Thermal expansion is the tendency of matter to increase in size with increasing temperature. Matter generally increases in length, area, and volume, changing its size and density, in response to an increase in temperature (usually excluding phase transitions). Substances usually contract with decreasing temperature which is called thermal contraction. The SI unit of thermal expansion is the inverse kelvin (K−1).
Temperature is a measure of the average kinetic energy of the molecules in a body. The faster the molecules are moving, the higher that body's temperature is. Specifically, it is a monotonic function of the average molecular kinetic energy of a substance. As the energy in the 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 them.
The relative expansion (also called strain) divided by the change in temperature is called the material's coefficient of linear thermal expansion. For small temperature changes, this is nearly constant, but for larger changes it varies with temperature. The coefficient of thermal expansion typically increases with temperature, as higher thermal energy reduces intermolecular forces and allows greater atomic displacement.Atkins, Peter; de Paula, Julio. Physical Chemistry. Oxford University Press. 10th ed. (2014).
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 the coefficient of thermal expansion and specific heat. These discontinuities allow detection of the glass transition temperature where a supercooled liquid transforms into 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 the 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.Callister, William D.; Rethwisch, David G. Materials Science and Engineering: An Introduction. Wiley. 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
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 . For liquids, usually the coefficient of volumetric expansion is listed and linear expansion is calculated here for comparison.
For common materials, such as many metals and compounds, the thermal expansion coefficient is inversely proportional to the melting point. In particular, for metals the relation is:
for halides and oxides
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.
The formula is usually used for solids. Volumetric coefficients shown which do not follow that rule are highlighted.
+Thermal expansion coefficients at 20 °CMaterialMaterial typeLinearcoefficient CLTEα(×10−6 K−1)VolumetriccoefficientαV(×10−6 K−1)NotesALLVAR Alloy 30Metal alloy−30AnisotropicExhibits negative thermal expansion in broad range of temperaturesAluminiumMetal23.169BrassMetal alloy1957Carbon steelMetal alloy10.832.4CFRPComposite–0.8AnisotropicFiber directionConcreteAggregate1236CopperMetal1751DiamondNonmetal13Douglas firBiological2775Radial45Tangential 3.5Parallel to grainEthanolLiquid250750GasolineLiquid317950GlassGlass8.525.5Glass (borosilicate)Glass3.39.9Matched sealing partner for tungsten, molybdenum and kovar.GlycerineLiquid485GoldMetal1442GraniteRock35–43105–129IceNonmetal51InvarMetal Alloy1.23.6IronMetal11.835.4KaptonPolymer2060DuPont Kapton 200ENLeadMetal2987Macor9.3MercuryLiquid60.4181NickelMetal1339OakBiological54Perpendicular to the grainPlatinumMetal927Polypropylene (PP)Polymer150450PVCPolymer52156Quartz (alpha)Nonmetal12–16/6–9Parallel to a-axis/c-axis T = –50 to 150 °CQuartz (fused)Nonmetal0.591.77RubberBiologicalDisputedDisputedSee TalkRock saltRock40120SapphireNonmetal5.3Parallel to C axis, or Silicon carbideNonmetal2.778.31SiliconNonmetal2.569SilverMetal1854SitallGlass-ceramic 0±0.150±0.45Average for −60 °C to 60 °CStainless steelMetal alloy10.1 ~ 17.330.3 ~ 51.9SteelMetal alloy11.0 ~ 13.033.0 ~ 39.0Depends on compositionTitaniumMetal8.626TungstenMetal4.513.5WaterNonmetal69207ZerodurGlass-ceramic ≈0.007–0.1From 0 °C to 50 °C
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:
where is a particular length measurement and 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:
This estimation works well as long as the linear-expansion coefficient does not change much over the change in temperature , and the fractional change in length is small . If either of these conditions does not hold, the exact differential equation (using ) 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 and defined as:
where is the length before the change of temperature and is the length after the change of temperature.
For most solids, thermal expansion is proportional to the change in temperature:
Thus, the change in either the strain or temperature can be estimated by:
where
is the difference of the temperature between the two recorded strains, measured in degrees Fahrenheit, degrees Rankine, degrees Celsius, or kelvin, and is the linear coefficient of thermal expansion in "per degree Fahrenheit", "per degree Rankine", "per degree Celsius", or "per kelvin", denoted by , , , or , 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:
where is some area of interest on the object, and is the rate of change of that area per unit change in temperature.
The change in the area can be estimated as: