Coefficient of thermal expansion

All materials change their size when subjected to a temperature change as long as the pressure is held constant. In the special case of solid materials, the pressure does not appreciably affect the size of an object, and so for solids, it usually not necessary to specify that the pressure be held constant. The coefficient of thermal expansion describes how the size of an object changes with a change in temperature. Specifically, it measures the fractional change in volume per degree change in temperature at a constant pressure.

The volumetric thermal expansion coefficient is the most basic thermal expansion coefficient. All substances expand or contract when their temperature changes, and the expansion or contraction always occurs in all directions. Substances that expand at the same rate in any direction are called isotropic. Unlike gases or liquids, solid materials tend to keep their shape. For solids, one might only be concerned with the change along a length, or over some area. Expansion coefficients are specially defined for these cases, and they are known as the linear and area expansion coefficients. However, they all come from the volume expansion coefficient, which explains how the substance expands in any direction.

Some substances expand when cooled, such as freezing water, so they have negative thermal expansion coefficients.

The volumetric thermal expansion coefficient for solids

The thermal expansion coefficient for a solid is a thermodynamic property of that solid. For a solid, we can ignore the effects of pressure on the material and the volumetric thermal expansion coefficient can be written :

$\alpha _{V}={\frac {1}{V}}\,{\frac {dV}{dT}}$

where V is the volume of the material, and $dV/dT$ is the rate of change of that volume with temperature.

What this basically means is that the volume of a material changes by some fixed fractional amount. For example, a steel block with a volume of 1 cubic foot might expand to 1.02 cubic feet when the temperature is raised by 50 degrees. This is an expansion of two percent. If we had a block of steel with a volume of 2 cubic feet, then under the same conditions, it would expand to 2.04 cubic feet, again an expansion of two percent. The volumetric expansion coefficient would be two percent for 50 degrees or 0.04 percent per degree, or 0.0004 per degree.

If we already know the expansion coefficient, then we can calculate the change in volume

${\frac {\Delta V}{V}}=\alpha _{V}\Delta T$

where $\Delta V/V$ is the fractional change in volume (0.02) and $\Delta T$ is the change in temperature (50 degrees).

The above example assumes that the expansion coefficient did not change as the temperature changed by 50 degrees. This is not always true, but for small changes in temperature, it is a good approximation. If the volumetric expansion coefficient does change appreciably with temperature, then the above equation will have to be integrated:

${\frac {\Delta V}{V}}=\int _{T_{0}}^{T_{0}+50}\alpha _{V}(T)\,dT$

where $T_{0}$ is the starting temperature and $\alpha _{V}(T)$ is the volumetric expansion coefficient as a function of temperature T.

Linear thermal expansion coefficient for a solid

The linear thermal expansion coefficient relates the change in a material's linear dimensions to a change in temperature. It is the fractional change in length per degree of temperature change. Again, ignoring pressure, we may write:

$\alpha _{L}={\frac {1}{L}}\,{\frac {dL}{dT}}$

where L is the linear dimension (e.g. length) and $dL/dT$ is the rate of change of that linear dimension per unit change in temperature. Just as with the volumetric coefficient, the change in the linear dimension can be estimated as:

${\frac {\Delta L}{L}}=\alpha _{L}\Delta T$

Again, this equation works well as long as the linear expansion coefficient does not change much over the change in temperature $\delta T$. If it does, the equation must be integrated.

For exactly isotropic materials, the linear thermal expansion coefficient is one third the volumetric coefficient.

$\alpha _{V}=3\alpha _{L}$

This ratio arises because volume is composed of three mutually orthogonal directions. Thus, in an isotropic material, one-third of the volumetric expansion is in a single axis (a very close approximation for small differential changes). As an example, take a cube of steel that has sides of length L. The original volume will be $V=L^{3}$ and the new volume, after a temperature increase, will be

$V+\Delta V=(L+\Delta L)^{3}$

We can make the subsitutions $\Delta V=\alpha _{V}L^{3}\Delta T$ and, for isotropic materials, $\Delta L=\alpha _{L}L\Delta T$. We now have:

$L^{3}+L^{3}\alpha _{V}\Delta T=L^{3}+3L^{3}\alpha _{L}\Delta T+3L^{3}\alpha _{L}^{2}\Delta T^{2}+L^{3}\alpha _{L}^{3}\Delta T^{3}$

Since the volumetric and linear coefficients are defined only for extremely small temperature and dimensional changes, the last two terms can be ignored and we get the above relationship between the two coefficients. If we are trying go back and forth between volumetric and linear coefficients using larger values of $\Delta T$ then we will need to take into account the third term, and sometimes even the fourth term.

Area thermal expansion coefficient for a solid

The area thermal expansion coefficient relates the change in a material's area dimensions to a change in temperature. It is the fractional change in area per degree of temperature change. Again, ignoring pressure, we may write:

$\alpha _{A}={\frac {1}{L}}\,{\frac {dA}{dT}}$

where L is some area on the object, and $dA/dT$ is the rate of change of that area per unit change in temperature. Just as with the volumetric coefficient, the change in the linear dimension can be estimated as:

${\frac {\Delta A}{A}}=\alpha _{A}\Delta T$

Again, this equation works well as long as the linear expansion coefficient does not change much over the change in temperature $\delta T$. If it does, the equation must be integrated.

For exactly isotropic materials, the area thermal expansion coefficient is 2/3 of the volumetric coefficient.

$\alpha ={\frac {2}{3}}\alpha _{V}$

This ratio can be found in a way similar to that in the linear example above, noting that the area of a face on the cube is just $L^{2}$. Also, the same considerations must be made when dealing with large values of $\Delta T$

The general volumetric thermal expansion coefficient

In the general case of a gas, liquid, or solid, the coefficient of thermal expansion is given by

$\alpha ={\frac {1}{V}}\,\left({\frac {\partial V}{\partial T}}\right)_{p}$

where partial derivatives must now be used, the subscript p indicating that the pressure is held constant during the expansion. In the case of a gas, the fact that the pressure is held constant is important, because the volume of a gas will vary appreciably with pressure as well as temperature. For a gas of low density this can be seen from the ideal gas law.

Anisotropic materials

In anisotropic materials the total volumetric expansion is distributed unequally among the three axes and if the crystal symmetry is monoclinic or triclinic even the angles between these axes are subject to thermal changes. In such cases it is necessary to treat thermal expansion as a tensor that has up to six independent elements. A good way to determine the elements of the tensor is to study the expansion by powder diffraction.

Thermal expansion coefficients for various materials

The expansion and contraction of material must be considered when designing large structures, when using tape or chain to measure distances for land surveys, when designing molds for casting hot material, and in other engineering applications when large changes in dimension due to temperature are expected. The range for α is from 10⁻⁷/°C for hard solids to 10⁻³/°C for organic liquids. α varies with the temperature and some materials have a very high variation.

Theoretically, the coefficient of linear expansion can be approximated from the coefficient of volumetric expansion (β≈3α). However, for liquids, α is calculated through the experimental determination of β, so it is more accurate to state β here, rather than α. (The formula β≈3α is usually used for solids.)

coefficient of linear thermal expansion α coefficient of volumetric thermal expansion β material α in 10−6/C° at 20 °C β(≈3α) in 10−6/C° at 20 °C Gasoline ~317 950 Ethanol ~250 750 Water ~69 207 Mercury ~61 182 Rubber Penis 77 231 PVC 52 156 Benzocyclobutene 42 126 Lead 29 87 Magnesium 26 78 Aluminium 23 69 Brass 19 57 Silver 18 54 Stainless steel 17.3 51.9 Copper 17 51 Gold 14 42 Nickel 13 39 YbGaGe 0 0 Concrete 12 36 Steel, depends on composition 11.0 ~ 13.0 33.0 ~ 39.0 Iron 11.1 33.3 Carbon steel 10.8 32.4 Platinum 9 27 Glass 8.5 25.5 Gallium(III) arsenide 5.8 17.4 Molybdenum 4.8 14.4 Indium phosphide 4.6 13.8 Tungsten 4.5 13.5 Glass, borosilicate 3.3 9.9 Silicon 3 9 Silicon Carbide 2.77 8.31 Invar 1.2 3.6 Diamond 1 3 Quartz (fused) 0.59 1.77 Oak (perpendicular to the grain) 54 162 Pine (perpendicular to the grain) 34 102 Applications For applications using the thermal expansion property, see bi-metal and mercury thermometer. Thermal expansion is also used in mechanical applications to fit parts over one another, e.g. a bushing can be fitted over a shaft by making its inner diameter slightly smaller than the diameter of the shaft, then heating it until it fits over the shaft, and allowing it to cool after it has been pushed over the shaft, thus achieving a 'shrink fit'. Induction shrink fitting is a common industrial method to pre-heat metal components between 150 °C and 300 °C thereby causing them to expand and allow for the insertion or removal of another component. There exist some alloys with a very small linear expansion coefficient, used in applications that demand very small changes in physical dimension over a range of temperatures. One of these is Invar 36, with α approximately equal to 0.6×10−6/°C. These alloys are useful in aerospace applications where wide temperature swings may occur. Pullinger's apparatus is used to determine linear expansion of a metallic rod in laboratory. The apparatus consists of a metal cylinder closed at both ends (called steam jacket). It is provided with an inlet and outlet for the steam.The steam for heating the rod is supplied by a boiler which is connected by a rubber tube to the inlet. The center of cylinder contains a hole to insert a thermometer. The rod, under investigation, is enclosed in a steam jacket. Its one end is free, but the second end is pressed against a fixed screw. The position of the rod is determined by a micrometer screw gauge or spherometer. See also Autovent Thermal expansion coefficients of the elements (data page) Notes and references External links Engineering Toolbox – List of coefficients of Linear Expansion for some common materials Article on how β is determined MatWeb: Free database of engineering properties for over 64,000 materials Clemson University Physics Lab: Linear Thermal Expansion USA NIST Website - Temperature and Dimensional Measurement workshop Hyperphysics: Thermal expansion