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{{Material properties (thermodynamics)}}
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When the [[temperature]] of a substance changes, the energy that is stored in the [[intermolecular bond]]s between atoms changes. When the stored energy increases, so does the length of the molecular bonds. As a result, solids typically expand in response to heating and contract on cooling; this dimensional response to temperature change is expressed by its '''coefficient of thermal expansion (CTE)'''.
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Different coefficients of thermal expansion can be defined for a substance depending on whether the expansion is measured by:
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*linear thermal expansion ('''CLTE''')
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*area thermal expansion
*volumetric thermal expansion

These characteristics are closely related. The volumetric thermal expansion coefficient can be defined for both liquids and solids. The linear thermal expansion can only be defined for solids, and is common in engineering applications.

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

==Thermal expansion coefficient==

The '''thermal expansion coefficient''' is a [[thermodynamic properties|thermodynamic property]] of a substance.

It relates the change in temperature to the change in a material's linear dimensions. It is the fractional change in length per degree of temperature change.

:<math>
\alpha={1\over L_0}{\partial L \over \partial T}
</math>

dL = L<sub>0</sub> x ( alpha x dT )

where <math>L_0\ </math> is the original length, <math>L\ </math> the new length, and <math>T\ </math> the temperature.

==Linear thermal expansion==

: <math>{\Delta L \over L_0} = \alpha_L \Delta T</math>

or equivalently:

: <math>L_1 = L_0(1 + \alpha_L (T_2 - T_1))</math>
The linear thermal expansion is the one-dimensional length change with temperature. This equation works reasonably well as long as <math>\alpha_L << 1</math> and <math>\Delta T</math> are relatively small.

Consider a rod of length <math>L = 1m</math> with <math>\alpha = 0.1</math>,
if the change in temperature <math>\Delta T = 2</math> (say from 0 to 2 degree C)
<math>\Delta L</math> will be 0.1 x 1 x 2 = 0.2

On the other hand if we consider the same temperature change in two steps
that is:<br />
1) <math>\Delta T = 1</math> (from 0 to 1 degree C), with <math>L_0</math> = 1m at 0 degree C
<math>\Delta L</math> will be 0.1 x 1 x 1 = 0.1 m
and<br />
2) <math>\Delta T = 1</math> (from 1 to 2 degree C), with <math>L_0</math> = 1+0.1 = 1.1 m at 1 degree C
<math>\Delta L</math> will be 0.1 x 1.1 x 1 = 0.11m

adding the two we get a total change in length of 0.21 m

and hence a contradiction. This occurred because our example <math>\alpha_L</math> was very large, compared to typical values of the order <math>10^{-5}</math>. If <math>\alpha_L</math> is much smaller, this discrepancy will be very small.

=== Proof that the discrepancy will be small ===

Consider a change of temperature from <math>T_0</math> to <math>T_1</math> to <math>T_2</math>, resulting in a change in length from <math>L_0</math> to <math>L_1</math> to <math>L_2</math>:

: <math>L_1 = L_0(1 + \alpha_L (T_1 - T_0))</math>
: <math>L_2 = L_1(1 + \alpha_L (T_2 - T_1))</math>
Combining these equations

: <math>
\begin{alignat}{2}
L_2 & = L_0(1 + \alpha_L (T_1 - T_0))(1 + \alpha_L (T_2 - T_1))\\
& = L_0(1 + \alpha_L (T_2-T_0)) + \alpha_L^2 L_0 (T_1 - T_0) (T_2 - T_1)\\
\end{alignat}
</math>

The second term becomes negligible in the case <math>\alpha_L << 1</math> and that the temperature changes aren't too large, resulting in:

: <math> L_2 = L_0(1 + \alpha_L (T_2-T_0))</math>

As expected.

==Area thermal expansion==

The change in area with temperature can be written:

:<math>
{\Delta A \over A_0} = \alpha_A \Delta T
</math>

For exactly [[Isotropy|isotropic]] materials, the area thermal expansion coefficient is very closely approximated as twice the linear coefficient.

:<math>\alpha_A\cong 2\alpha_L</math>

:<math>
{\Delta A \over A_0} = 2 \alpha_L\Delta T
</math>

==Volumetric thermal expansion==

The change in volume with temperature can be written<ref>{{cite book | first = Donald L. | last = Turcotte | coauthors = Schubert, Gerald | year = 2002 | title = Geodynamics | edition = 2nd Edition | publisher = Cambridge | isbn = 0-521-66624-4 }}</ref>:

:<math>
{\Delta V \over V_0} = \alpha_V \Delta T
</math>

The volumetric thermal expansion coefficient can be written

:<math>
\alpha_V =\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p=-{1\over\rho} \left(\frac{\partial \rho}{\partial T}\right)_{p}
</math>

where <math>T\ </math> is the temperature, <math>V\ </math> is the volume, <math>\rho\ </math> is the density, derivatives are taken at constant pressure <math>p\ </math>; <math>\beta\ </math> measures the fractional change in density as temperature increases at constant pressure.

For exactly [[Isotropy|isotropic]] materials, the volumetric thermal expansion coefficient is very closely approximated as three times the linear coefficient.

:<math>\alpha_V\cong 3\alpha_L</math>

:<math>
{\Delta V \over V_0} = 3 \alpha \Delta T
</math>

Proof:

:<math>
\alpha_V = \frac{1}{V} \frac{\partial V}{\partial T} = \frac{1}{L^3} \frac{\partial L^3}{\partial T} = \frac{1}{L^3}\left(\frac{\partial L^3}{\partial L} \cdot \frac{\partial L}{\partial T}\right) = \frac{1}{L^3}\left(3L^2 \frac{\partial L}{\partial T}\right) = 3 \cdot \frac{1}{L}\frac{\partial L}{\partial T} = 3\alpha_L
</math>

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). Note that the partial derivative of volume with respect to length as shown in the above equation is exact, however, in practice it is important to note that the differential change in volume is only valid for small changes in volume (i.e., the expression is not linear). As the change in temperature increases, and as the value for the linear coefficient of thermal expansion increases, the error in this formula also increases.
For non-negligible changes in volume:

:<math>
({L + }{\Delta L})^3 = {L^3 + 3L^2}{\Delta L} + {3L}{\Delta L}^2 + {\Delta L}^3 \,</math>

Note that this equation contains the main term, <math> 3L^2\ </math>, but also shows a secondary term that scales as <math> 3L{\Delta L}^2 = {3L^3}{\alpha}^2{\Delta T}^2\,</math>, which shows that a large change in temperature can overshadow a small value for the linear coefficient of thermal expansion. Although the coefficient of linear thermal expansion can be quite small, when combined with a large change in temperature the differential change in length can become large enough that this factor needs to be considered. The last term, <math>{\Delta L}^3\ </math> is vanishingly small, and is almost universally ignored.

==Anisotropy==

In [[anisotropic]] materials the total volumetric expansion is distributed unequally among the three axes and if the 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#Expansion_tensors.2C_bulk_modulus|powder diffraction]].

==Thermal expansion coefficients for some common 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<sup>-7</sup> for hard solids to 10<sup>-3</sup> for organic liquids. α varies with the temperature and some materials have a very high variation. Some values for common materials, given in parts per million per [[Celsius]] degree: (NOTE: This can also be in [[kelvin]]s as the changes in temperature are a 1:1 ratio)

NOTE: Theoretically, the coefficient of linear expansion can be found from the coefficient of volumetric expansion (β=3α). However, for liquids, α is calculated through the experimental determination of β, hence it is more accurate to state β here than α.
(The formula β=3α is usually used for solids)<ref name="thermex1">[http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Thermal/ThermExpan.html Thermal Expansion<!-- Bot generated title -->]</ref>

<!-- when adding/editing values, please arrange them in decreasing numerical value. thank you. -->
{|
| valign=top |
{| class="wikitable"
|- {{highlight1}}
! colspan=2 | coefficient of linear thermal expansion ''α''
! colspan=1 | coefficient of volumetric thermal expansion ''β''
|- {{highlight1}}
!material
!α in 10<sup>-6</sup>/K at 20 °C
!β(=3α) in 10<sup>-6</sup>/K at 20 °C
|-
|[[Gasoline]]
|~317
|950<ref name="thermex1" />
|-
|[[Ethanol]]
|~250
|750<ref>Textbook: Young and Geller College Physics, 8e</ref>
|-
|[[Water]]
|~69
|207<ref name="thermex2">[http://www.efunda.com/materials/common_matl/Common_Matl.cfm?MatlPhase=Liquid&MatlProp=Thermal Properties of Common Liquid Materials<!-- Bot generated title -->]</ref>
|-
|[[Mercury (element)|Mercury]]
|~61
|182<ref name="thermex2" />
|-
|[[Rubber]]
|77
|231
|-
|[[PVC]]
|52
|156
|-
|[[Benzocyclobutene]]
|42
|126
|-
|[[Lead]]
|29
|87
|-
|[[Magnesium]]
|26
|78
|-
|[[Aluminium]]
|23
|69
|-
|[[Brass]]
|19
|57
|-
|[[Silver]]
|18<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/tables/thexp.html#c1 Thermal Expansion Coefficients<!-- Bot generated title -->]</ref>
|54
|-
|[[Stainless steel]]
|17.3
|51.9
|-
|[[Copper]]
|17
|51
|-
|[[Gold]]
|14
|42
|-
|[[Nickel]]
|13
|39
|-
|[[Concrete]]
|12
|36
|-
|[[Steel]], depends on composition
|11.0 ~ 13.0
|33.0 ~ 39.0
|-
|[[Iron (element)|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
|-
|[[Indium phosphide]]
|4.6
|13.8
|-
|[[Tungsten]]
|4.5
|13.5
|-
|[[Glass]], [[borosilicate]]
|3.3
|9.9
|-
|[[Silicon]]
|3
|9
|-
|[[Silicon Carbide]]
|2.77 <ref>[http://www.ioffe.rssi.ru/SVA/NSM/Semicond/SiC/basic.html Basic Parameters of Silicon Carbide (SiC)<!-- Bot generated title -->]</ref>
|8.31
|-
|[[Invar]]
|1.2
|3.6
|-
|[[Diamond]]
|1
|3
|-
|[[Quartz]] ([[Fused quartz|fused]])
|0.59
|1.77
|-
|[[Oak]] (perpendicular to the grain)
|54 <ref>[http://www.forestry.caf.wvu.edu/programs/woodindustries/wdsc340_7.htm WDSC 340. Class Notes on Thermal Properties of Wood<!-- Bot generated title -->]</ref>
|162
|-
|[[Pine]] (perpendicular to the grain)
|34
|102
|}

==Applications==
For applications using the thermal expansion property, see [[bi-metal]] and [[mercury-in-glass thermometer|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 CTE, used in applications that demand very small changes in physical dimension over a range of temperatures. One of these is [[Invar]] 36, with a coefficient in the 0.6x10<sup>-6</sup> range<!-- This value does not match with the one in the table above (1.2x10^-6) -->. 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 consist 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, bu the second end is pressed against a fixed screw. The position of the rod is determined by a micrometer [[screw gauge]] or [[spherometer]].

==External links==
*[http://www.engineeringtoolbox.com/linear-expansion-coefficients-d_95.html Engineering Toolbox – List of coefficients of Linear Expansion for some common materials]
*[http://www.leybold-didactic.com/literatur/hb/e/p2/p2121_e.pdf Article on how β is determined]
*[http://www.matweb.com MatWeb: Free database of engineering properties for over 64,000 materials]
*[http://phoenix.phys.clemson.edu/labs/223/expansion/index.html Clemson University Physics Lab: Linear Thermal Expansion]
*[http://emtoolbox.nist.gov/Temperature/Slide1.asp#Slide1 USA NIST Website - Temperature and Dimensional Measurement workshop]
*[http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thexp.html Hyperphysics: Thermal expansion]

==References==
<references/>

==See also==
*[[Autovent]]
*[[Thermal expansion coefficients of the elements (data page)]]

[[Category:Thermodynamics]]
[[Category:Heat transfer]]
[[Category:Physical quantities]]

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Revision as of 16:10, 15 June 2009

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