A new combined approach to investigate stacking faults in lamellar ...

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The presence of stacking faults in lamellar compounds can be greatly ... In order to validate this combined method, the lamellar structure β-Ni(OH)2 was investi-.
Z. Kristallogr. Proc. 1 (2011) 49-54 / DOI 10.1524/zkpr.2011.0007 © by Oldenbourg Wissenschaftsverlag, München

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A new combined approach to investigate stacking faults in lamellar compounds Romain Gautier*, E. Furet, Régis Gautier, N. Audebrand, E. Le Fur Sciences Chimiques de Rennes, UMR 6226, CNRS – Université de Rennes 1 - Ecole Nationale Supérieure de Chimie de Rennes, Avenue du Générale Leclerc, CS 50837, 35708 Rennes Cedex 7, France * Contact author; e-mail: [email protected] Keywords: stacking faults, DFT calculations, DIFFaX+, lamellar vanadophosphate Abstract. A new combined approach based on DFT calculations and X-ray powder diffraction allowing the investigation of stacking faults is herein presented. In a first step, most probable stacking faults vectors are computed using first-principles calculations. The favoured layer translations are then considered for the crystal structure refinement using the DIFFaX+ program. As a test case, this method was validated on β-Ni(OH)2 lamellar structure in which the stacking faults have been unambiguously described. Some lamellar vanadophosphates have been investigated using this combined approach.

1. Introduction The presence of stacking faults in lamellar compounds can be greatly correlated to the properties of these materials. In order to study stacking faults, emergent tools are available. Programs such as DIFFaX [1-4], DIFFaX+ [5, 6], and FAULTS [7, 8] allow a better comprehension of numerous diffraction patterns by employing translations vectors and their probabilities. In particular, DIFFaX+ allows the refinement of translation vectors. However, in some cases, it is difficult or nearly impossible to find translation vectors using DIFFAX+ without providing an initial guess of reliable translation vectors. Thus, the literature reports numerous compounds, of which the crystal structures present stacking faults, but the induced translation vectors keep or hardly modify the symmetry. As an example, symmetry of the hexagonal system remains the same when layer translations (1/3 ; 2/3) or (2/3 ; 1/3) are considered [3]. Conversely, tetragonal systems become monoclinic when layer translations (x ; 0), (0 ; y) or (x ; x), are taken into account [2]. To our knowledge, the general case of layer translation (x ; y) was never considered because the determination of this vector, before simulating diffraction pattern with DIFFaX+ code or other programs, is essential but not necessarily obvious. Coupling DFT calculations and structural determination using DIFFaX+ is relevant because it helps finding layer translation vectors before refinement.

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2. A combined method for the study of stacking faults This approach that combines DFT calculations and DIFFaX+ refinements is a multi-step approach: - Modeling an isolated layer using DFT calculations. This needs a preliminary model of the isolated layer. It can be issued from a structure showing no stacking faults or an isotype structure. The initial model of the isolated layer is computed using a supercell with fixed cell parameters. The parameters α and β are kept equal to 90° in order to avoid any structure distortion. The stacking parameter c is fixed to a large enough value so that neighbouring layers can be considered as isolated. Stacking parameter is not chosen too large in order to avoid large computing resources. Remaining cell parameters, namely a, b and γ are free to relax. Atomic positions are optimized as well, using DFT calculations. - Determination of interlayer distance. This can be done using diffraction methods. For a good description of interlayer interactions, this parameter has to be known the most accurately. Without preliminary indexing the diffraction pattern, the determination of interlayer distance can be facilitated by considering preferential orientation. - Creation of a set of triclinic structures corresponding to different layer translations. The model of the isolated layer optimised by DFT calculations and the interlayer distance are used to create a set of triclinic structures resulting from possible translations (x ; y) (with 0 ≤ x ≤ a and 0 ≤ y ≤ b). - DFT calculations for the set of triclinic structures. First-principles calculations are used to compute the corresponding potential energy surface. Depending upon the symmetry of the layer, the investigated region of the energy surface can be reduced. The influence of interlayer distance on the computed energy surfaces can also be studied. The energy surface can be computed using fixed atomic positions. Atomic positions can also be optimized for each layer translations (x;y) that are considered. Some information about atoms movement during layer glides can be obtained. However, since this approach is more CPU-time consuming, its use must be restricted to small systems. Favoured stable translations can be identified. Estimation of their probabilities can be done using a Boltzmann distribution. - Refinement of translation vectors and associate probabilities using DIFFaX+ code.

3. Validation of the approach for β-Ni(OH)2 In order to validate this combined method, the lamellar structure β-Ni(OH)2 was investigated. This structure crystallizing in a brucite type (P-3m1, a = 3.126 Å and c = 4.593 Å) has been largely studied and stacking faults were described using programs such as DIFFaX.[3, 12]. In most cases, this compound is prepared by addition of nickel nitrate to a NaOH solution. The stacking faults are described by translation vectors (0 ; 0) or (1/3 ; 2/3) and can be explained on the basis of steric effects (Figure 1).

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(a)

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(b)

Figure 1. View of Ni(OH)2 layer (a) without translation and (b) with layer translation (1/3 ; 2/3).

The two stacking sketched in Figure 1 can both be described in the same hexagonal system. Numerous simulations of X-ray diffraction pattern by different authors using various codes are reported in the literature. Therefore our validation study is herein focused on the identification of translation vectors using DFT calculations. First-principles calculations were carried out using the CASTEP code [9]. The exchange and correlation potentials were calculated using the GGA in the parameterization of PerdewBurke-Ernzerhof [10]. Pseudo-potentials were generated using the OTF_ultra-soft_pseudopotential generator included in the program. The electronic wave functions were sampled in the irreducible Brillouin zone (BZ) using the Monkhorst-Pack method [11]. All calculations were checked for convergence with respect to the kinetic energy cut-off of the plane waves basis set (up to 800 eV) and the k points grid (up to 112) used for integration over the BZ. The supercell sketched in Figure 2 that has been used to model the Ni(OH)2 layer, presents a and b cell parameters that are equal to the experimental values whereas c has been fixed to 10 Å. This latter value, that is large enough to avoid interaction between neighbouring layers, remains fixed during optimisation.

Figure 2. Representations of the isolated layer of β-Ni(OH)2 in a supercell. Nickel atoms are linked to six oxygen atoms that form octahedral.

The potential energy surface computed for different translation vector is shown on Figure 3. This shows that structural arrangements where no layer translation occurs as well as the one that takes into account the layer translation (1/3 ; 2/3) are stable cases. Conversely the layer translation (2/3 ; 1/3) is strongly unfavoured. Indeed, (2/3 ; 1/3) translation leads to strong repulsions between hydroxide groups. Although interlayer interactions in β-Ni(OH)2 are of Van der Waals type, this method based on DFT calculations does not fail in dealing weak interactions. This method that is used to identify translation vectors is also able to confirm the nature of stacking faults in layered

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Translation b (Å)

Ni(OH)2. In some cases such as this nickel hydroxyde, considering stacking faults in hexagonal structures does not modify the crystal system. That is not the case for the compounds studied in the next section, the crystal system of which changing from tetragonal to triclinic.

Energy Energie (eV/unit cell) (eV/maille)

Translation a (Å)

Figure 3. Energy surface function of β-Ni(OH)2 layer translation. Translations (0 ; 0) and (1/3 ; 2/3) are energetically stables.

4. Application to the study of lamellar vanadium phosphates VOPO4 compounds are particularly used for industrial conversion of butane into maleic anhydride [13]. Among the seven varieties of VOPO4 involved during industrial catalytic process, two polymorphs (α1-VOPO4 and α2-VOPO4) can be prepared from the dehydration of VOPO4•2H2O. During this dehydration an intermediate phase appears, the monohydrate VOPO4•H2O. All these compounds present layered structures but the calculated X-ray diffraction powder patterns from the proposed structural models are generally not in good agreement with the experimental ones. In order to investigate the stacking faults in the lamellar α1-VOPO4, the previous method was used. On the basis of DFT calculations, five possible layer translations were identified (Figure 4). Using DIFFaX+ program, the refinement of translation probabilities and vectors given in Table 1 leads to a better agreement between the measured diffraction pattern and the calculated one if stacking faults are considered, as shown on Figure 5. Table 1. Translation vectors and probabilities refined by DIFFaX+ program.

Translation vectors Refined probabilities

(0 ; 0)

(0.5 ; 0.174)

(0.5 ; 0.826)

(0.174 ; 0.5)

(0.826 ; 0.5)

0.00

0.00

0.97

0.03

0.00

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(0,5; 0,79) Energy (eV/unit cell)

(0,21; 0,5)

(0,79; 0,5)

(0,5; 0,21) (0; 0)

(a)

(b)

Figure 4. (a) Energy surface function of α1-VOPO4 layer translations and (b) energetically favoured translations.

λ = 1.5406 Å

Experimental data Calculated with stacking faults Calculated without stacking faults

Figure 5. Experimental and calculated X-ray powder diffraction patterns for α1-VOPO4. Rwp values obtained with DIFFaX+ for the model without and with stacking faults are equal to 37.5% and 15.3%, respectively.

5. Concluding remarks Stacking faults in lamellar compounds can be investigated using a combined approach based on DFT calculations and X-ray diffraction experiments. The preliminary step that consists in the identification of translation vectors using DFT calculations is useful for complex systems. In particular, it allows the treatment of structures which present no obvious layer translation vectors. The second step is the refinement of translation probabilities and the stackingsequence extraction using DIFFaX+ code. This new approach could be used to reinvestigate diffraction patterns that have not been completely solved because of the occurrence of stacking faults.

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