Solution Properties of the System ZrSiO4–HfSiO4: A Computational

Apr 22, 2013 - ZrSiO4 and HfSiO4 are of considerable interest because of their low thermal expansions, thermal conductivities, and the optical propert...
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Solution Properties of the System ZrSiO4−HfSiO4: A Computational and Experimental Study Agustín Cota,†,‡ Benjamin P. Burton,§ Pablo Chaín,† Esperanza Pavón,∥ and María D. Alba*,† †

Instituto Ciencia de los Materiales de Sevilla (CSIC-US), Avda. Americo Vespucio 49, 41092-Sevilla, Spain Laboratorio de rayos-X (CITIUS), Avda. Reina Mercedes 4b, 41012-Sevilla, Spain § Material Measurements Laboratory, Metallurgy Division, National Institute of Standards and Technology (NIST), Gaithersburg, Maryland 20899, United States ∥ Unité de Catalyse et de Chimie du Solide, UCCS, CNRS, UMR8181, Université Lille Nord de France, 59655 Villeneuve d’Ascq, France ‡

S Supporting Information *

ABSTRACT: ZrSiO4 and HfSiO4 are of considerable interest because of their low thermal expansions, thermal conductivities, and the optical properties of HfSiO4. In addition, silicate phases of both are studied as model radioactive waste disposal materials. Previous first principles calculations reported near ideal mixing in the Zr1−xHfxSiO4 system, with a very weak propensity for phase separation. Density functional theory (DFT)/cluster-expansion first principles calculations presented in this work indicate near ideal mixing with a very weak propensity for ordering. Zr1−xHfxSiO4 samples (x = 0, 0.25, 0.5, 0.75, and 1.0) were synthesized from intimate stoichiometric mixtures of constituent-oxides and annealing at 1823 K for 20 days in a platinum crucible. Samples were characterized by X-ray diffraction (XRD; Rietveld analysis) and 29Si MAS NMR. The XRD data exhibited a pronounced negative deviation from Vegard’s law in the excess volume of mixing, and the 29Si MAS NMR spectra also suggest nonideal mixing. Given the very weak energetics that favor cation ordering, it is clear that there must be some other cause(s) for the observed deviations from ideal mixing behavior.

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al.10 determined the tetragonal lattice parameters (a and c) from the interplanar spacing of a set hkl reflections and concluded that ZrSiO4 and HfSiO4 are completely soluble. However, only the structure of ZrSiO4 doped with Hf 1 mol % was published.11 These results are consistent with first principles calculations by Ferriss et al.12 that suggested nearly ideal mixing with a very weak tendency for phase separation. The first principles phase diagram calculation reported below also predicts nearly ideal mixing, but with a very weak tendency for ordering, and a negative deviation from ideality in the excess volume of mixing, VxS(x) = V(x) − (1 − x)VZrSiO4 − xVHfSiO4, as a function of bulk composition. The aim of this work is to clarify the solution properties of the ZrSiO4−HfSiO4 system, via a combined experimental and computational study.

INTRODUCTION It is well-known that two of the most difficult elements to separate are Zr and Hf. Silicates of Zr and Hf are of considerable interest because of their low thermal expansion coefficients, thermal conductivities, and the optical properties of HfSiO4.1 In addition, Zr4+ and Hf4+ are commonly used to simulate the most important tetravalent rare earth cations in radioactive waste, and silicate phases of both are studied as model radioactive waste disposal materials.2,3 Most of the earth’s crust is made of silicates, in which Si is tetrahedrally coordinated [(SiO4)−4 tetrahedra], and the Si−O bonds have mixed ionic and covalent character. ZrSiO4 and HfSiO4 are isostructural and crystallize in the tetragonal orthosilicate zircon structure: space group I41/amd;4 with isolated (SiO4)−4 tetrahedra; a Q0 silicate in Liebau’s classification.5 In a polyhedral description, the zircon structure is a combination of isolated (SiO4)−4 tetrahedra that share edges with pseudohexagonal ZrO8 bypiramids along the tetragonal c-axis and share vertices with ZrO8 bypiramids along the a and b axes. The Shannon and Prewitt6 ionic radii for VIII-coordinated Zr4+ and Hf4+ are 0.83 Å and 0.84 Å, respectively, and electronegativities are 1.33 for Zr4+ and 1.30 for Hf4+;7 hence one expects nearly ideal mixing with a slightly larger cell volume for ZrSiO4 than for HfSiO4.8 Khalezova and Chernittsova9 predicted isomorphous substitution of Hf4+ for Zr4+ up to 10 mol %, based on the observation of a gradual change in the (301) X-ray diffraction peak. Ramanakirshnan et © 2013 American Chemical Society

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EXPERIMENTAL AND COMPUTATIONAL DETAILS Synthesis. Zr1−xHfxSiO4 samples with (x = 0, 0.25, 0.5, 0.75, and 1.0) were synthesized from intimate stoichiometric mixtures of: SiO2, ZrO2, and HfO2. Precursor oxides were heated at 1273 K before mixing and maintained in a dry atmosphere to avoid carbonation and/or hydration. The stoichiometric mixtures were ground in acetone and calcined Received: February 12, 2013 Revised: April 22, 2013 Published: April 22, 2013 10013

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at 1823 K for 20 days in a platinum crucible. Every 5 days the samples were ground in acetone. After thermal treatment, the furnace was turned off, and samples were cooled to room temperature in the furnace. X-ray Powder Diffraction (XRD). XRD patterns were recorded at the X-ray Laboratory of CITIUS (University of Seville, Spain), using a D8 Advance Bruker instrument equipped with a Cu Kα radiation source, operating at 40 kV and 40 mA, and a graphite monochromator. The diffractometer was calibrated mechanically according to the manufacturer specifications, and corundum and silicon standards were used to check the resolution in a wide range of angles. Powder samples were prepared using a side-packed methacrylate holder to ensure random microcrystalline orientation. Diffractograms were collected from 10° to 120° 2θ, with a scanning speed of 0.02° and a counting time of 10 s. Crystalline phase determinations were performed with the DIFFRAC PLUS Evaluation Software,13 and the structural refinements were carried out by Rietveld methodology14 using TOPAS 2.1 software.15 Nuclear Magnetic Resonance Spectroscopy (MAS NMR). Single pulse spectra were recorded at the Spectroscopy Service of ICMS (CSIC-US) using a Bruker AVANCE SB400 spectrometer equipped with a multinuclear probe. Powdered samples were packed into 4 mm zirconia rotors and spun at 10 kHz. 29Si MAS NMR spectra were collected at a frequency of 79.49 MHz, using a π/6 pulse width of 2.66 μs and a pulse space of 600 s. Spectra were simulated using the DMFIT software16 assuming infinite spinning speed. A Gaussian−Lorentzian model was used for all the peaks, and fitted parameters were: amplitude, position, line width, and Gaussian−Lorenztian ratios. 91 Zr and 177Hf spectra were not measurable because of their low gyromagnetic ratios (γ(91Zr) = 2.4975 × 10−7 rad·T−1·s−1 and γ(177Hf) = 1.086 × 10−7 rad·T−1·s−1) and low natural abundance, 11.22% and 18.6%, respectively. In addition, both nuclei have a strong quadrupole moment (Q(91Zr) = −0.21 × 10−28 m2 and Q(177Hf) = 4.5·10−28 m2) which contribute to broadening of central transitions over 350 kHz under the 9.4 T magnetic field.17 First-Principles Calculations. A first principles phase diagram calculation was performed for the Zr1−xHfxSiO4 system. The density functional theory (DFT) based Vienna ab initio simulation program (VASP, version 44518) was used with projector-augmented plane-wave pseudopotentials and the generalized gradient approximation for exchange and correlation energies. Formation energies for ZrSiO4, HfSiO4, and 62 ZrmHfn(SiO4)m+n supercells were calculated. Electronic degrees of freedom were optimized using a conjugate gradient algorithm, and both cell constant and ionic positions were fully relaxed. Pseudopotential valence electron configurations were: Zrsv: 4s24p65s24d2; Hfpv: 5s25p6s25d2; Os: 3s23p4; Si:4s24p2. Zrsv: 4s24p65s24d2; Hfpv: 5s25p66s25d2; Os: 3s23p4; Si: 4s24p2. K-point meshes were converged for each structure, and the energy cutoff was set at 500 eV. These settings yield converged values of formation energies to within a few meV/ exchangable-cation (Hf4+,Zr4+).

Figure 1. XRD patterns of Zr1−xHfxSiO4. * = cristobalite (PDF no. 750923) and MO2 (M = Zr, Hf) (PDF no. 37-1484).

patterns showed similar numbers and shapes of reflections with the only differences being in intensities, and in shifts of the reflections to lower angles, in HfSiO4-rich samples, consistent within the slightly larger (Shannon and Prewitt) ionic radius of Zr4+.19 The fitting parameters are shown in Table 1. Qualitatively, intermediate compositions (x = 0.25, 0.50, and 0.75) have diffraction patterns that are similar to those for ZrSiO4 and HfSiO4 with a progressive position and intensity shifts in the XRD reflections as functions of composition, Figure 1. The 101 reflection shifted to a lower 2θ angle, and 101, 211, and 220 reflection intensities decreased as Zr content increased. Additionally, the XRD patterns of the x = 0.50 and 0.75 samples showed two low-intensity reflections (marked with asterisk) which corresponded to cristobalite (PDF 750923) and Hf and/or Zr oxides (PDF 37-1484). Total deviations of unit cell parameters between the extreme compositions are less than 0.47%20 owing to the small difference between Zr and Hf ionic radii. Hence, XRD is not a sensitive technique for phase or composition determinations. The Zr0.5Hf0.5SiO4 XRD patterns for both a random solution phase configuration and an equimolar mechanical mixture of ZrSiO4 + HfSiO4 were simulated with the GSAS software (Figure 2).21 The comparison between simulated and experimental patterns indicates that a solid solution is the best model for Zr0.5Hf0.5SiO4. Simulated XRD patterns of solid solutions for all compositions were compared with raw data in Figure 3; good agreement between them suggests that this model is a good starting point for structure refinements. Rietveld refinements were performed with the TOPAS Software15 using ZrSiO4 crystallographic parameters22 as starting values for samples with x < 0.5 and HfSiO4 structure parameters23 for samples with x ≥ 0.5. Refined parameters were: background coefficient, histogram scale, frame parameters, reflections profile, occupation factors of Zr and Hf, and temperature factors for all atoms.24 Refined unit cell parameters, unit cell volume, Hf/Zr site occupancies, and goodness-of-fit parameters (GOF and Rwp) are listed in Table 2. The occupancy factors of the M site (M = Zr or Hf) agreed with the nominal composition, the maximum difference being 0.04. Unit cell parameters vary smoothly as functions of composition, in Figure 4, consistent with complete solid solution. Figure 4 shows plots of unit cell constants as functions of x; a(x), c(x), and V(x), all of which exhibit sigmoidal trends similar to (Na,K) nephelite25 and (Mg,Fe) amphibole26 with an

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RESULTS AND DISCUSSION XRD Data. XRD diagrams of extreme composition (x = 0 and 1), Figure 1, showed patterns compatible with HfSiO4 (PDF 06-266) and ZrSiO4 (PDF 20-467), respectively. Both 10014

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Table 1. XRD Fitting Parameters of Zr1−xHfxSiO4a x

M site

Zr1−xHfxSiO4

Rwp

GOF

0 0.25 0.50 0.75 1.00

14.8 19.4 15.3 12.9 9.7

1.4 2.2 1.8 1.6 1.3

a (Å) 6.606 6.597 6.586 6.578 6.576

± ± ± ± ±

0.001 0.001 0.001 0.001 0.001

V (Å3)

c (Å) 5.983 5.978 5.974 5.971 5.970

± ± ± ± ±

0.001 0.001 0.001 0.001 0.001

261.053 260.126 259.118 258.411 258.121

± ± ± ± ±

0.012 0.003 0.003 0.018 0.009

Zr

Hf

1.00 0.79 ± 0.03 0.52 ± 0.06 0.26 ± 0.06 0.00

0.00 0.21 ± 0.03 0.48 ± 0.06 0.74 ± 0.06 1.00

a

ZrSiO4 crystallographic parameters (ref 22) was used as starting values for samples with x < 0.5 and HfSiO4 structure parameters (ref 23) was used as starting values for samples with x ≥ 0.5.

Figure 2. XRD patterns of x = 0.5: (a) experimental; (b) calculated for a solid solution ZrHfSiO4; and (c) calculated for a 50% mixture of ZrSiO4−HfSiO4. * = cristobalite (PDF no. 75-0923) and MO2 (M = Zr, Hf) (PDF no. 37-1484).

Figure 4. (a) Lattice parameters and (b) unit cell volume of Zr1−xHfxSiO4 as a function of Hf content.

inflection point between 0.50 < x < 0.75. The greater negative deviation of a(x) than c(x) reflects greater structural rigidity along the c-axis. Vegard’s law,8 posits linear variation of unit cell volume as a function of bulk composition, V(x) (dotted lines, Figure 4b), and the observed trend clearly exhibits a negative deviation from Vegard’s law, which is suggestive of nonideal mixing.

Figure 3. XRD patterns of Zr1−xHfxSiO4: Raw (dots) and simulation of solid solution (line). * = cristobalite (PDF no. 75-0923) and MO2 (M = Zr, Hf) (PDF no. 37-1484).

Table 2. Relative Intensities of the Different Si Environments and the Gravity Center of the 29Si NMR Peaks of Zr1−xHfxSiO4 percentage x 0 0.25 0.5 0.75 1.00

0Zr·6Hf 14.2 25.8 38.6 100

1Zr·5Hf 7.4 9.0 22.9

2Zr·4Hf 7.6 16.3 18.3

3Zr·3Hf 2.8 9.0 4.1

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4Zr·2Hf 11.1 13 7.0

5Zr·1Hf

6Zr·0Hf

δCG (ppm)

21.8 12.5 5.1

100 35.2 14.5 4.0

−81.7 −80.4 −79.6 −78.8 −78.2

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First Principle Calculations. A cluster expansion (CE) Hamiltonian27 was fit to the DFT-calculated formation energies using the Alloy Theoretic Automated Toolkit (ATAT)28 which automates most of the tasks associated with construction of a CE Hamiltonian. The CE Hamiltonian was used as input for Monte Carlo calculations of the phase diagram (Figure 5),

Figure 5. First principles calculated phase diagram. Cation order− disorder is predicted to occur at such a low temperature that it is unlikely to be realized experimentally.

which indicates a very weak tendency for ordering with at least two ordered ground-states (at x = 0.5 and x = 0.75). A molar volume CE (CEV) was also fit to DFT-calculated molar volumes, and the calculated theoretical trend is plotted in Figure 6: solid circles indicate DFT values for VxS/cation; open squares indicate CEV calculated values; near-continuous curves (blue and red online) indicate VxS(x) trends for calculated cation distributions at 1823 and 100 K, respectively (essentially from a near-random distribution at 1823 K to a strongly shortrange ordered distribution at 100 K). Qualitatively, the negative deviation from Vegard’s Law (Figure 4b) in V(x) and VxS(x) is consistent with the first principles calculation (Figures 6) which also predicts a negative deviation from Vegard’s law, even with a near-random cation distribution. Quantitatively, however, the agreement with experiment is poor (Figure 6b). The calculated negative deviation is about an order of magnitude smaller than the experimentally determined deviation, and Zr:Hf short-range ordering during the furnace cool cannot explain such a large discrepancy. Experimentally, the samples were furnace-cooled from 1823 K to room temperature, so the quenched cation distribution should be characteristic of the blocking-temperature for cation diffusion: between 1823 K and 100 K, and probably in excess of 1000 K. Note, however, that even if one compares the totally unrealistic 100 K curve to experiment, there is still nearly an order of magnitude difference. Effects of Postulated Cation Distributions on the Zr1−xHfxSiO4 NMR Spectra. The ZrSiO4−HfSiO4 29Si MAS NMR spectra are shown in Figure 7. Spectra of pure phases (x = 0 and 1) exhibit a unique symmetric peak in the Q0 region,29 which is compatible with a unique silicon site surrounded by four unshared oxygen atoms. 29Si signal of ZrSiO4 was centered at −81.7 ppm, as previously reported in the literature.30 29Si signal of HfSiO4 was centered at −78.2 ppm. No reference of 29 Si spectrum of this structure has been published, but this 29Si chemical shift agrees with the few 29Si MAS NMR studies of Hf silicates which demonstrated that when Zr is substituted by Hf the signal shifts 2−3 ppm to higher frequencies.31−33 29Si MAS

Figure 6. DFT-calculated excess molar volumes (ΔVxS/cation; solid circles), cluster expansion (CEV) calculated ΔVxS/cation (open squares), and CEV-calculated theoretical trends (near-continuous curves, blue and red). Trends were calculated cation distributions at 1823 K and 100 K, respectively [essentially from a near-random distribution at 1823 K (upper curve) to a strongly short-range ordered distribution at 100 K (lower curve)].

Figure 7. 29Si MAS NMR spectra of Zr1−xHfxSiO4.

NMR spectra of intermediate composition samples displayed a complex profile with contributions centered in the chemical shift ranges of the pure phases. Janes and Oldfield’s34 mathematical model for predicting the 29 Si chemical shift has been used to interpret the complex 10016

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spectra of the ZrSiO4−HfSiO4 system. This model was based on group electronegativities of ligands bonded to Si. Janes and Oldfield34 defined three types of silicon: (i) Type S silicon: all ligands are σ-bonds only and lack the lone-pair p-hybridized orbitals necessary for further conjugations; (ii) Type P silicon: all ligands are capable of σ- and π-bonding; and (iii) Type M silicon: the ligands are mixed, so silicon is coordinated to both S and P ligand types of σ- and π-bonds are implied. The OM (M = Zr or Hf) group is a P ligand type, and the 29Si chemical shift correlation proposed was:

δ = −97.344[(1 − x)ENOZr + x ENOHf ] + 279.27

where x is the mole fraction of Hf (x = 0.5 in this sample). (ii) Random distribution of cations allows seven different silicon environments: nOZr, (6 − n)OHf with 0 ≤ n ≤ 6. Thereby, seven Lorentzian contributions of 0.71 ppm fwhh will be observed in the 29Si spectra. Relative signal intensities are dominated by probability law, and the chemical shift of each Si site can be calculated by: δ = −16.224[(6 − n)ENOZr + n ENOHf ] + 279.27

δ = −24.336 ∑ EN + 279.27

The 29Si NMR spectra of the intermediate composition were deconvoluted in seven contributions, and the chemical shifts and intensities of the lines are listed in Table 2. The comparison between the percentage of each nOZr, (6 − n)OHf environment for the Zr1−xHfxSiO4 (0.25 ≤ x ≤ 0.75) with those calculated from a random distributions, Figure 9, showed that:

where ΣEN is the sum of the group electronegativity of the ligand OM and can be calculated with the following equation:

∑ EN = 4 ∑ (ENnf znf / ∑ znf ) where ENnf and znf represent the nonframework ligands electronegativity (OM) and the cation effective charge (M = Zr, Hf), respectively. Group electronegativity of the OZr and OHf group could be obtained from the experimental 29Si chemical shift of the pure ZrSiO4 and HfSiO4 phases and from the above equations (ENOZr = 3.70819 and ENOHf = 3.67254). The comparison of experimental and simulated spectra, with homogeneous and random distributions of Zr and Hf, spectra for the x = 0.5 sample is shown in Figure 8a−c. The simulation

Figure 8. 29Si MAS NMR spectra of x = 0.50 composition: (a) experimental, (b) random distribution model, and (c) homogeneous distribution model; and (d) local environment of SiO4 tetrahedron surrounded by six M atoms (M = Zr, Hf).

of the spectra for both cation distribution models was performed using the chemical shifts calculated from Janes and Oldfield’s equations and taking into account that each silicon is surrounded by six MO8 groups (Figure 8d). For Zr0.5Hf0.5SiO4 composition, two different scenarios should be expected. (i) Homogenous distribution (3OZr, 3OHf): only one Si environment is possible and the 29Si spectrum is characterized by a unique Lorentzian signal of 0.71 ppm fwhh as seen in pure sample spectra. The chemical shift depends on composition and could be calculated from the Janes and Oldfieldś equation:

Figure 9. Si-environment distribution in of Zr1−xHfxSiO4: calculated from a random Zr:Hf-distribution (white: calculated from nominal xvalue, and, gray: calculated from Rietveld x-value) and from 29Si MAS NMR spectra (black). 10017

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parameters with which to fit the data. A more likely explanation is that positional disorder of Zr- and Hf in the MO8 units is responsible for the observed variations in NMR spectra; i.e. that the Zr:Hf cation distribution is close to random, but that local deviations from randomness coupled to Zr- and Hf-positional disorder create the apparent variation in Si environments.

(i) For all compositions, the assumption of chemical shortrange order improved the fit relative to that for a random Zr:Hf-distribution. (ii) For the x = 0.25 sample, a Si-site surrounded by a Zr-rich environment (n ≥ 4) yielded the best fit. (iii) For the x = 0.75 sample, a Si-site surrounded by a Hf-rich environment (n ≤ 2) yielded the best fit. The centers of gravity for the 29Si signals in all samples were calculated by as:

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CONCLUSIONS The ZrSiO4−HfSiO4 system forms a complete solid solution that exhibits large and unexplained negative deviations from Vegard’s law in VxS(x). Qualitatively, first principles calculations predict a negative deviation in VxS(x), for both a random cation distribution and for a distribution that is strongly short-range ordered. However, even maximal short-range Zr:Hf-cation ordering does not adequately explain the magnitude of the experimentally observed trend. Fits to 29Si MAS NMR spectra are improved by assuming Zr:Hf-short-range cation order, but because the first principles results indicate such weak energetics for cation order−disorder, it seems necessary to invoke some other phenomenon to explain observed deviations from ideal solution (randomcation) behavior.

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δCG =

∑n = 0 {(Inδn)} 6

∑n = 0 In

where In and δn are the relative intensity and the chemical shift of any nOZr, (6 − n)OHf Si-environment, respectively. The correlation between the δCG for the 29Si signal and the calculated δCG from Janes and Oldfieldś equation34 is plotted in Figure 10a. A linear correlation with R2 = 0.9885 was obtained with a mean deviation of 0.31 ppm, which is much lower than those accepted by the Janes and Oldfield’s model (1.96 ppm).34

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ASSOCIATED CONTENT

S Supporting Information *

Plots of the Rietveld refinement of the XRD patter (Suppl. 1− 5) and the fits of the 29Si MAS NMR spectra (Suppl. 6). This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Fax: +34 954460665. Tel.: +34 954489546. E-mail: alba@ icmse.csic.es. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We would like to thank the DGICYT and FEDER funs (project no. CTQ 2010-14874) for their financial support. REFERENCES

(1) Schelz, S.; Branland, N.; Plessis, D.; Minot, B.; Guillet, F. Laser Treatment of Plasma-Sprayed ZrSiO4 Coatings. Surf. Coat. Technol. 2006, 200, 6384−6388. Ozel, E.; Turan, S. Production of Coloured Zircon Pigments from Zircon. J. Eur. Ceram. Soc. 2007, 27, 1751− 1757. Sancho, M.; Arnal, J. M.; Pineda, A.; Catala, R.; Lora, J. Application of Membrane Technology for the Treatment of Effluent from a Zirconium Silicate Production Process. Desalination 2005, 178, 361−367. Xiong, K.; Du, Y.; Tse, K.; Robertson, J. Defect States in the High-Dielectric-Constant Gate Oxide HfSiO4. J. Appl. Phys. 2007, 101, Art No. 024101. Lin, Y. H.; Chien, C. H.; Chou, T. H.; Chao, T. S.; Lei, T. F. Low-temperature Polycrystalline Silicon Thin-Film Flash Memory with Hafnium Silicate. IEEE Trans. Electron Devices 2007, 54, 531−536. (2) Devanathan, R.; Corrales, L. R.; Weber, W. J.; Chartier, A.; Meis, C. Molecular Dynamics Simulation of Energetic Uranium Recoil Damage in Zircon. Mol. Simul. 2006, 32, 1069−1077. (3) Cherniak, D. J.; Hanchar, J. M.; Watson, E. B. Diffusion of Tetravalent Cations in Zircon. Contr. Miner. Petr. 1997, 127, 383−390. Cherniak, D. J.; Hanchar, J. M.; Watson, E. B. Rare-Earth Diffusion in Zircon. Chem. Geol. 1997, 134, 289−301. (4) Hazen, R. M.; Finger, L. W. Crystal-Structure and Compressibility of Zircon at High-Pressure. Am. Mineral. 1979, 64, 196−201.

Figure 10. (a) Graph showing correlation between experimentally determined center of gravity for the 29Si-signal and the center of gravity predicted from Janes and Oldfield’s model. (b) 29Si NMR center of gravity for Zr1−xHfxSiO4 as a function of Hf content. Dots are the experimental values. The dashed line indicates the theoretical value for a random Zr:Hf cation distribution.

The first principles calculations (Figure 6) indicate that a near random Zr:Hf cation distribution is expected, and shortrange order is clearly not the primary cause of the observed negative deviation from Vegard’s law in VxS(x), Figure 4. Invoking Zr:Hf-short-range order allows one to generate improved fits to the NMR spectra of samples with 0.25 ≤ x ≤ 0.75, but given the first principles results, it seems likely that the improvement is only a consequence of having more 10018

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dx.doi.org/10.1021/jp401539g | J. Phys. Chem. C 2013, 117, 10013−10019