Entropic elasticity and negative thermal expansion in a simple cubic crystal

TL;DR

This paper demonstrates that negative thermal expansion in a simple cubic crystal, ScF3, is driven by entropic elasticity similar to polymers, challenging previous beliefs that its origin was different.

Contribution

It provides a quantitative theory linking NTE in solids to entropic elasticity, supported by neutron scattering experiments on ScF3.

Findings

Correlation in fluorine atom positions fades with temperature

Uncorrelated thermal motion constrained by rigid bonds

Quantitative agreement between theory and experiment

Abstract

While most solids expand when heated, some materials show the opposite behavior: negative thermal expansion (NTE). In polymers and biomolecules, NTE originates from the entropic elasticity of an ideal, freely-jointed chain. The origin of NTE in solids has been widely believed to be different. Our neutron scattering study of a simple cubic NTE material, ScF3, overturns this consensus. We observe that the correlation in the positions of the neighboring fluorine atoms rapidly fades on warming, indicating an uncorrelated thermal motion constrained by the rigid Sc-F bonds. This leads us to a quantitative theory of NTE in terms of entropic elasticity of a floppy network crystal, which is in remarkable agreement with experimental results. We thus reveal the formidable universality of the NTE phenomenon in soft and hard matter.

Figures

Equations

$$H=K+\frac{3N(6-M)e^{2}}{4\pi\epsilon_{0}r}-\sum_{i}{\frac{\gamma e^{2}}{4\pi\epsilon_{0}r^{3}}\frac{\mbox{\boldmath$u$}_{i\perp}^{2}}{2}},$$

Full text

Near zero, or negative thermal expansion is well known in metallic alloys of the invar (Fe0.64Ni0.36) family, where it is closely related to electronic magnetism Guillaume_CompRend1897 . These alloys are widely used in applications requiring dimensional stability of metallic parts, e.g. in precision instruments, watches, and engines. Until recently, much less attention was paid to insulating NTE ceramics, which hold promise for numerous applications in electronics, optics and medicine Sleight_AnnuRev1998 ; Barrera_etal_JPCM2005 ; Lind_Materials2012 ; Dove_RepProgPhys2016 . Somewhat surprisingly, thanks to the specific crystal lattice geometry, NTE in these materials can have the same physical origin as a more common, positive thermal expansion: atomic thermal motion.

Interest in such systems was renewed with the observation of large isotropic NTE in zirconium tungstate, ZrW2O8, and then in the structurally related AM2O8, AM2O7 and A2M3O12 phases (A = Zr, Hf, Sc, Y, … and M = W, V, Mo, P, …) and their solid solutions Sleight_AnnuRev1998 ; Mary_Science1996 , which opened avenues for designing ceramic materials with tailored thermal expansion Sleight_AnnuRev1998 ; Barrera_etal_JPCM2005 ; Lind_Materials2012 ; Dove_RepProgPhys2016 . These compounds have complex crystal structures, which can be viewed as three-dimensional (3D) networks of AX6 octahedra and MX4 tetrahedra (X = O) that share the corner X atoms and, most importantly, contain nearly straight, two-fold-coordinated M-X-M and M-X-A linkages (the so-called open framework structures Sleight_AnnuRev1998 ).

NTE in such a structure can be explained by the transverse thermal motion of anion atoms, X, in the presence of the strong M-X bond, which has small, or negligible thermal expansion: the so-called “guitar string effect” Sleight_AnnuRev1998 ; Hu_JACS2016 ; Ernst_Nature1998 . As the amplitude of the anion transverse vibration increases with temperature, the metal atoms in M-X-M linkages are pulled closer together, thus causing the net contraction of the structure. While this simple picture does not consider the correlated motion of nearby X anions caused by their interactions in the lattice, in what follows we show that it provides an accurate description of NTE in ScF3 Greve_JACS2010 .

An appealing model for including anion correlation considers vibrations that preserve the structure of the MXn polyhedra, which thus move as rigid bodies, without deforming the anion-anion bonds Sleight_AnnuRev1998 ; Pryde_JPCM1996 ; TaoSleight_JSolStChem2003 ; Tucker_PRL2005 ; Tucker_JPCM2007 . The relative importance of such rigid unit modes (RUM) has been rationalized by arguing that vibrations distorting the high symmetry of the polyhedron must have a high energy cost and therefore contribute relatively little to NTE Dove_RepProgPhys2016 . A priori, such an ad-hoc assumption is not required for the NTE effect and its relevance has been a matter of debate Sleight_AnnuRev1998 ; Dove_RepProgPhys2016 ; Pryde_JPCM1996 ; TaoSleight_JSolStChem2003 ; Tucker_PRL2005 ; Tucker_JPCM2007 ; Cao_PRL2002 ; Hancock_PRL2004 ; Schlesinger_PRL2008 ; Li_PRL2011 ; Handunkanda_PRB2015 ; Handunkanda_PRB2016 ; Young_JMatChem2011 . In a structure that is under-constrained, RUM correspond to zero-energy floppy phonon modes Dove_RepProgPhys2016 ; Pryde_JPCM1996 . In a fully constrained structure, such as a cubic network of corner-sharing octahedra in ScF3 (Fig. 1a), vibrations that do not distort polyhedra are only present on special low-dimensional manifolds occupying zero volume fraction of the system’s phase space Dove_RepProgPhys2016 ; Pryde_JPCM1996 . Nevertheless, it has been argued that phonons in the vicinity of these manifolds, quasi-RUM, which involve only small distortions of the polyhedra have special importance for NTE. In fact, ScF3 was suggested to be a perfect example of an NTE system where the tension effect is enabled by RUM (Fig. 1b) Dove_RepProgPhys2016 .

Pair distribution function (PDF) analysis Young_JMatChem2011 ; EgamiBillinge_book2012 of neutron total scattering is a powerful and direct experimental method for studying average local atomic structures and their relevance for NTE Dove_RepProgPhys2016 ; EgamiBillinge_book2012 . The PDF, $g(r)$, which is obtained from the measured scattering intensity, $S(Q)$, gives the probability distribution of inter-atomic distances weighted by the scattering lengths of the constituent atoms and thus is uniquely sensitive to local structural patterns. Figure 1c-e presents the PDF of ScF3 measured on NPDF (c,d) and NOMAD (e) neutron diffractometers at temperatures from 2 K to 1099 K. These measurements are complementary and show good agreement in the temperature range where they overlap.

An inspection of $g(r)$ curves reveals several remarkable features, of which the most important is the distinct behavior of Sc-F and F-F pair distributions. NTE of the average crystal structure is manifested by the systematic negative shift to smaller $r$ of PDF peaks from atomic pairs with large separation, $r$, with increasing temperature. It is most clearly seen in Fig. 1d. The nearest F-F ($\approx 2.8$ Å) peak shows similar NTE behavior. On the other hand, the nearest-neighbor Sc-F ($\approx 2$ Å) peak shifts on heating to slightly larger $r$, consistent with the conventional positive thermal expansion (PTE) Hu_JACS2016 ; Piskunov_PRB2016 . This peak broadens only moderately with temperature, by about 20% at 450 K (this accounts for the decrease of peak maximum in Fig. 1), indicating a very stiff Sc-F bond. In contrast, the width of the nearest F-F peak increases markedly, revealing rapid loss of F-F correlation with increasing thermal motion. Even more dramatic is the behavior of further-neighbor F-F distributions. The corresponding peaks (marked by dashed lines in Fig. 1c,e) are only present at T $\lesssim$ 300 K and entirely disappear at higher temperatures, suggesting complete loss of positional correlation between further-neighbor F atoms. Such a liquid-like F-F PDF pattern indicates randomly phased transverse local motion of F atoms and is inconsistent with the RUM model where a large number of F-F distances are constrained by the rigid unit geometry Dove_RepProgPhys2016 ; Handunkanda_PRB2015 ; Handunkanda_PRB2016 ; Supplementary .

We quantify the observed behaviors by fitting the first several PDF peaks, which are well resolved and can be uniquely associated with distance distributions of particular atomic pairs, to Gaussian distributions (Fig. 2a-c). The results of this analysis are summarized in Figure 2d-f. While the Sc-F bond shows PTE of $\approx 5$ ppm at 1000 K, both Sc-Sc, $r_{Sc-Sc}=a$ (LRD, lattice repeat distance), and the nearest F-F, $r_{F-F}$, distances exhibit NTE about twice larger in magnitude (Fig. 2d). The large error bars on $r_{F-F}$ reflect dramatic broadening of the F-F peak with temperature. While the full width at half maximum (FWHM) of Sc-F and Sc-Sc (LRD) peaks increase by less than 50% at 1000 K, the width of the nearest F-F distribution shows nearly an order-of-magnitude larger change, increasing to nearly 1 Å (Fig. 2e). This indicates an uncertainty of the nearest F-F distance that is comparable to the $r_{F-F}$ distance itself, clearly invalidating the RUM model assumption of quasi-rigid ScF6 octahedra.

The loss of F-F pair correlation is further revealed by the temperature dependence of the intensity of the LRD ($\approx 4$Å) peak (Fig. 2f). It contains partial contributions from both nearest-neighbor Sc-Sc and next-nearest-neighbor F-F pairs, in proportion $\sigma_{Sc}:3\sigma_{F}\approx 1.6$, where $\sigma_{Sc}$ and $\sigma_{F}$ are coherent scattering cross-sections of Sc and F, respectively. A PDF peak presents the probability distribution of inter-atomic distance and therefore its integral intensity must be temperature-independent. This roughly holds for Sc-F peak. A small systematic drift of its intensity, which is likely caused by T-dependent background, is within the error bar of the average value. In contrast, the LRD peak rapidly loses a substantial part of its intensity above $\approx$ 300 K, where F-F correlations disappear. The decrease is consistent with the loss of the entire $\approx$ 40% partial contribution of F-F pairs, which above $\approx$ 300 K contribute to broad background rather than to the narrow LRD peak described by the fit.

Motivated by these observations, we use a simple model for the probability distribution of the nearest F-F distance, which is presented in Figure 3. It assumes un-correlated thermal motion of individual F atoms, which is subject to a single constraint of the rigid Sc-F bond. If Sc atoms were fixed at the nodes of ScF3 lattice, the constraint would result in F atoms following ring trajectories with the Sc-F bond sweeping a cone. The resulting $r_{F-F}$ probability distribution would be that of a distance between two points randomly positioned on the two nearest rings. This model has no adjustable parameters because the radius of the rings, $r_{\perp}$ (the average transverse deviation of F), and $r_{Sc-Sc}=a$ are obtained from the Rietveld refinement of the coherent Bragg scattering contained in our data (Fig. 4b,c). The model can also be set up using Sc-F and Sc-Sc distances refined from PDF peaks (Fig. 2), but the accuracy of this refinement is lower. Surprisingly, when broadened by convolution with the Gaussian of the same width as Sc-F peak at 2 K to account for experimental resolution (truncation), our over-simplified model provides adequate description of the measured F-F distribution for all temperatures where NTE is observed (dashed lines in Figs. 3a and 4a). In this model, the peak maximum follows lattice NTE, in agreement with Fig. 2d. The model can be further improved if instead of rings (or conventional Gaussian TD ellipsoids, Fig. 3b), F atoms are randomly positioned on a torus-shaped Gaussian distribution peaked at the same major diameter, $2r_{\perp}$, and with the minor diameter representing the F part of the Sc-F peak FWHM (Fig. 3c). This improved model conjectured by Sleight Sleight_AnnuRev1998 indeed provides slightly better agreement with the data (Fig. 3a), as quantified by the reduced mean square deviation, $\chi^{2}$, presented in Fig. 4a. The $\chi^{2}$ analysis is a standard way to evaluate the goodness of fit: where $\chi^{2}\approx 1$, the data is indistinguishable from the model. With $N\approx 11$ effectively independent data points used in our comparison, $\chi^{2}<3$ places the model within $3\sigma$ interval, or above 99.5% likelihood level. The model begins to fail above $\approx$ 700 K, where $\chi^{2}$ increases to $\sim 10$, but NTE fades, too. Below $\approx$ 200 K the broadening of the F-F peak is small and $\chi^{2}\lesssim 1$, which means that within the experimental error our model cannot be distinguished from other models, such as RUM.

The essential implication of our analysis is that thermal motion of even the nearest F atoms is uncorrelated rather than in RUM (Fig. 3d,e). The spread of the F-F PDF peak with temperature simply follows from the increase in size of the manifold (circle, or torus) populated by each F atom in the course of its thermal motion. Underlying this model is the phenomenon of energy scales separation, where two very different energies govern longitudinal and transverse motion of the F ion. Then, in some temperature range transverse modes can be thermally excited and the corresponding degrees of freedom be equipartitioned, while longitudinal vibrations are still frozen out. In this case, Sc-F bonds are rigid, while F transverse vibrations are uncorrelated. Inspection of the vibrational spectra measured in ScF3 Li_PRL2011 ; Handunkanda_PRB2015 ; Piskunov_PRB2016 indeed reveals two major phonon groups, which give rise to maxima in the density of states below $\hbar\omega_{t}\approx 22$ meV and above $\hbar\omega_{l}\approx 62$ meV. These correspond to transverse and longitudinal vibrations, respectively. Such separation of energy scales implies that transverse degrees of freedom are thermally excited and equipartitioned at $T>\hbar\omega_{t}/k_{B}\approx 260$ K ($k_{B}$ is Boltzmann constant), while longitudinal rigidity of Sc-F bond persists up to at least $T\approx\hbar\omega_{l}/k_{B}\approx 710$ K. This is exactly the temperature range where NTE is observed and where our model provides very good description of the PDF data (Fig. 4).

The exceptional longitudinal rigidity of the Sc-F bond, which underlies the NTE mechanism in ScF3 is rooted in covalence, where the hybridization of Sc and F electronic orbitals that lie deep inside the valence band is responsible for the large energy cost of changing the Sc-F distance Supplementary ; Pauling_book1960 ; Shaik2016 ; Teterin_RussChemRev2002 ; Bocharov_LTP2016 . Such Lewis-type dative bonding where paired electrons delocalize between ions to lower their kinetic energy has recently been described as a “charge transfer bond” Shaik2016 . Although it has long been known that ScF3 is anomalous among supposedly ionic metal trifluorides, MF3 (M = Al, Sc, Fe, In, …,) Kury_JACS1959 , only relatively recently has the exceptional strength of the Sc-F bond been traced to the covalent nature of the valence molecular orbitals (MO). The comparative analysis of the X-ray photoemission spectra (XPS) and the density functional theory (DFT) electronic structure calculations Teterin_RussChemRev2002 for the (ScF6)3- cluster has indicated a large contribution to the Sc-F bonding energy of a specific ($5a_{1}$) $d-p$ MO at $\approx-5$ eV. A modest hybridization ($|5a_{1}\rangle\approx 0.83|F2p\rangle+0.16|Sc3d\rangle$) does not lead to sizeable charge transfer, such that Sc3+ and F*-* appear to be close to their nominal ionic oxidation states. We performed DFT electronic structure calculations in ScF3, which further support these findings, indicating slightly hybrid valence bands below $\sim-4$ eV and strong anisotropy of the effective potential of F ions Supplementary ; Bocharov_LTP2016 .

These observations immediately suggest a simple theoretical description of the NTE effect, where each Sc-F bond is treated as a rigid monomer link and the entire ScF3 crystal structure as a floppy network of such freely jointed monomers, a direct 3D analogue of the celebrated model of polymer chains Flory_book1969 ; deGennes_book1979 (Fig. 3b,d). Without electrostatic interactions, the network is under-constrained (floppy): the number of constraints imposed by rigid Sc-F links is 6 per unit cell, while the number of degrees of freedom is 12. In particular, the motion of the Sc ion is constrained by rigid bonds in all 3 directions while each of the F ions has two zero-energy displacement modes corresponding to motion orthogonal to the Sc-F bond. In the absence of external tension the system has no rigidity and would collapse. In ScF3, net Coulomb repulsion between charged ions provides tension (negative pressure), which stabilizes the system and balances its entropic elasticity Supplementary .

We thus separate interactions in the system into a sum of the nearest-neighbor pair potentials, which include the cumulative effect of electrostatic Coulomb attraction, core repulsion, and covalent bonding, and in the simplest approximation are treated as rigid links, and the remaining Coulomb potential of non-nearest-neighbor ions. The resulting effective Hamiltonian for the fluorine transverse motion is Supplementary

[TABLE]

where $K$ is kinetic energy, $M\approx 2.98$ is the Madelung constant for ScF3 lattice, $r=a/2$ is half of the lattice repeat, $N$ is the number of sites, $e$ is electron charge, $\epsilon_{0}$ is vacuum permittivity, \mbox{\boldmathu}_{i\perp} is transverse displacement of the F ion at lattice site $i$, and $\gamma\approx 1.8$ was obtained by lattice summation of Coulomb interactions, similarly to the Madelung constant Supplementary . On account of the rigid link constraint, Hamiltonian (1) describes the F floppy modes as independent Einstein type oscillators with the frequency, $\hbar\omega_{0}=\hbar e\sqrt{\frac{(6-M-\gamma)}{4\pi\epsilon_{0}r^{3}m_{F}}}\approx 21.6$ meV. This value compares very favorably with the phonon dispersions measured in ScF3, where low-energy peaks in the density of states are observed below $\hbar\omega_{t}\approx 22$ meV.

Although the Einstein approximation is expected to perform poorly at low temperatures, where the exact phonon dispersions are important for determining the bulk thermodynamic properties such as heat capacity, or thermal expansion, it works well at $k_{B}T>\frac{1}{2}\hbar\omega_{0}$, where the equipartition theorem sets the thermal average of an oscillator Hamiltonian to $k_{B}T$ per degree of freedom. In this regime, heat capacity obeys the Dulong-Petit law, which is equally well described by both Einstein and Debye models. We thus expect our description, Eq. (1), to be applicable for $T\gtrsim 130$ K. Since each of the F floppy modes has a Hamiltonian of an oscillator with frequency $\omega_{0}$, the value of $\langle u_{\perp}^{2}\rangle=\langle u_{i\perp}^{2}\rangle$ can be found with the full account for quantum effects and for an arbitrary temperature Supplementary . We thus obtain the NTE effect, $\frac{r-r_{0}}{r_{0}}\approx-\frac{\langle u_{\perp}^{2}\rangle}{2r_{0}^{2}}\approx\alpha T$, where $r_{0}$ is $r$ at $T=0$ and $\alpha=-\frac{4\pi\epsilon_{0}r_{0}k_{B}}{(6-M-\gamma)e^{2}}\approx-10.1\cdot 10^{-6}$ K*-1*, in an impressive agreement with the experimental value Greve_JACS2010 (Fig. 4c).

An account for the finite rigidity of the Sc-F bond is done by replacing the rigid link constraint with an interaction potential, $V_{b}(r_{b})\approx V_{b}(r_{0})+f_{0}(r_{b}-r_{0})+\frac{1}{2}k(r_{b}-r_{0})^{2}$. Here, $r_{b}$ is the bond length, $f_{0}$ is the tension force that Sc-F linkage provides to compensate the negative electrostatic pressure, and $k$ is the effective harmonic spring constant Supplementary . From the measured frequency of the longitudinal Sc-F phonon mode, $\hbar\omega_{l}\approx 62$ meV, we estimate, $kr_{0}^{2}\approx 26$ eV, in good agreement with our DFT results Supplementary . The minimization of the resulting free energy, which includes the entropic term, electrostatics, and bond potential, $V(r_{b})$, with respect to both $r$ and $r_{b}$, yields the equilibrium values $r=r(T)$, $r_{b}=r_{b}(T)$. We thus obtain the relation $\frac{r-r_{0}}{r_{0}}-\frac{r_{b}-r_{0}}{r_{0}}\approx\alpha T$, which means that the net entropic tension effect is split between PTE of the Sc-F bond and NTE of the lattice. It further yields the relation, $\frac{r_{b}-r_{0}}{r_{0}}=-\beta\frac{r-r_{0}}{r_{0}}$, where $\beta\approx 0.36$, which determines the relative split between the two effects, $\frac{r_{b}-r_{0}}{r_{0}}=\frac{\alpha\beta}{1+\beta}T\approx 2.7\cdot 10^{-6}T$ and $\frac{r-r_{0}}{r_{0}}=\frac{\alpha}{1+\beta}T\approx-7.4\cdot 10^{-6}T$ Supplementary . These predictions are shown by dashed lines in Figs. 2d and 4c, which demonstrate remarkable agreement of our simple theory with experiment. We note that in our estimates we neglected the covalent reduction of the ionic charge on F and Sc ions, which would increase the predicted NTE effect by $\approx 20$%. This provides a ballpark estimate for the accuracy of our predictions.

We conclude that floppy vibration modes associated with the transverse fluorine displacement in an under-constrained network crystal structure of ScF3 give rise to both negative thermal expansion of the lattice and positive expansion of Sc-F bond. The latter effect is distinct from the conventional positive thermal expansion based on cubic anharmonism of Sc-F bond potential, $V_{b}(r_{b})$ Barron_book1999 . Instead, it originates from entropic elasticity via floppy modes and is already present for the harmonic Sc-F bond. RUM, which are of crucial importance for understanding the stability of cubic crystal structure, appear not to be of primary importance for NTE.

In ZrW2O8, the RUM model was challenged by X-ray absorption fine structure (XAFS) studies Cao_PRL2002 but later was argued to be consistent with neutron PDF measurements Dove_RepProgPhys2016 ; Tucker_PRL2005 ; Tucker_JPCM2007 . However, for a complex material with more than 3 different atom types PDF analysis has a degree of uncertainty because the measured PDF is a sum of PDFs from all atomic pairs where some features may overlap, meaning that it is difficult to identify individual peaks with specific atomic pairs Dove_RepProgPhys2016 . This problem is absent in ScF3, which has simple cubic structure with only two atom types. In agreement with the earlier molecular dynamics (MD) simulations Li_PRL2011 , our present results do not support the presence of RUM, indicating that the only rigid unit is the Sc-F bond. This makes the ScF3 structure an under-constrained 3D analog of a freely jointed polymer chain.

Based on our experimental observations, we developed a simple theoretical description of the NTE effect in ScF3, which is rooted in entropic elasticity of an underconstrained floppy network, similar in spirit to the celebrated Flory-deGennes theory of polymer elasticity Flory_book1969 ; deGennes_book1979 . Our approach presents a paradigm shift, where instead of focusing on peculiar energetics of low-energy lattice vibrations, such as RUM Barron_book1999 ; Sleight_AnnuRev1998 ; Barrera_etal_JPCM2005 ; Lind_Materials2012 ; Dove_RepProgPhys2016 ; Greve_JACS2010 ; Hu_JACS2016 ; Pryde_JPCM1996 ; TaoSleight_JSolStChem2003 ; SimonVarma_PRL2001 ; HeCvetkovicVarma_PRB2010 ; Occhialini_PRB2017 , these vibrations are approximated by Einstein local phonon modes and the focus is on their entropic contribution to free energy. Not only our results provide clear understanding of the entropic elasticity origin of the NTE effect in the practically important class of materials and temperature range, including at and above room temperature, they also provide an accurate, quantitative, textbook description of NTE, thus opening new avenues for predictive modeling of this effect in solids.

Note. A recent study (https://arxiv.org/abs/1905.09250) reproducing some of our measurements appeared after our results were presented at the 2019 APS March meeting (http://meetings.aps.org/Meeting/MAR19/Session/B33.8). This study also finds substantial flexibility of ScF6 octahedra inconsistent with RUM and supporting our results.