Tendency of matter to change volume in response to a change in temperature
Thermal expansion is the tendency of matter to increase in length, area, or volume, changing its size and density, in response to an increase in temperature (usually excluding phase transitions).
Substances usually contract with decreasing temperature (thermal contraction), with rare exceptions within limited temperature ranges ("negative thermal expansion").
Temperature is a monotonic function of the average molecular kinetic energy of a substance. As energy in particles increases, they start moving faster and faster, weakening the intermolecular forces between them and therefore expanding the substance.
When a substance is heated, molecules begin to vibrate and move more, usually creating more distance between themselves. 
The relative expansion (also called strain) divided by the change in temperature is called the material's coefficient of linear thermal expansion and generally varies with temperature. 
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Prediction.
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.
Contraction effects (negative expansion).
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 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.
Other materials are also known to exhibit negative thermal expansion. Fairly pure silicon has a negative coefficient of thermal expansion for temperatures between about . ALLVAR Alloy 30, a titanium alloy, exhibits anisotropic negative thermal expansion across a wide range of temperatures.
Factors.
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 slightly higher compared to that of crystals. 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.
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.
Effect on density.
Thermal expansion changes the space between particles of a substance, which changes the volume of the substance while negligibly changing its mass (the negligible amount comes from mass–energy equivalence), thus changing its density, which has an effect on any buoyant forces acting on it. This plays a crucial role in convection of unevenly heated fluid masses, notably making thermal expansion partly responsible for wind and ocean currents.
Coefficients.
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 size per degree change in temperature at a constant pressure, such that lower coefficients describe lower propensity for change in size. Several types of coefficients have been developed: volumetric, area, and linear. The choice of coefficient depends on the particular application and which dimensions are considered important. For solids, one might only be concerned with the change along a length, or over some area.
The volumetric thermal expansion coefficient is the most basic thermal expansion coefficient, and the most relevant for fluids. In general, substances expand or contract when their temperature changes, with expansion or contraction occurring in all directions. Substances that expand at the same rate in every direction are called isotropic. For isotropic materials, the area and volumetric thermal expansion coefficient are, respectively, approximately twice and three times larger than the linear thermal expansion coefficient.
In the general case of a gas, liquid, or solid, the volumetric coefficient of thermal expansion is given by
formula_1
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.
For various materials.
This section summarizes the coefficients for some common materials.
For isotropic materials the coefficients linear thermal expansion "α" and volumetric thermal expansion "αV" are related by "αV" = 3"α".
For liquids usually the coefficient of volumetric expansion is listed and linear expansion is calculated here for comparison.
For common materials like many metals and compounds, the thermal expansion coefficient is inversely proportional to the melting point.
In particular, for metals the relation is:
formula_2
for halides and oxides
formula_3
In the table below, the range for "α" is from 10−7 K−1 for hard solids to 10−3 K−1 for organic liquids. The coefficient "α" varies with the temperature and some materials have a very high variation; see for example the variation vs. temperature of the volumetric coefficient for a semicrystalline polypropylene (PP) at different pressure, and the variation of the linear coefficient vs. temperature for some steel grades (from bottom to top: ferritic stainless steel, martensitic stainless steel, carbon steel, duplex stainless steel, austenitic steel). The highest linear coefficient in a solid has been reported for a Ti-Nb alloy.
In solids.
When calculating thermal expansion it is necessary to consider whether the body is free to expand or is constrained. If the body is free to expand, the expansion or strain resulting from an increase in temperature can be simply calculated by using the applicable coefficient of thermal expansion.
If the body is constrained so that it cannot expand, then internal stress will be caused (or changed) by a change in temperature. This stress can be calculated by considering the strain that would occur if the body were free to expand and the stress required to reduce that strain to zero, through the stress/strain relationship characterised by the elastic or Young's modulus. In the special case of solid materials, external ambient pressure does not usually appreciably affect the size of an object and so it is not usually necessary to consider the effect of pressure changes.
Common engineering solids usually have coefficients of thermal expansion 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, practical calculations can be based on a constant, average, value of the coefficient of expansion.
Length.
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, one may write:
formula_4
where formula_5 is a particular length measurement and formula_6 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:
formula_7
This estimation works well as long as the linear-expansion coefficient does not change much over the change in temperature formula_8, and the fractional change in length is small formula_9. If either of these conditions does not hold, the exact differential equation (using formula_6) must be integrated.
Effects on strain.
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 formula_11 and defined as:
formula_12
where formula_13 is the length before the change of temperature and formula_14 is the length after the change of temperature.
For most solids, thermal expansion is proportional to the change in temperature:
formula_15
Thus, the change in either the strain or temperature can be estimated by:
formula_16
where
formula_17
is the difference of the temperature between the two recorded strains, measured in degrees Fahrenheit, degrees Rankine, degrees Celsius, or kelvin, and formula_18 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, thermal expansion and its effects are treated as eigenstrain and eigenstress.
Area.
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, one may write:
formula_19
where formula_20 is some area of interest on the object, and formula_21 is the rate of change of that area per unit change in temperature.
The change in the area can be estimated as:
formula_22
This equation works well as long as the area expansion coefficient does not change much over the change in temperature formula_8, and the fractional change in area is small formula_24. If either of these conditions does not hold, the equation must be integrated.
Volume.
For a solid, one can ignore the effects of pressure on the material, and the volumetric (or cubical) thermal expansion coefficient can be written:
formula_25
where formula_26 is the volume of the material, and formula_27 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 a block of steel has 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 the expansion coefficient is known, the change in volume can be calculated
formula_28
where formula_29 is the fractional change in volume (e.g., 0.002) and formula_8 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:
formula_31
formula_32
where formula_33 is the volumetric expansion coefficient as a function of temperature "T", and formula_34 and formula_35 are the initial and final temperatures respectively.
Isotropic materials.
For isotropic materials the volumetric thermal expansion coefficient is three times the linear coefficient:
formula_36
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 formula_37 and the new volume, after a temperature increase, will be
formula_38
We can easily ignore the terms as Δ"L" is a small quantity which on squaring gets much smaller and on cubing gets smaller still.
So
formula_39
The above approximation holds for small temperature and dimensional changes (that is, when formula_8 and formula_41 are small), but it does not hold if trying to go back and forth between volumetric and linear coefficients using larger values of formula_8. 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:
formula_43
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 formula_44. Also, the same considerations must be made when dealing with large values of formula_8.
Put more simply, if the length of a cubic solid expands from 1.00 m to 1.01 m, then the area of one of its sides expands from 1.00 m2 to 1.02 m2 and its volume expands from 1.00 m3 to 1.03 m3.
Anisotropic materials.
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