Negative thermal expansion: a review

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J Mater Sci (2009) 44:5441–5451
DOI 10.1007/s10853-009-3692-4


Negative thermal expansion: a review
W. Miller Æ C. W. Smith Æ D. S. Mackenzie Æ
K. E. Evans

Received: 4 March 2009 / Accepted: 15 June 2009 / Published online: 2 July 2009
Ó Springer Science+Business Media, LLC 2009

Abstract Most materials demonstrate an expansion upon
heating, however a few are known tocontract, i.e. exhibit a
negative coefficient of thermal expansivity (NTE). This
naturally occurring phenomenon has been shown to occur
in a range of solids including complex metal oxides,
polymers and zeolites, and opens the door to composites
with a coefficient of thermal expansion (CTE) of zero. The
state of the art in NTE solids is reviewed, and understanding of the driving mechanisms of theeffect is considered along with experimental and theoretical evidence.
The various categories of solids with NTE are explored,
and experimental methods for their experimental characterisation and applications for such solids are proposed. An
abstraction for an underlying mechanism for NTE at the
supramolecular level and its applicability at the molecular
level is discussed.

Ingeneral, solids expand upon heating, i.e. they exhibit
positive coefficients of thermal expansivity (CTE), denoted
as a herein. However, a minority of solids show the inverse
effect, i.e. of contracting upon heating, and thus exhibit
negative thermal expansion (NTE). There has been an
increasing amount of interest in these solids and their
potential applications. The underlying mechanisms forNTE have been found to be complex.

W. Miller Á C. W. Smith (&) Á D. S. Mackenzie Á K. E. Evans
School of Engineering, Computing and Mathematics,
University of Exeter, Exeter EX4 4QF, UK

The reason that most solids have positive CTEs is well
understood which is due primarily to an increase in the
interatomic bond length, which manifests at the macroscopiclevel as an overall increase in a dimension or volume. Bond lengthening is perhaps best explained by the
potential energy versus interatomic distance diagram, see
Fig. 1. On heating, the vibrational energy rises, and due to
the asymmetry of the potential energy curve, shown in
Fig. 1 (which can be considered typical of most strong
bonds), the mean interatomic distance increases [1]. The
rate ofchange (slope) of the potential energy curve is lower
on the lengthening side of the curve than on the shortening
side; thus, the mean bond length tends to increase with
temperature. The so called ‘stronger’ bonds have steeper
and narrower potential wells resulting in a slower rate of
increase in interatomic distance, and hence a smaller
thermal expansion coefficient (a). The CTE a is ameasure
of volumetric (av) or linear (al) change with temperature
and is defined as

ð 1Þ

l0 DT

ð 2Þ

aV ¼
al ¼

where DV, Dl are the changes in volume and length,
respectively, V0, l0 are the initial volume and length,
respectively, and DT is the change in temperature. In
crystalline solids, aV can be split to show the extent of
expansion/contraction of individualcrystal axial directions.
In the case of isotropic solids, Eqs. 1 and 2 are related by
av = 3al. However, in anisotropic solids, the relationship
between al and aV is not so simple as each crystal axis
potentially has a different magnitude and sign of a giving
three distinct values, aa, ab and ac, contributing to aV.



J Mater Sci (2009) 44:5441–5451

r interatomicdistance



Fig. 1 Graph of a general anharmonic potential energy well,
expanded for clarity, where Ei is the energy level and ri is the
interatomic distance

Several solids have now been identified with NTE
behaviour, some associated with phase transitions and
some stable over large temperature ranges. Examples
include metal oxides [2–4], zeolites [5],...
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