Toward an Understanding of the Local Origin of Negative Thermal

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Toward an Understanding of the Local Origin of Negative Thermal Expansion in ZrW2O8: Limits and Inconsistencies of the Tent and Rigid Unit Mode Models Andrea Sanson Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/cm501107w • Publication Date (Web): 23 May 2014 Downloaded from http://pubs.acs.org on May 31, 2014

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Toward an Understanding of the Local Origin of Negative Thermal Expansion in ZrW2O8: Limits and Inconsistencies of the Tent and Rigid Unit Mode Models A. Sanson∗ Department of Physics and Astronomy, University of Padova, Padova (Italy) E-mail: [email protected]

Abstract Although zirconium tungstate (ZrW2 O8 ) is the most popular negative thermal expansion (NTE) material, the exact mechanism responsible for its NTE still remains controversial. Specifically, the "Tent" model [Cao et al., Phys. Rev. Lett., 2002, 89, 215902, Bridges et al., Phys. Rev. Lett., 2014, 112, 045505] and the "rigid unit mode" (RUM) model [Tucker et al., Phys. Rev. Lett., 2005, 95, 255501] were subject of debate during recent years. This work aims to shed light on this issue by means of molecular dynamics simulations which allow us to separate, for each bond distance, the "true" thermal expansion from the "apparent" thermal expansion, as well as to study the effective bond strength and the anisotropy of relative thermal motion. In spite of the good agreement with the experimental data of Cao, Bridges and co-workers, it has been observed a decrease of the "true" W-Zr distances accompanied by large transverse vibrations of the O atoms in the middle of the W-O-Zr linkage, in sharp contrast to the "tent" model. Moreover, in contrast to the RUM model, it has been found that the WO4 ∗ To

whom correspondence should be addressed

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and ZrO6 polyhedra are strongly distorted by thermal motion, and, more importantly, that intra-polyhedra contributions to NTE are present. Accordingly, we can conclude that both the tent and RUM models are inadequate to explain NTE in ZrW2 O8 , and a more flexible model, simply based on rigid nearest W-O and Zr-O bonds and tension effect, should be adopted.

Introduction Negative thermal expansion (NTE), i.e., material contraction on heating over a certain temperature range, is relatively rare but has important technological applications. 1 Since the discovery in 1996 that zirconium tungstate (ZrW2 O8 ) exhibits large isotropic NTE over a wide temperature range (from 0.3 to 1050 K), 2 the interest in this compound has rapidly grown to become the most popular NTE material, often used as the key representative of NTE. 3–6 The crystal structure of ZrW2 O8 is cubic with space group P21 3 between 0 and about 430 K (the α -phase), while at higher temperatures the space group changes to Pa3 (the β -phase) through an order-to-disorder phase transition due to the disordering of the WO4 tetrahedra. 7,8 A sketch of the crystal structure of α -ZrW2 O8 is shown in Fig. 1: it consists of a network of corner-linked ZrO6 octahedra and WO4 tetrahedra, with one non-bridging W-O bond on each tetrahedron which confers great flexibility to the ZrW2 O8 framework. Many works on ZrW2 O8 have aimed at identifying the mechanism for the NTE. In the early studies, Mary et al. 2 attributed the NTE of ZrW2 O8 to the existence of large transverse vibrations of the O atoms in the middle of the Zr-O-W linkage, which combined with the presence of rigid Zr-O and W-O bonds pull the W and Zr atoms closer together. This model was further developed by Pryde et al. to include the linkage between the ZrO6 and WO4 polyhedra through the concept of "rigid unit modes" (RUMs): 9,10 the transverse vibrations perpendicular to the Zr-O-W linkage are the result of whole-body rotations and translations of the linked ZrO6 and WO4 polyhedra with essentially no distortion. These vibrations of the undistorted polyhedra are low-frequency modes 0a

Department of Physics and Astronomy, University of Padova, Via Marzolo n. 8, I-35131, Padova, Italy. E-mail: [email protected]

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which describe the correlation between the thermal motion of the oxygen atoms; with increasing temperature they increase in amplitude leading to negative thermal expansion. Evidence of this low-frequency lattice dynamics, relevant to negative thermal expansion, was observed in neutron scattering experiments, 11 low-temperature specific heat measurements, 12 infrared and Raman spectroscopic studies, 13,14 and in the temperature dependence of the α -ZrW2 O8 lattice constant. 15 However, although the RUM model provides the essential features for understanding the NTE, the exact nature of the local mechanism responsible for NTE in ZrW2 O8 still remains controversial. In this regard, Cao et al. 16,17 proposed, by means of extended x-ray absorption fine structure (EXAFS) measurements, that in zirconium tungstate the Zr-O-W linkage should be stiff enough so that the essentially rigid components of the structure are larger and composed of a WO4 tetrahedron and its three nearest ZrO6 octahedra, thereby invalidating the RUM model for ZrW2 O8 . According to this model (the "Tent model"), the NTE in α -ZrW2 O8 was explained as due to the translational motion of WO4 tetrahedra along the axis and the correlated motion of the three nearest ZrO6 octahedra. In contrast, Tucker et al., 18 by reverse Monte Carlo analysis of neutron total scattering data for α -ZrW2 O8 , have concluded that the Zr-W linkages are not particularly stiff and the WO4 and ZrO6 polyhedra move as rigid units and that the NTE arises from RUMs. Moreover, according to Tucker et al., a larger rigid structural component involving the Zr-O-W linkage, as proposed by Cao et al., would have the effect of stiffening the structure inhibiting the NTE. More later, Gava et al., 19 by combining first-principles calculations and far infrared absorption spectroscopy, have suggested that the lowest energy optic modes, which contribute significantly to the NTE of α -ZrW2 O8 , exhibit features characteristic of both the RUM and tent models, but without pointing out the weaknesses of one or the other model. Very recently, Bridges and co-workers 20 have published a new EXAFS study combined with X-ray pair distribution function analysis, where they confirm that the Zr-W linkage is relatively stiff and does not permit bending of the Zr-O-W linkage, thus again supporting the idea of the "tent model". This work aims to shed light on this controversy by using molecular dynamics simulations to

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Figure 1: Sketch 21 of the crystal structure of α -ZrW2 O8 showing the corner-linked ZrO6 octahedra and WO4 tetrahedra. The latter can be distinguished in W1O4 (green) and W2O4 (blue) tetrahedra having different bond lengths (see Tab. 1). The atoms are labeled as in Refs., 17,18 i.e., bridging oxygens as O1 and O2, non-bridging oxygens as O3 and O4, respectively.

study the local vibrational dynamics of α -ZrW2 O8 . This approach allows us to directly investigate the thermal expansion of neighboring bonds and the degree of rigidity of corner-sharing polyhedra, including the correlation of atomic motion and separating the "true" bond length hri=h|r1 − r2 |i from the "apparent" bond length R=|hr1 i − hr2 i|. 5 This is important because the rigid unit modes, which cause rigid rotations of the basic polyhedral units, lead to NTE of the "apparent" intrapolyhedral bond distances but leave unchanged the "true" intra-polyhedra bond distances. The thermal expansion of the true bond lengths was exploited to rule out the possibility of explaining NTE in cuprite structures by a simple RUM model. 22,23

Computational details Molecular dynamics (MD) simulations have been performed using the General Utility Lattice Program (GULP) code. 25 The MD calculations were carried out over the temperature range of 0-300 K on a simulation box of 2×2×2 supercell dimension (608 atoms/shells), with periodic boundary 4



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conditions, within the canonical NVT ensemble. The cell parameter was adjusted at each temperature according to the crystallographic data. 2,7 Integration of the Newton equations of motion was performed using the Verlet leapfrog algorithm 27 with a time step of 0.5 fs. At each temperature, the MD data have been collected for a total time of 8 ps after an initial equilibration time of 1 ps. An in-house software was developed to determine, from MD trajectories, the vibrational properties needed for this study. The interatomic potential derived by Pryde et al. 9 for ZrW2 O8 has been used for the present simulations. It includes short-range Buckingham potentials for the W-O, Zr-O and O-O pairs, the Coulomb interaction, a shell model for the oxygen atoms, and a bond-bending potential for the OW-O and O-Zr-O bonds. The quality of this interatomic potential was assessed by comparing the calculated crystal structure with the experimental data. More importantly, Pryde et al. calculated the thermal expansion of α -ZrW2 O8 by this potential within the quasi-harmonic approximation, and the results were in excellent agreement with the NTE obtained by experiment. 9 The reliability of this model potential was also validated by simulation studies of ZrW2 O8 at high pressure. 26 As a consequence, it makes sense to use this interatomic potential also for the present study. It is acknowledged that to reproduce the exact vibrational dynamics, one should know the real crystal potential. Nevertheless, the overall agreement between the present simulations and the experimental results of Cao, Bridges and co-workers 16,17,20 (see Fig. 3 below) is a further confirmation that the Pryde’s potential is a good approximation of the real crystal potential. In particular, the agreement for the W-O and Zr-O pairs (black color in Fig. 3) confirms the validity of the corresponding W-O and Zr-O short-range potentials, while the agreement for the W-W, ZrZr and Zr-W pairs (red, blue and green colors in Fig. 3) supports the reliability of the O-W-O and O-Zr-O bending potentials. The top panel of Fig. 2 shows the instantaneous "true" bond length hri=h|r1 − r2 |i between some pairs of atoms, calculated from MD trajectories at 300 K, as a function of the simulation time. By Fourier transform analysis, the corresponding vibrational density of states (VDOS) can be obtained for each distance, as shown in the bottom panel of Fig. 2. It can be observed that

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the calculated VDOS cover the entire range reported by neutron-scattering measurements (0-140 meV), including the low-energy modes below 5 meV. 11 This ensures that the choice of a 2×2×2 simulation cell, whose size is larger than 18 Å, allow us to capture the low-energy phonon modes important for NTE. 13,14

Istantaneous true distance (Å)

7 Zr-Zr

6 5 4

W1-W2(b) W1-W2(a) Zr-W2 Zr-W1

3

O1-O2(a)

2

Zr-O1 W1-O1

1

0

2

4

6

8

Time (ps)

Zr-Zr W1-W2(b)

Projected VDOS

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W1-W2(a) Zr-W2 Zr-W1 O1-O2(a) Zr-O1 W1-O1

0

20

40

60

80

100

120

140

160

hQ (meV)

Figure 2: Instantaneous "true" bond length hri=h|r1 − r2 |i between some pairs of atoms, calculated from MD trajectories at 300 K, as a function of the simulation time (top panel), and corresponding projected density of vibrational states (bottom panel).

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Results and Discussion For each bond (or linkage) of α -ZrW2 O8 , the following key parameters, that need to be defined here, have been calculated from MD trajectories: a) the thermal expansion coefficients αR and αr of the "apparent" and "true" bond length, respectively

αR =

1 δR R δT

αr =

1 δ hri hri δ T

(1)

Owing to the effect of relative vibrations perpendicular to the bond direction, the true bond length hri and the corresponding thermal expansion αr are larger than the apparent bond length R and the corresponding thermal expansion αR . 5 For completeness, also the "apparent" and "true" volume thermal expansion coefficient, βR and βr , respectively, have been calculated for each polyhedron. b) the variance σ 2 of the atom-pair distance distribution. With u¯1 and u¯2 indicating the instantaneous thermal displacements of a pair of atoms, it can be shown that σ 2 corresponds to a very good approximation to the projection of the Mean-Square Relative Displacement (MSRD) along ˆ i.e. the bond direction R, ˆ 2i σ 2 ' h∆u2k i = h[(u¯1 − u¯2 ) · R]

(2)

Therefore, within the Einstein approximation in the classical limit, σ 2 can be connected to an "effective" bond-stretching force constant kk by the relation σ 2 ∼ KB T /kk . 28 Its value can be utilized to estimate and compare the strength of different bonds. For the bond distances of polyhedral units which rotate rigidly, it is obvious that αR < 0, αr = 0, σ 2 = 0 and kk → +∞. c) by calculating the perpendicular MSRD, say the projection of the total MSRD in the plane normal to the bond direction, 28 i.e., h∆u2⊥ i = h[(u¯1 − u¯2 ) · Rˆ ⊥ ]2 i, we can determine the anisotropy of the relative thermal vibrations by the ratio γ = h∆u2⊥ i/h∆u2k i. For perfect isotropy, γ =2. Large values for γ can be correlated to the tension effect giving rise to the negative contribution to thermal expansion. 29 Let us remember that in the tension effect (sometimes referred to as "guitar string effect") the bonds between some pairs of atoms are so strong that they are as rigid. Therefore relative motion of the atoms along the bond direction is prohibited, so there are no "bond-stretching" but 7


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"transverse motion" to the bond direction which induces a NTE. 5 The MD results obtained for these parameters are presented in Tab. 1. In the first instance, we observe that i) the nearest W-O and Zr-O bonds are very rigid, their relative atomic motion is well correlated along the bond direction and exhibits a strong anisotropy ii) also the Zr-W atom pairs seem to be quite rigid with a large anisotropy of relative atomic motion iii) in contrast, the nearest W-W and Zr-Zr pairs are not rigid, their relative atomic motion is poorly correlated and close to the isotropic case, with along-bond anisotropy (γ < 2) for the W1-W2(a) pairs, consistent with the elongated thermal ellipsoids observed by neutron powder diffraction 4 iv) the O-O distances (edges of the WO4 and ZrO6 polyhedral units) are not particularly stiff and, some of these, show a surprising negative thermal expansion of the "true" bond length. These results will be discussed below. First of all, let us focus on the "tent model" proposed by Cao and co-workers 16,17 and corroborated by the recent work of Bridges and co-workers. 20 In their EXAFS study, from the very small temperature dependence of σ 2 for the nearest W-O and Zr-O distances, Cao et al. stated that the WO4 tetrahedra are rigid and the ZrO6 octahedra are stiff but not rigid. The W-W and Zr-Zr pairs, on the contrary, showed a much larger temperature dependence of σ 2 and consequently the vibrations of these atom pairs are practically uncorrelated. More importantly, the σ 2 for the Zr-W pairs showed a small temperature dependence, comparable to that for the nearest Zr-O bond. Therefore Cao et al. assumed that nearest Zr-O-W linkage is quite stiff and suggested the existence of a larger quasi-rigid unit involving a WO4 tetrahedron and its three nearest ZrO6 octahedra. Hence they concluded that the transverse vibrations of O in the Zr-O-W linkage cannot be the primary origin of NTE in ZrW2 O8 and proposed the "tent model" to explain it. As it can be observed from Fig. 3 (and Tab. 1), our calculated σ 2 are in good agreement with the EXAFS results of Cao et al. and with recent experimental results of Bridges et al., because they confirm the high rigidity of the W-O and Zr-O bonds, the more flexibility of the W-W and Zr-Zr distances, and the small σ 2 for the Zr-W pairs. However, a more careful analysis shows that Cao’s et al. interpretation and the subsequent tent model are inconsistent with the other MD results

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Table 1: Summary of the bond properties of α -ZrW2 O8 at 300 K calculated from MD simulations: thermal expansion coefficient of the "apparent" (αR ) and "true" (αr ) bond lengths, variance (σ 2 ) of the atom-pair distance distribution and corresponding "effective" bond-stretching force constant (kk ), perpendicular to parallel anisotropy of relative motion (γ ). At the bottom of the table are given the "apparent" and "true" volume thermal expansion coefficient (βR and βr , respectively) calculated for each polyhedral unit.

W1–O1 W1–O4 W2–O2 W2–O3 Zr–O1 Zr–O2 Zr–W1 Zr–W2 W1–W2 (a) W1–W2 (b) Zr–Zr W1–O3 O1–O2 (a) O1–O2 (b) O1–O1 (a) O2–O2 (a) O1–O1 (b) O1–O4 O2–O2 (b) O2–O3

3×1.797 Å, W1O4 bond 1×1.707 Å, W1O4 bond 3×1.782 Å, W2O4 bond 1×1.733 Å, W2O4 bond 3×2.042 Å, ZrO6 bond 3×2.109 Å, ZrO6 bond 3×3.744 Å, ZrO6 –W1O4 linkage 3×3.880 Å, ZrO6 –W2O4 linkage 1×4.119 Å, W1O4 –W2O4 linkage 3×4.639 Å, W1O4 –W2O4 linkage 6×6.473 + 6×6.481 Å, ZrO6 –ZrO6 linkage 1×2.386 Å, W1–O3 non-bonding distance 3×2.873 Å, ZrO6 edge 3×2.924 Å, ZrO6 edge 3×2.924 Å, ZrO6 edge 3×3.019 Å, ZrO6 edge 3×3.050 Å, W1O4 edge 3×2.716 Å, W1O4 edge 3×2.914 Å, W2O4 edge 3×2.866 Å, W2O4 edge

W1O4 tetrahedron W2O4 tetrahedron ZrO6 octahedron

αR (10−5 K−1 ) -1.18 -1.45 -1.03 -0.84 -0.71 -0.70 -0.79 -0.82 -0.75 -1.32 -1.07 -0.81 -1.44 +0.31 -1.16 -0.83 -0.78 -1.35 -1.11 -0.87 βR (10−5 K−1 ) -3.27 -2.96 -2.42

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αr (10−5 K−1 ) -0.01 +0.51 -0.02 +0.27 +0.24 +0.18 -0.51 -0.62 -0.59 -1.13 -0.99 +0.19 -0.67 +1.11 -0.29 -0.25 +0.05 +0.20 -0.35 +0.02 βr (10−5 K−1 ) +0.39 -0.50 +0.16

σ2 (Å2 ) 0.0009 0.0007 0.0006 0.0006 0.0010 0.0007 0.0020 0.0011 0.0139 0.0123 0.0072 0.0141 0.0120 0.0101 0.0046 0.0101 0.0105 0.0041 0.0041 0.0032

kk (eV/Å2 ) 30.1 38.4 40.1 42.3 26.6 34.9 13.2 22.9 1.9 2.1 3.6 1.8 2.2 2.6 5.6 2.6 2.5 6.3 6.3 8.0

γ 25.9 53.0 29.2 32.8 26.8 31.8 12.3 15.7 1.2 2.0 2.5 2.5 3.1 4.1 10.1 3.5 4.4 17.2 8.9 13.7

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0.016

V 7 V Å

0.012

Experimental W-O Cao W-O Bridges W-Zr Cao W-Zr Bridges W1-W2 Cao W1-W2 Bridges W1-W2 (a) Cao

Calculated (present work) W-O W-Zr W1-W2 W1-W2 (a)

Experimental Zr-O Cao Zr-W Cao Zr-Zr Cao

Calculated (present work) Zr-O Zr-W Zr-Zr

0.008

0.004

0.000 0.016

0.012

Zr-O Bridges Zr-W Bridges

0.008

V 7 V Å

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0.004

0.000 0

100

200

300

0

100

T (K)

200

300

400

T (K)

Figure 3: Comparison between calculated and experimental 16,17,20 temperature-dependence of σ 2 (right and left panels, respectively) for the W-O, W-Zr, W1-W2 and W1-W2(a) distances (top panels), and for the Zr-O, Zr-W and Zr-Zr distances (bottom panels). The data are plotted with respect to the lowest temperature to avoid possible errors on the absolute value due to the inelastic term S02 of EXAFS.

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listed in Tab. 1. Indeed we observe that: – both the "true" Zr-W1 and Zr-W2 distances display a negative thermal expansion (αr ' −(0.5-0.6)·10−5 K−1 ) in contrast to the tent model, which is based on the assumption that the WO4 tetrahedron and its three nearest ZrO6 octahedra form a rigid (or quasi-rigid) unit, so that it should be αr = 0 for the Zr-W distances; – the shrink of the Zr-W distances is accompanied by the presence of large transverse vibrations of the O atoms in the middle of the Zr-O-W linkage (see Tab. 1, on average γ ' 28 for the W1O1 and Zr-O1 bonds, γ ' 37 for the W2-O2 and Zr-O2 bonds). Therefore, contrary to what was claimed by Cao et al., the transverse vibrations of the O atom in the middle of the Zr-O-W linkage must have an important role in the NTE of ZrW2 O8 . In the light of this result, the RUM model 9,10,18 seems to be more appropriate to explain the origin of NTE in ZrW2 O8 , because the transverse vibrations perpendicular to the Zr-O-W linkage could be considered as the result of whole-body rotations and translations of the linked WO4 and ZrO6 rigid units. However, the MD results for the O-O pairs (see Tab. 1), edges of the WO4 and ZrO6 polyhedra, also highlight the limits of the RUM model. This because: – the relatively high values of σ 2 for the O-O pairs, in part similar to that of the W-W and Zr-Zr distances, indicate that the WO4 and ZrO6 polyhedra are strongly deformed by thermal motion, in contrast to the RUM model which assumes undistorted polyhedral units; – more importantly, some of the edges of the polyhedral units, in particular for the ZrO6 and W2O4 polyhedra, exhibit a NTE coefficient of the "true" bond distance. This corresponds to intrapolyhedra contributions to NTE, in contrast to the RUM model, which only gives inter-polyhedra contributions to NTE. Further confirmation of the presence of intra-polyhedra contributions to NTE comes from the "true" occupied volume of the W2O4 tetrahedra, which has a NTE coefficient

βr < 0 (see Tab. 1). Similar results were reached in cuprite structures, where a distortion of the M4 O tetrahedra (M=Cu, Ag), accompanied by a contraction of the M-M edges, were observed by EXAFS spectroscopy, ruling out the possibility of explaining NTE by a simple RUM model. 23,24 Significant

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distortions of the ScF6 octahedra were also observed by MD simulations in cubic scandium trifluoride (ScF3 ), a recently discovered material which exhibits surprisingly large and isotropic NTE over a wide temperature range. 30 As a result, the present findings indicate a rather complex local dynamics in ZrW2 O8 which is inconsistent with both the tent and RUM models. On the other side, the high stiffness and the large anisotropy of relative motion observed for the nearest W-O and Zr-O bonds, as well as the shrink of the O-O edges, suggest a more flexible model not based on rigid polyhedral units, but simply on rigid W-O and Zr-O bonds, thus giving rise to the tension mechanism, 5 the true responsible for NTE in ZrW2 O8 . In this regard, it is important to point out that the small σ 2 for the Zr-W pairs observed by Cao, Bridges and co-workers (and confirmed in this study) is consistent with the presence of a tension mechanism for the Zr-O-W linkage. In fact, if we assume that the WO and Zr-O bonds are perfectly rigid, the resulting tension mechanism would give rise to large NTE and small σ 2 for the Zr-W distances (by a numerical simulation, it can be shown that for uncorrelated Zr-W motion is αR ' −2.4 · 10−5 K−1 and αr ' −1.2 · 10−5 K−1 with ∆σ 2 ' 0.002 Å2 ). In short, no large rigid structural units are needed to explain the NTE of ZrW2 O8 , but only rigid nearest-neighbor bonds.

Conclusions The emergence of RUM and tent models represented major steps along the path to understanding the nature of the crystal dynamics leading to NTE. The present work aimed to, and succeeds, in showing how some features of each model arise from a more thorough analysis. This is accomplished by means of molecular dynamics study of local lattice vibrations in producing NTE. Valid and inconsistent elements of both RUM and tent models are elucidated, and it emerges that neither model is adequate in reproducing the features that are required to explain quantitatively the NTE of ZrW2 O8 . Accordingly, a more flexible model, simply based on rigid nearest-neighbor bonds and tension effect, should be adopted. This work provides one more major step along the path to

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comprehensive modeling of NTE.

Acknowledgement The author is grateful to J. D. Gale for supplying the GULP program package.

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W2

O

W1

Zr O O W2 O W1

Figure 4: TOC graphic.

15


Toward an Understanding of the Local Origin of Negative Thermal

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