction during growth. In contrast, the calculations could not reproduce the experimental morphologies obtained at acidic pH, even when a model that accounts for pH effects on surface energies was applied to the DFT results. This discrepancy was attributed to the growth rate of the (100) surface being more strongly affected by low pH conditions than that of the other three surfaces. Dynamical simulations will be required to investigate the growth mechanism at the root of this phenomenon. Finally, the DFT results were used to correct calorimetric measurements of the surface enthalpy of boehmite by providing a quantitative estimate of the energetics of excess water present as chemisorbed water molecules. When combined with measurements of particle morphologies, this approach has the potential to remove uncertainties introduced in thermodynamic analyses by excess water present on hydrophilic surfaces and thus to provide a reliable and quantitative determination of surface energetics.


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ASSOCIATED CONTENT Supporting Information. The following file is available free of charge. Lattice parameters of ice XI calculated with all XC functionals, structural model of the reconstructed (101) surface, Al−O bond distances in boehmite calculated with all XC functionals, surface energies of and water adsorption energies on the (001) and (101) surfaces as a function of the length of the Al−O bonds cleaved to generate the surfaces as obtained with all XC functionals, surface energies of the four boehmite surfaces and their fractional surface areas from equilibrium morphologies as calculated with all XC functionals with and without pH effects, surface energies of the four boehmite surfaces as a function of pH based on the results of the PBE0 calculations, and calculated enthalpies of water removal from boehmite particles, particle/crystallite dimensions of boehmite samples synthesized in this work and of one REDOX tank waste sludge sample, corrected experimental enthalpies of boehmite formation from the oxides, and corrected experimental surface energies obtained based on the water adsorption energies and growth and equilibrium morphologies obtained with all XC functionals (PDF). AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]; [email protected] Notes The authors declare no competing financial interest. ACKNOWLEDGMENT


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This research was supported by the Laboratory Directed Research and Development (LDRD), Nuclear Process Science Initiative (NPSI) at Pacific Northwest National Laboratory (PNNL). The computational work was performed using PNNL Institutional Computing at Pacific Northwest National Laboratory. Part of the research was performed using the Environmental Molecular Sciences Laboratory (EMSL), a national scientific user facility sponsored by the DOE's Office of Biological and Environmental Research and located at PNNL. PNNL is a multiprogram national laboratory operated for the U.S. Department of Energy (DOE) by Battelle Memorial Institute under Contract No. DE-AC06-76RLO-1830. The authors acknowledge Dr. Edgar C. Buck for providing TEM micrographs of boehmite particles in samples from Hanford’s nuclear waste tanks. REFERENCES (1) Majzlan, J.; Navrotsky, A.; Casey, W. H. Surface Enthalpy of Boehmite. Clays Clay Miner. 2000, 48, 699-707. (2) Snow, L. A.; Lumetta, G. J.; Fiskum, S.; Peterson, R. A. Boehmite Actual Waste Dissolution Studies. Separ. Sci. Technol. 2008, 43, 2900-2916. (3) Chiche, D.; Chanéac, C.; Revel, R.; Jolivet, J.-P. Size and Shape Control of Γ-AlOOH Boehmite Nanoparticles, a Precursor of Γ-Al2O3 Catalyst. Stud. Surf. Sci. Catal. 2006, 162, 393400. (4) Karouia, F.; Boualleg, M.; Digne, M.; Alphonse, P. The Impact of Nanocrystallite Size and Shape on Phase Transformation: Application to the Boehmite/Alumina Transformation. Adv. Power Technol. 2016, 27, 1814-1820.


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(5) Pardo, P.; Serrano, F. J.; Vallcorba, O.; Calatayud, J. M.; Amigó, J. M.; Alarcón, J. Enhanced Lateral to Basal Surface Ratio in Boehmite Nanoparticles Achieved by Hydrothermal Aging. Cryst. Growth Des. 2015, 15, 3532-3538. (6) Fleming, S.; Rohl, A.; Lee, M.-Y.; Gale, J. D.; Parkinson, G. Atomistic Modelling of Gibbsite: Surface Structure and Morphology. J. Cryst. Growth 2000, 209, 159-166. (7) Fleming, S.; Rohl, A. L.; Parker, S. C.; Parkinson, G. M. Atomistic Modeling of Gibbsite: Cation Incorporation. J. Phys. Chem. B 2001, 105, 5099-5105. (8) Raybaud, P.; Digne, M.; Iftimie, R.; Wellens, W.; Euzen, P.; Toulhoat, H. Morphology and Surface Properties of Boehmite (Γ-AlOOH): A Density Functional Theory Study. J. Catal. 2001, 201, 236-246. (9) Frenzel, J.; Oliveira, A. F.; Duarte, H. A.; Heine, T.; Seifert, G. Structural and Electronic Properties of Bulk Gibbsite and Gibbsite Surfaces. Z. Anorg. Allg. Chem. 2005, 631, 1267-1271. (10) Motta, A.; Gaigeot, M.-P.; Costa, D. Ab Initio Molecular Dynamics Study of the AlOOH Boehmite/Water Interface: Role of Steps in Interfacial Grotthus Proton Transfers. J. Phys. Chem. C 2012, 116, 12514-12524. (11) Mercuri, F.; Costa, D.; Marcus, P. Theoretical Investigations of the Relaxation and Reconstruction of the Γ-AlO(OH) Boehmite (101) Surface and Boehmite Nanorods. J. Phys. Chem. C 2009, 113, 5228-5237. (12) Xia, Y.; Jiao, X.; Liu, Y.; Chen, D.; Zhang, L.; Qin, Z. Study of the Formation Mechanism of Boehmite with Different Morphology Upon Surface Hydroxyls and Adsorption of Chloride Ions. J. Phys. Chem. C 2013, 117, 15279-15286.


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(13) Xia, Y.; Zhang, L.; Jiao, X.; Chen, D. Synthesis of Γ-AlOOH Nanocrystals with Different Morphologies Due to the Effect of Sulfate Ions and the Corresponding Formation Mechanism Study. Phys. Chem. Chem. Phys. 2013, 15, 18290-18299. (14) de Souza Santos, P.; Vieira Coelho, A. C.; de Souza Santos, H.; Kunihiko Kiyohara, P. Hydrothermal Synthesis of Well-Crystallised Boehmite Crystals of Various Shapes. Mater. Res. 2009, 12, 437-445. (15) He, T.; Xiang, L.; Zhu, S. Different Nanostructures of Boehmite Fabricated by Hydrothermal Process: Effects of pH and Anions. Cryst. Eng. Comm. 2009, 11, 1338-1342. (16) Chen, X. Y.; Huh, H. S.; Lee, S. W. Hydrothermal Synthesis of Boehmite (Γ-AlOOH) Nanoplatelets and Nanowires: pH-Controlled Morphologies. Nanotech. 2007, 18, 285608. (17) Casassa, S.; Demichelis, R. Relative Energy of Aluminum Hydroxides: The Role of Electron Correlation. J. Phys. Chem. C 2012, 116, 13313-13321. (18) Demichelis, R.; Civalleri, B.; Ferrabone, M.; Dovesi, R. On the Performance of Eleven DFT Functionals in the Description of the Vibrational Properties of Aluminosilicates. Int. J. Quantum Chem. 2010, 110, 406-415. (19) Demichelis, R.; Civalleri, B.; D'Arco, P.; Dovesi, R. Performance of 12 DFT Functionals in the Study of Crystal Systems: Al2SiO5 Orthosilicates and Al Hydroxides as a Case Study. Int. J. Quantum Chem. 2010, 110, 2260-2273. (20) Noel, Y.; Demichelis, R.; Pascale, F.; Ugliengo, P.; Orlando, R.; Dovesi, R. Ab Initio Quantum Mechanical Study of Γ-AlOOH Boehmite: Structure and Vibrational Spectrum. Phys. Chem. Minerals 2009, 36, 47-59.


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(21) Demichelis, R.; Noël, Y.; Ugliengo, P.; Zicovich-Wilson, C. M.; Dovesi, R. PhysicoChemical Features of Aluminum Hydroxides as Modeled with the Hybrid B3LYP Functional and Localized Basis Functions. J. Phys. Chem. C 2011, 115, 13107-13134. (22) Grebille, D.; Bérar, J.-F. Calculation of Diffraction Line Profiles in the Case of a Major Size Effect: Application to Boehmite AlOOH. J. Appl. Cryst. 1985, 18, 301-307. (23) Chiche, D.; Digne, M.; Revel, R.; Chanéac, C.; Jolivet, J.-P. Accurate Determination of Oxide Nanoparticle Size and Shape Based on X-Ray Powder Pattern Simulation: Application to Boehmite AlOOH. J. Phys. Chem. C 2008, 112, 8524-8533. (24) Hill, R. J. Hydrogen Atoms in Boehmite: A Single Crystal X-Ray Diffraction and Molecular Orbital Study. Clays Clay Miner. 1981, 29, 435-445. (25) Blöchl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 17953-17979. (26) Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B 1999, 59, 1758-1775. (27) Hamann, D. R.; Schlüter, M.; Chiang, C. Norm-Conserving Pseudopotentials. Phys. Rev. Lett. 1979, 43, 1494-1497. (28) Kleinman, L.; Bylander, D. M. Efficacious Form for Model Pseudopotentials. Phys. Rev. Lett. 1982, 48, 1425-1428. (29) Hamann, D. R. Generalized Norm-Conserving Pseudopotentials. Phys. Rev. B 1989, 40, 2980-2987. (30) Troullier, N.; Martins, J. L. Efficient Pseudopotentials for Plane-Wave Calculations. Phys. Rev. B 1991, 43, 1993-2006. (31) Troullier, N.; Martins, J. L. Efficient Pseudopotentials for Plane-Wave Calculations. II. Operators for Fast Iterative Diagonalization. Phys. Rev. B 1991, 43, 8861-8869.


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Predicting Surface Energies and Particle Morphologies of Boehmite (Î³

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