# Negative thermal expansion

Negative thermal expansion (NTE) is an unusual physicochemical process in which some materials contract upon heating, rather than expand as most other materials do. Materials which undergo NTE have a range of potential engineering, photonic, electronic, and structural applications. For example, if one were to mix a negative thermal expansion material with a "normal" material which expands on heating, it could be possible to make a zero expansion composite material.

## Origin of negative thermal expansion

There are a number of physical processes which may cause contraction with increasing temperature, including transverse vibrational modes, Rigid Unit Modes and phase transitions.

Recently, Liu et al.[1] showed that the NTE phenomenon originates from the existence of high pressure, small volume configurations with higher entropy, with their configurations present in the stable phase matrix through thermal fluctuations. They were able to predict both the colossal positive thermal expansion (In cerium) and zero and infinite negative thermal expansion (in Fe
3
Pt
) [2]

## Negative thermal expansion in close-packed systems

Negative thermal expansion is usually observed in non-close-packed systems with directional interactions (e.g. ice, graphene, etc.) and complex compounds (e.g. Cu
2
O
, ZrW
2
O
8
, beta-quartz, some zeolites, etc.). However, in a paper,[3] it was shown that negative thermal expansion (NTE) is also realized in single-component close-packed lattices with pair central force interactions. The following sufficient condition for potential giving rise to NTE behavior is proposed for the interatomic potential, ${\displaystyle \Pi (x)}$, at the equilibrium distance ${\displaystyle a}$:

${\displaystyle \Pi '''(a)>0,}$

Where ${\displaystyle \Pi '''(a)}$ is shorthand for the third derivative of the interatomic potential at the equilibrium point:

${\displaystyle \Pi '''(a)=\left.{\frac {d^{3}\Pi (x)}{dx^{3}}}\right|_{x=a}}$

This condition is (i) necessary and sufficient in 1D and (ii) sufficient, but not necessary in 2D and 3D. An approximate necessary and sufficient condition is derived in a paper[4]

${\displaystyle \Pi '''(a)a>-(d-1)\Pi ''(a),}$

where ${\displaystyle d}$ is the space dimensionality. Thus in 2D and 3D negative thermal expansion in close-packed systems with pair interactions is realized even when the third derivative of the potential is zero or even negative. Note that one-dimensional and multidimensional cases are qualitatively different. In 1D thermal expansion is caused by anharmonicity of interatomic potential only. Therefore, the sign of thermal expansion coefficient is determined by the sign of the third derivative of the potential. In multidimensional case the geometrical nonlinearity is also present, i.e. lattice vibrations are nonlinear even in the case of harmonic interatomic potential. This nonlinearity contributes to thermal expansion. Therefore, in multidimensional case both ${\displaystyle \Pi ''}$ and ${\displaystyle \Pi '''}$ are present in the condition for negative thermal expansion.

## Applications

There are many potential applications for materials with controlled thermal expansion properties, as thermal expansion causes many problems in engineering, and indeed in everyday life. One simple example of a thermal expansion problem is the tendency of dental fillings to expand by an amount different from the teeth, for example when drinking a hot drink, causing toothache. If dental fillings were made of a composite material containing a mixture of materials with positive and negative thermal expansion then the overall expansion could be precisely tailored to that of tooth enamel.

## Materials

Perhaps one of the most studied materials to exhibit negative thermal expansion is zirconium tungstate (ZrW
2
O
8
). This compound contracts continuously over a temperature range of 0.3 to 1050 K (at higher temperatures the material decomposes).[5] Other materials that exhibit this behaviour include: other members of the AM
2
O
8
family of materials (where A = Zr or Hf, M = Mo or W) and ZrV
2
O
7
. A
2
(MO
4
)
3
also is an example of controllable negative thermal expansion.

Ordinary ice shows NTE in its hexagonal and cubic phases at very low temperatures (below –200 °C).[6] In its liquid form, pure water also displays negative thermal expansivity below 3.984 °C.

Rubber elasticity shows NTE at normal temperatures, but the reason for the effect is rather different from that in most other materials. Put simply, as the long polymer chains absorb energy, they adopt a more contorted configuration, reducing the volume of the material.[7]

Quartz (SiO
2
) and a number of zeolites also show NTE over certain temperature ranges.[8][9] Fairly pure silicon (Si) has a negative coefficient of thermal expansion for temperatures between about 18 K and 120 K.[10] Cubic Scandium trifluoride has this property which is explained by the quartic oscillation of the fluoride ions. The energy stored in the bending strain of the fluoride ion is proportional to the fourth power of the displacement angle, unlike most other materials where it is proportional to the square of the displacement. A fluorine atom is bound to two scandium atoms, and as temperature increases the fluorine oscillates more perpendicularly to its bonds. This draws the scandium atoms together throughout the material and it contracts.[11] ScF
3
exhibits this property from 10 to 1100 K above which it shows the normal positive thermal expansion.[12] Shape memory alloys such as NiTi are a nascent class of materials that exhibit zero and negative thermal expansion.[13][14]

## References

1. ^ Liu, Zi-Kui; Wang, Yi; Shang, Shun-Li (2011). "Origin of negative thermal expansion phenomenon in solids". Scripta Materialia. 65 (8): 664–667. doi:10.1016/j.scriptamat.2011.07.001.
2. ^ Liu, Zi-Kui; Wang, Yi; Shang, Shunli (2014). "Thermal Expansion Anomaly Regulated by Entropy". Scientific Reports. 4: 7043. Bibcode:2014NatSR...4E7043L. doi:10.1038/srep07043. PMC 4229665. PMID 25391631.
3. ^ Rechtsman, M.C.; Stillinger, F.H.; Torquato, S. (2007), "Negative thermal expansion in single-component systems with isotropic interactions", The Journal of Physical Chemistry A, 111 (49): 12816–12821, arXiv:0807.3559, Bibcode:2007JPCA..11112816R, doi:10.1021/jp076859l, PMID 17988108
4. ^ Kuzkin, Vitaly A. (2014), "Comment on 'Negative Thermal Expansion in Single-Component Systems with Isotropic Interactions'", The Journal of Physical Chemistry A, 118 (41): 9793–4, Bibcode:2014JPCA..118.9793K, doi:10.1021/jp509140n, PMID 25245826
5. ^ Mary, T. A.; Evans, J. S. O.; Vogt, T.; Sleight, A. W. (1996). "Negative Thermal Expansion from 0.3 to 1050 Kelvin in ZrW
2
O
8
". Science. 272 (5258): 90–92. Bibcode:1996Sci...272...90M. doi:10.1126/science.272.5258.90.
6. ^ Röttger, K.; Endriss, A.; Ihringer, J.; Doyle, S.; Kuhs, W. F. (1994). "Lattice constants and thermal expansion of H
2
O
and D
2
O
ice Ih between 10 and 265 K". Acta Crystallographica Section B. 50 (6): 644–648. doi:10.1107/S0108768194004933.
7. ^ "Heating a rubber band: negative coefficient of thermal expansion | Lecture Demonstrations". Berkeleyphysicsdemos.net. Retrieved 2015-05-10.
8. ^ Lightfoot, Philip; Woodcock, David A.; Maple, Martin J.; Villaescusa, Luis A.; Wright, Paul A. (2001). "The widespread occurrence of negative thermal expansion in zeolites". Journal of Materials Chemistry. 11: 212–216. doi:10.1039/b002950p.
9. ^ Attfield, Martin P. (1998). "Strong negative thermal expansion in siliceous faujasite". Chemical Communications (5): 601–602. doi:10.1039/A707141H.
10. ^ Bullis, W. Murray (1990). "Chapter 6". In O'Mara, William C.; Herring, Robert B.; Hunt, Lee P. (eds.). Handbook of semiconductor silicon technology. Park Ridge, New Jersey: Noyes Publications. p. 431. ISBN 978-0-8155-1237-0. Retrieved 2010-07-11.
11. ^ Woo, Marcus (7 November 2011). "An incredible shrinking material: Engineers reveal how scandium trifluoride contracts with heat". Physorg. Retrieved 8 November 2011.
12. ^ Greve, Benjamin K.; Kenneth L. Martin; Peter L. Lee; Peter J. Chupas; Karena W. Chapman; Angus P. Wilkinson (19 October 2010). "Pronounced negative thermal expansion from a simple structure: cubic ScF
3
". Journal of the American Chemical Society. 132 (44): 15496–15498. doi:10.1021/ja106711v. PMID 20958035.
13. ^ Röttger, K.; Endriss, A.; Ihringer, J.; Doyle, S.; Kuhs, W. F. (1994). "Lattice constants and thermal expansion of H2O and D2O ice Ihbetween 10 and 265 K". Acta Crystallographica Section B. 50 (6): 644–648. doi:10.1107/S0108768194004933.
14. ^ Ahadi, A.; Matsushita, Y.; Sawaguchi, T.; Sun, Q.P.; Tsuchiya, K. (2017). "Origin of zero and negative thermal expansion in severely-deformed superelastic Ni Ti alloy". Acta Materialia. 124: 79–92. doi:10.1016/j.actamat.2016.10.054.

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2
O
8
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