Thermal expansion
Contents 1 Overview 1.1 Predicting expansion 1.2 Contraction effects (negative thermal expansion) 1.3 Factors affecting thermal expansion 2 Coefficient of thermal expansion 2.1 General volumetric thermal expansion coefficient 3 Expansion in solids 3.1 Linear expansion 3.1.1 Effects on strain 3.2
Area
3.3.1 Isotropic materials 3.4
Anisotropic
4 Isobaric expansion in gases 5 Expansion in liquids 5.1 Apparent and absolute expansion of a liquid 6 Examples and applications
7
Thermal expansion
Overview[edit]
Predicting expansion[edit]
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 thermal expansion)[edit]
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 subzero weather. Also, fairly pure silicon has a
negative coefficient of thermal expansion for temperatures between
about 18 and 120 kelvins.[2]
Factors affecting thermal expansion[edit]
Unlike gases or liquids, solid materials tend to keep their shape when
undergoing thermal expansion.
Thermal expansion
α V = 1 V ( ∂ V ∂ T ) p displaystyle alpha _ V = frac 1 V ,left( frac partial V partial T right)_ p The subscript p 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. Expansion in solids[edit] 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. Linear expansion[edit] 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 "linear expansion coefficient". It is the fractional change in length per degree of temperature change. Assuming negligible effect of pressure, we may write: α L = 1 L d L d T displaystyle alpha _ L = frac 1 L , frac dL dT where L displaystyle L is a particular length measurement and d L / d T 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: Δ L L = α L Δ T displaystyle frac Delta L L =alpha _ L Delta T This equation works well as long as the linearexpansion coefficient does not change much over the change in temperature Δ T displaystyle Delta T , and the fractional change in length is small Δ L / L ≪ 1 displaystyle Delta L/Lll 1 . If either of these conditions does not hold, the equation must be integrated. Effects on strain[edit] 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 ϵ t h e r m a l displaystyle epsilon _ mathrm thermal and defined as: ϵ t h e r m a l = ( L f i n a l − L i n i t i a l ) L i n i t i a l displaystyle epsilon _ mathrm thermal = frac (L_ mathrm final L_ mathrm initial ) L_ mathrm initial where L i n i t i a l displaystyle L_ mathrm initial is the length before the change of temperature and L f i n a l displaystyle L_ mathrm final is the length after the change of temperature. For most solids, thermal expansion is proportional to the change in temperature: ϵ t h e r m a l ∝ Δ T displaystyle epsilon _ mathrm thermal propto Delta T Thus, the change in either the strain or temperature can be estimated by: ϵ t h e r m a l = α L Δ T displaystyle epsilon _ mathrm thermal =alpha _ L Delta T where Δ T = ( T f i n a l − T i n i t i a l ) displaystyle Delta T=(T_ mathrm final T_ mathrm initial ) is the difference of the temperature between the two recorded strains, measured in degrees Celsius or kelvins, and α L displaystyle alpha _ L is the linear coefficient of thermal expansion in “per degree
Celsius” or “per kelvin”, denoted by °C−1 or K−1,
respectively. In the field of continuum mechanics, the thermal
expansion and its effects are treated as eigenstrain and eigenstress.
Area
α A = 1 A d A d T displaystyle alpha _ A = frac 1 A , frac dA dT where A displaystyle A is some area of interest on the object, and d A / d T 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: Δ A A = α A Δ T 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 Δ T displaystyle Delta T , and the fractional change in area is small Δ A / A ≪ 1 displaystyle Delta A/All 1 . If either of these conditions does not hold, the equation must be
integrated.
Volume
α V = 1 V d V d T displaystyle alpha _ V = frac 1 V , frac dV dT where V displaystyle V is the volume of the material, and d V / d T 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 Δ V V = α V Δ T displaystyle frac Delta V V =alpha _ V Delta T where Δ V / V displaystyle Delta V/V is the fractional change in volume (e.g., 0.002) and Δ T 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: ln ( V + Δ V V ) = ∫ T i T f α V ( T ) d T displaystyle ln left( frac V+Delta V V right)=int _ T_ i ^ T_ f alpha _ V (T),dT Δ V V = exp ( ∫ T i T f α V ( T ) d T ) − 1 displaystyle frac Delta V V =exp left(int _ T_ i ^ T_ f alpha _ V (T),dTright)1 where α V ( T ) displaystyle alpha _ V (T) is the volumetric expansion coefficient as a function of temperature T, and T i displaystyle T_ i , T f displaystyle T_ f are the initial and final temperatures respectively. Isotropic materials[edit] For isotropic materials the volumetric thermal expansion coefficient is three times the linear coefficient: α V = 3 α L displaystyle alpha _ V =3alpha _ L This ratio arises because volume is composed of three mutually orthogonal directions. Thus, in an isotropic material, for small differential changes, onethird 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 V = L 3 displaystyle V=L^ 3 and the new volume, after a temperature increase, will be V + Δ V = ( L + Δ L ) 3 = L 3 + 3 L 2 Δ L + 3 L Δ L 2 + Δ L 3 ≈ L 3 + 3 L 2 Δ L = V + 3 V Δ L L . displaystyle V+Delta V=(L+Delta L)^ 3 =L^ 3 +3L^ 2 Delta L+3LDelta L^ 2 +Delta L^ 3 approx L^ 3 +3L^ 2 Delta L=V+3V Delta L over L . So Δ V V = 3 Δ L L = 3 α L Δ T . displaystyle frac Delta V V =3 Delta L over L =3alpha _ L Delta T. The above approximation holds for small temperature and dimensional changes (that is, when Δ T displaystyle Delta T and Δ L 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 Δ T 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: α A = 2 α L displaystyle alpha _ A =2alpha _ 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 L 2 displaystyle L^ 2 . Also, the same considerations must be made when dealing with large values of Δ T displaystyle Delta T .
Anisotropic
α L displaystyle frac 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 xray powder diffraction. Isobaric expansion in gases[edit] For an ideal gas, the volumetric thermal expansion (i.e., relative change in volume due to temperature change) depends on the type of process in which temperature is changed. Two simple cases are where the pressure is held constant (Isobaric process), or when the volume (Isochoric process) is held constant. The derivative of the ideal gas law, P V = T displaystyle PV=T , is P d V + V d P = d T displaystyle PdV+VdP=dT where P displaystyle P is the pressure, V displaystyle V is the specific volume, and T displaystyle T is temperature measured in energy units. By definition of an isobaric thermal expansion, we have d P = 0 displaystyle dP=0 , so that P d V = d T displaystyle PdV=dT , and the isobaric thermal expansion coefficient is α P = C t e ≡ 1 V ( d V d T ) = 1 V ( 1 P ) = 1 P V = 1 T displaystyle alpha _ P=C^ te equiv frac 1 V left( frac dV dT right)= frac 1 V left( frac 1 P right)= frac 1 PV = frac 1 T . Similarly, if the volume is held constant, that is if d V = 0 displaystyle dV=0 , we have V d P = d T displaystyle VdP=dT , so that the isovolumic thermal expansion is α V = C t e ≡ 1 P ( d P d T ) = 1 P ( 1 V ) = 1 P V = 1 T displaystyle alpha _ V=C^ te equiv frac 1 P left( frac dP dT right)= frac 1 P left( frac 1 V right)= frac 1 PV = frac 1 T . Expansion in liquids[edit] This section needs expansion. You can help by adding to it. (August 2010) Theoretically, the coefficient of linear expansion can be found from
the coefficient of volumetric expansion (αV ≈ 3α). For
liquids, α is calculated through the experimental determination of
αV. Liquids, unlike solids have no definite shape and they take the
shape of the container. Consequently, liquids have no definite length
and area, so linear and areal expansions of liquids have no
significance.
Liquids in general, expand on heating. However water is an exception
to this general behaviour: Below 4 °C it contracts on heating.
For higher temperature it shows the normal positive thermal expansion.
The thermal expansions of liquid is usually higher than in solids
because of weak intermolecular forces present in liquids.
Thermal expansion
Thermal expansion
The expansion and contraction of materials 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.
Thermal expansion
Drinking glass with fracture due to uneven thermal expansion after pouring of hot liquid into the otherwise cool glass The control of thermal expansion in brittle materials is a key concern
for a wide range of reasons. For example, both glass and ceramics are
brittle and uneven temperature causes uneven expansion which again
causes thermal stress and this might lead to fracture. Ceramics need
to be joined or work in concert with a wide range of materials and
therefore their expansion must be matched to the application. Because
glazes need to be firmly attached to the underlying porcelain (or
other body type) their thermal expansion must be tuned to 'fit' the
body so that crazing or shivering do not occur. Good example of
products whose thermal expansion is the key to their success are
CorningWare
Metal framed windows need rubber spacers
Rubber
Thermometers are another application of thermal expansion – most
contain a liquid (usually mercury or alcohol) which is constrained to
flow in only one direction (along the tube) due to changes in volume
brought about by changes in temperature. A bimetal mechanical
thermometer uses a bimetallic strip and bends due to the differing
thermal expansion of the two metals.
Metal pipes made of different materials are heated by passing steam
through them. While each pipe is being tested, one end is securely
fixed and the other rests on a rotating shaft, the motion of which is
indicated with a pointer. The linear expansion of the different metals
is compared qualitatively and the coefficient of linear thermal
expansion is calculated.
Thermal expansion
Volumetric thermal expansion coefficient for a semicrystalline polypropylene. Linear thermal expansion coefficient for some steel grades. 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 = 3α. 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.[10] In particular for metals the relation is: α ≈ 0.020 M P displaystyle alpha approx frac 0.020 M_ P for halides and oxides α ≈ 0.038 M P − 7.0 ⋅ 10 − 6 K − 1 displaystyle alpha approx frac 0.038 M_ P 7.0cdot 10^ 6 ,mathrm K ^ 1 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 TiNb alloy [11]. (The formula αV ≈ 3α is usually used for solids.)[12] Material Linear coefficient α at 20 °C (10−6 K−1) Volumetric coefficient αV at 20 °C (10−6 K−1) Notes Aluminium 23.1 69
Aluminium
Benzocyclobutene 42 126 Brass 19 57 Carbon steel 10.8 32.4 CFRP – 0.8[13] Anisotropic Fiber direction Concrete 12 36 Copper 17 51 Diamond 1 3 Ethanol 250 750[14] Gallium(III) arsenide 5.8 17.4 Gasoline 317 950[12] Glass 8.5 25.5 Glass, borosilicate 3.3 [15] 9.9 matched sealing partner for tungsten, molybdenum and kovar.
Glass
Glycerine 485[16] Gold 14 42 Helium 36.65[16] Ice 51 Indium phosphide 4.6 13.8 Invar 1.2 3.6 Iron 11.8 35.4 Kapton
20[17]
60
DuPont
Kapton
Lead 29 87 Macor 9.3[18] Magnesium 26 78 Mercury 61 182[16][19] Molybdenum 4.8 14.4 Nickel 13 39 Oak 54[20] Perpendicular to the grain Douglasfir 27[21] 75 radial Douglasfir 45[21] 75 tangential Douglasfir 3.5[21] 75 parallel to grain Platinum 9 27 PP 150 450 [citation needed] PVC 52 156
Quartz
alphaQuartz 1216/69[22] Parallel to aaxis/caxis T = 50 to 150 C Rubber disputed disputed see Talk Sapphire 5.3[23] Parallel to C axis, or [001]
Silicon
Silicon 2.56[25] 9 Silver 18[26] 54 Sitall 0±0.15[27] 0±0.45 average for −60 °C to 60 °C Stainless steel 10.1 ~ 17.3 30.3 ~ 51.9 Steel 11.0 ~ 13.0 33.0 ~ 39.0 Depends on composition Titanium 8.6 26[28] Tungsten 4.5 13.5 Turpentine 90[16] Water 69 207[19] YbGaGe ≐0 ≐0[29] Refuted[30] Zerodur ≈0.0070.1[31] at 0...50 °C See also[edit] Negative thermal expansion
MieGruneisen equation of state
Autovent
Grüneisen parameter
Apparent molar property
Heat
References[edit] ^ Tipler, Paul A.; Mosca, Gene (2008). Physics for Scientists and
Engineers 
Volume
External links[edit] Wikimedia Commons has media related to Thermal expansion.
Glass
