Thermal expansion

If an equation of state is available, it can be used to predict the values of the thermal expansion at all the required temperatures and pressures, along with many other state functions.

A number of materials contract on heating within certain temperature ranges; this is usually called negative thermal expansion, rather than "thermal contraction". For example, the coefficient of thermal expansion of water drops to zero as it is cooled to 3.983 °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. Also, fairly pure silicon has a negative coefficient of thermal expansion for temperatures between about 18 and 120 kelvins.[2]

Unlike gases or liquids, solid materials tend to keep their shape when undergoing thermal expansion.

Thermal expansion generally decreases with increasing bond energy, which also has an effect on the melting point of solids, so, high melting point materials are more likely to have lower thermal expansion. In general, liquids expand slightly more than solids. The thermal expansion of glasses is higher compared to that of crystals.[3] At the glass transition temperature, rearrangements that occur in an amorphous material lead to characteristic discontinuities of coefficient of thermal expansion and specific heat. These discontinuities allow detection of the glass transition temperature where a supercooled liquid transforms to a glass.[4]

Absorption or desorption of water (or other solvents) can change the size of many common materials; many organic materials change size much more due to this effect than due to thermal expansion. Common plastics exposed to water can, in the long term, expand by many percent.

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

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

The subscript "p" to the derivative indicates that the pressure is held constant during the expansion, and the subscript V stresses that it is the volumetric (not linear) expansion that enters this general definition. 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.

Change in length of a rod due to thermal expansion.

Linear expansion means change in one dimension (length) as opposed to change in volume (volumetric expansion). To a first approximation, the change in length measurements of an object due to thermal expansion is related to temperature change by a coefficient of linear thermal expansion (CLTE). It is the fractional change in length per degree of temperature change. Assuming negligible effect of pressure, we may write:

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

where ${\displaystyle L}$ is a particular length measurement and ${\displaystyle dL/dT}$ is the rate of change of that linear dimension per unit change in temperature.

The change in the linear dimension can be estimated to be:

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

This estimation works well as long as the linear-expansion coefficient does not change much over the change in temperature ${\displaystyle \Delta T}$, and the fractional change in length is small ${\displaystyle \Delta L/L\ll 1}$. If either of these conditions does not hold, the exact differential equation (using ${\displaystyle dL/dT}$) must be integrated.

For solid materials with a significant length, like rods or cables, an estimate of the amount of thermal expansion can be described by the material strain, given by ${\displaystyle \epsilon _{\mathrm {thermal} }}$ and defined as:

${\displaystyle \epsilon _{\mathrm {thermal} }={\frac {(L_{\mathrm {final} }-L_{\mathrm {initial} })}{L_{\mathrm {initial} }}}}$

where ${\displaystyle L_{\mathrm {initial} }}$ is the length before the change of temperature and ${\displaystyle L_{\mathrm {final} }}$ is the length after the change of temperature.

For most solids, thermal expansion is proportional to the change in temperature:

${\displaystyle \epsilon _{\mathrm {thermal} }\propto \Delta T}$

Thus, the change in either the strain or temperature can be estimated by:

${\displaystyle \epsilon _{\mathrm {thermal} }=\alpha _{L}\Delta T}$

where

${\displaystyle \Delta T=(T_{\mathrm {final} }-T_{\mathrm {initial} })}$

is the difference of the temperature between the two recorded strains, measured in degrees Fahrenheit, degrees Rankine, degrees Celsius, or kelvins, and ${\displaystyle \alpha _{L}}$ is the linear coefficient of thermal expansion in "per degree Fahrenheit", "per degree Rankine", “per degree Celsius”, or “per kelvin”, denoted by °F⁻¹, R⁻¹, °C⁻¹, or K⁻¹, respectively. In the field of continuum mechanics, the thermal expansion and its effects are treated as eigenstrain and eigenstress.

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. Ignoring pressure, we may write:

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

where ${\displaystyle A}$ is some area of interest on the object, and ${\displaystyle dA/dT}$ is the rate of change of that area per unit change in temperature.

The change in the area can be estimated as:

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

This equation works well as long as the area expansion coefficient does not change much over the change in temperature ${\displaystyle \Delta T}$, and the fractional change in area is small ${\displaystyle \Delta A/A\ll 1}$. If either of these conditions does not hold, the equation must be integrated.

For a solid, we can ignore the effects of pressure on the material, and the volumetric thermal expansion coefficient can be written:[5]

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

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

This means that the volume of a material changes by some fixed fractional amount. For example, a steel block with a volume of 1 cubic meter might expand to 1.002 cubic meters when the temperature is raised by 50 K. This is an expansion of 0.2%. If we had a block of steel with a volume of 2 cubic meters, then under the same conditions, it would expand to 2.004 cubic meters, again an expansion of 0.2%. The volumetric expansion coefficient would be 0.2% for 50 K, or 0.004% K⁻¹.

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

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

where ${\displaystyle \Delta V/V}$ is the fractional change in volume (e.g., 0.002) and ${\displaystyle \Delta T}$ is the change in temperature (50 °C).

The above example assumes that the expansion coefficient did not change as the temperature changed and the increase in volume is small compared to the original volume. 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, or the increase in volume is significant, then the above equation will have to be integrated:

${\displaystyle \ln \left({\frac {V+\Delta V}{V}}\right)=\int _{T_{i}}^{T_{f}}\alpha _{V}(T)\,dT}$

${\displaystyle {\frac {\Delta V}{V}}=\exp \left(\int _{T_{i}}^{T_{f}}\alpha _{V}(T)\,dT\right)-1}$

where ${\displaystyle \alpha _{V}(T)}$ is the volumetric expansion coefficient as a function of temperature T, and ${\displaystyle T_{i}}$,${\displaystyle T_{f}}$ are the initial and final temperatures respectively.

For isotropic materials the volumetric thermal expansion coefficient is three times the linear coefficient:

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

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

${\displaystyle V+\Delta V=(L+\Delta L)^{3}=L^{3}+3L^{2}\Delta L+3L\Delta L^{2}+\Delta L^{3}\approx L^{3}+3L^{2}\Delta L=V+3V{\Delta L \over L}.}$

We can easily ignore the terms as change in L is a small quantity which on squaring gets much smaller.

So

${\displaystyle {\frac {\Delta V}{V}}=3{\Delta L \over L}=3\alpha _{L}\Delta T.}$

The above approximation holds for small temperature and dimensional changes (that is, when ${\displaystyle \Delta T}$ and ${\displaystyle \Delta L}$ are small); but it does not hold if we are trying to go back and forth between volumetric and linear coefficients using larger values of ${\displaystyle \Delta T}$. In this case, the third term (and sometimes even the fourth term) in the expression above must be taken into account.

Similarly, the area thermal expansion coefficient is two times the linear coefficient:

${\displaystyle \alpha _{A}=2\alpha _{L}}$

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 ${\displaystyle L^{2}}$. Also, the same considerations must be made when dealing with large values of ${\displaystyle \Delta T}$.

Materials with anisotropic structures, such as crystals (with less than cubic symmetry, for example martensitic phases) and many composites, will generally have different linear expansion coefficients ${\displaystyle \alpha _{L}}$ in different directions. As a result, the total volumetric expansion is distributed unequally among the three axes. 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 the coefficient of thermal expansion as a tensor with up to six independent elements. A good way to determine the elements of the tensor is to study the expansion by x-ray powder diffraction. The thermal expansion coefficient tensor for the materials possessing cubic symmetry (for e.g. FCC, BCC) is isotropic.[6]

The expansion of liquids is usually measured in a container. When a liquid expands in a vessel, the vessel expands along with the liquid. Hence the observed increase in volume of the liquid level is not actual increase in its volume. The expansion of the liquid relative to the container is called its apparent expansion, while the actual expansion of the liquid is called real expansion or absolute expansion. The ratio of apparent increase in volume of the liquid per unit rise of temperature to the original volume is called its coefficient of apparent expansion.

For small and equal rises in temperature, the increase in volume (real expansion) of a liquid is equal to the sum of the apparent increase in volume (apparent expansion) of the liquid and the increase in volume of the containing vessel. Thus a liquid has two coefficients of expansion.

Measurement of the expansion of a liquid must account for the expansion of the container as well. For example, when a flask with a long narrow stem, containing enough liquid to partially fill the stem itself, is placed in a heat bath, the height of the liquid column in the stem will initially drop, followed immediately by a rise of that height until the whole system of flask, liquid and heat bath has warmed through. The initial drop in the height of the liquid column is not due to an initial contraction of the liquid, but rather to the expansion of the flask as it contacts the heat bath first. Soon after, the liquid in the flask is heated by the flask itself and begins to expand. Since liquids typically have a greater expansion over solids, the expansion of the liquid in the flask eventually exceeds that of the flask, causing the level of liquid in the flask to rise. A direct measurement of the height of the liquid column is a measurement of the apparent expansion of the liquid. The absolute expansion of the liquid is the apparent expansion corrected for the expansion of the containing vessel.[7]