# Thermal expansion

Material Properties
Specific heat $c={\frac {T}{N}}\left({\frac {\partial S}{\partial T}}\right)$ Compressibility $\beta =-{\frac {1}{V}}\left({\frac {\partial V}{\partial p}}\right)$ Thermal expansion $\alpha ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)$ Thermal Expansion is the tendency of matter to change in volume in response to a change in temperature. When a substance is heated, its constituent particles move around more vigorously and by doing so generally maintain a greater average separation. Materials that contract with an increase in temperature are very uncommon; this effect is limited in size, and only occurs within limited temperature ranges. The degree of expansion divided by the change in temperature is called the material's coefficient of thermal expansion and generally varies with temperature.

Common engineering solids usually have thermal expansion coefficients that do not vary significantly over the range of temperatures where they are designed to be used, so where extremely high accuracy is not required, calculations can be based on a constant, average, value of the coefficient of expansion.

Materials with anisotropic structures, such as crystals and composites, will generally have different expansion coefficients in different orientations.

To more accurately calculate thermal expansion of a substance a more advanced Equation of state must be used, which will then predict the values of the thermal expansion at all the required temperatures and pressures, along with many other state functions.

For solid materials with a significant length, like rods or cables, an estimate of the amount of thermal expansion can be described by the ${\frac {}{}}\epsilon _{thermal}$ ratio of strain:

$\epsilon _{thermal}={\frac {(L_{final}-L_{initial})}{L_{initial}}}$ ${\frac {}{}}L_{initial}$ is the initial length before the change of temperature and

${\frac {}{}}L_{final}$ the final length recorded after the change of temperature.

For most solids, thermal expansion relates directly with temperature:

$\epsilon _{thermal}\propto {\Delta T}$ Thus, the change in either the strain or temperature can be estimated by:

${\frac {}{}}\epsilon _{thermal}=\alpha \Delta T$ where

${\frac {}{}}\Delta T=(T_{final}-T_{initial})$ and

${\frac {}{}}\alpha$ is the coefficient of thermal expansion in inverse kelvins.
${\frac {}{}}\Delta T$ is the difference of the temperature between the two recorded strains, measured in celsius or kelvin.

A number of materials contract on heating within certain temperature ranges; we usually speak of negative thermal expansion, rather than thermal contraction, in such cases. For example, the coefficient of thermal expansion of water drops to zero as it is cooled to roughly 4 °C and then becomes negative below this temperature, this means that water has a maximum density at this temperature, and this leads to bodies of water maintaining this temperature at their lower depths during extended periods of sub-zero weather.

Common polymers expand roughly 4 times more than metals, which expand more than ceramics. Thermal expansion generally decreases with increasing bond energy, which also has an effect on the hardness of solids, so, harder materials are more likely to have lower thermal expansion. In general, liquids expand slightly more than solids.

In many common materials, changes in size can also be due to water (or other solvents) being absorbed/desorbed, and many organic materials change size much more due to this effect than they do to thermal expansion. Common plastics exposed to water can, in the long term, expand many percent.

Heat-induced expansion has to be taken into account in most areas of engineering. A few examples are:

• Metal framed windows need rubber spacers
• Metal hot water heating pipes should not be used in long straight lengths
• Large structures such as railways and bridges need expansion joints in the structures
• One of the reasons for the poor performance of cold car engines is that parts have inefficiently large spacings until the normal operating temperature is achieved.
• a Gridiron pendulum uses an arrangement of different metals to maintain a more temperature stable pendulum length.

This phenomenon can also be put to good use, for example in the process of thermal shrink-fitting parts are assembled with each at a different temperature, and sized such that when they reach the same temperature, the thermal expansion of the parts forces them together to form a stable joint.

Thermometers are another example of an application of thermal expansion — most contain a liquid which is constrained to flow in only one direction (along the tube) due to changes in volume brought about by changes in temperature. A bi-metal mechanical thermometer uses a bi-metal strip and registers changes based on the differing coefficient of thermal expansion between the two materials.

## Anisotropy

Many solid materials will expand evenly in all three directions, but this is not true for all. Graphite for example has a pronounced layer structure and the expansion in the direction perpendicular to the layers is quite different from that in the layers. In general the proper description of the thermal expansion of a solid must therefore include its symmetry. For cubic materials a single expansion coefficient suffices, but for a material with triclinic symmetry six parameters must be distinguished, three for each of the three axes (a,b,c) and three for the change in the angles (α,β,γ) between them. An excellent way of measuring the entire expansion tensor is to perform powder diffraction on the material during a heating or cooling run and monitor the position of its diffraction peaks. 