 |
2.3.5 Thermal Expansion
Coefficients of expansion
Coefficients of thermal expansion are normally described
either as the increase in length (or volume, esp. for liquids) per unit length
at a given temperature, known as the expansivity,
α. =
(l/L)(dL/dT), or as the mean expansion coefficient over a
temperature range,
 = (1/L0)(ΔL/ΔT),
where L is the instantaneous length, L0 is an initial
length, T is temperature, and ΔL and ΔT are
changes in length referenced to a temperature at which L0 was
measured. The latter form is more common than the former in engineering texts.
Significant differences in numerical data can arise between the two methods. In
this section, expansivity data only are given. The latter form can be
determined from the former by integration over ΔT. To a first
approximation, cubical or volume expansivities of solids are
three times the linear expansivity.
Coefficients of cubical expansion of liquids
The following table gives values for the cubical
(volume) expansivity (l/V) (dV/dT) at T = 293 K
(20 °C). Generally, the expansivity increases with increasing
temperature.
|
Liquid |
α/10−5K−1 |
Liquid |
α/10−5K−1 |
|
|
|
|
|
|
Acetic acid |
107 |
Ethyl
bromide |
141 |
|
Acetone |
143 |
Ethylene
glycol |
57 |
|
Alcohol, methyl |
118 |
Glycerol
(glycerine) |
49 |
|
Alcohol, ethyl |
109 |
Mercury* |
18.2 |
|
Aniline |
85 |
Methyl
iodide |
120 |
|
Benzene |
121 |
n-Pentane |
158 |
|
Bromine |
112 |
Sulphuric
acid (100%) |
56 |
|
Carbon disulphide |
119 |
Toluene |
107 |
|
Carbon tetrachloride |
122 |
Turpentine |
96 |
|
Chloroform |
127 |
m-Xylene |
99 |
|
Ether |
163 |
Water** |
21 |
|
|
|
|
|
* See also
section 2.2.1 (Density of mercury).
**
See also section 2.2.1 (Density of water).
Coefficients of linear expansion of
solids
The expansivities of the majority of solid materials increase
with increasing temperature, and can be represented by an equation of the
form α = a + bT +
cT2 over limited temperature ranges. The tables in
this section cover elements, metal alloys, ceramics and miscellaneous
materials.
Many materials exhibit anisotropic thermal expansion
behaviour. When single crystals are in common use, data in the respective
principal directions are given. Otherwise a homogeneous isotropic
polycrystal-line solid is assumed. High levels of anisotropy and/or phase
changes can lead to microcracking and thermal expansion hysteresis. Complex
multiphase materials possess thermal expansion characteristics which are
related to the expansion coefficients and elastic moduli of the individual
components. Only approximate ranges can be cited. Further, more detailed data
can be obtained from Touloukian et al., (1971).
|
Elements |
α/(10−6
K−1 |
|
100 K |
200 K |
293 K |
500 K |
800 K |
1100 K |
1500 K |
|
|
|
|
|
|
|
|
|
|
Aluminium |
12.2 |
20.3 |
23.1 |
26.4 |
34.0 |
– |
– |
|
Antimony* |
9.1 |
10.5 |
11.0 |
11.7 |
11.7 |
– |
– |
|
Beryllium* |
1.3 |
7.1 |
11.3 |
15.1 |
19.1 |
21.6 |
23.7 |
|
Bismuth* |
12.3 |
13.1 |
13.4 |
12.7 |
– |
– |
– |
|
Boron |
– |
– |
4.7 |
5.4 |
6.2 |
6.8 |
– |
|
Cadmium |
26.9 |
29.8 |
30.8 |
36.0 |
– |
– |
– |
|
Carbon, vitreous |
– |
– |
3.1 |
3.3 |
3.6 |
4.0 |
4.6 |
|
Carbon, diamond |
0.05 |
0.4 |
1.0 |
2.3 |
3.7 |
4.7 |
5.6 |
|
Carbon, graphite, polycrystalline** |
– |
– |
7.1 |
7.5 |
8.1 |
8.6 |
9.3 |
|
Carbon, pyrolytic, para. deposition |
– |
– |
23.1 |
24.4 |
25.9 |
27.2 |
28.6 |
|
perp. deposition |
– |
– |
−0.6 |
0.6 |
0.8 |
1.7 |
2.5 |
|
Chromium |
2.3 |
5.3 |
4.9a |
8.8 |
10.8 |
12.3 |
14.9 |
|
Cobalt* |
6.8 |
11.5 |
13.0 |
15.0b |
15.2 |
17.0 |
– |
|
Copper |
10.3 |
15.2 |
16.5 |
18.3 |
20.3 |
23.7 |
– |
|
Germanium |
2.4 |
4.9 |
5.7 |
6.5 |
7.2 |
7.8 |
– |
|
Gold |
11.8 |
13.7 |
14.2 |
15.4 |
17.0 |
19.7 |
– |
|
Indium* |
25.4 |
28.1 |
32.1 |
– |
– |
– |
– |
|
Iridium |
4.4 |
5.9 |
6.4 |
7.2 |
8.1 |
8.5 |
9.4 |
|
Iron |
5.6 |
10.1 |
11.8 |
13.9 |
16.2 |
16.7c |
23.3c |
|
Lead |
25.6 |
27.5 |
28.9 |
33.3 |
– |
– |
– |
|
Magnesium* |
−0.2 |
0 |
8.2 |
11.3 |
13.2 |
14.4 |
– |
|
Molybdenum |
2.8 |
4.6 |
4.8 |
5.1 |
5.7 |
6.5 |
7.5 |
|
Nickel |
6.6 |
11.3 |
13.4 |
15.3 |
16.8 |
17.8 |
20.3 |
|
Niobium |
5.2 |
6.8 |
7.3 |
7.8 |
8.2 |
8.7 |
9.3 |
|
Palladium |
8.0 |
10.7 |
11.8 |
13.2 |
14.5 |
16.3 |
– |
|
Platinum |
6.6 |
8.5 |
8.8 |
9.6 |
10.3 |
11.1 |
12.8 |
|
Rhodium |
5.0 |
7.3 |
8.2 |
9.3 |
10.8 |
12.5 |
14.8 |
|
Silicon |
−0.4 |
1.5 |
2.6 |
3.5 |
4.1 |
4.5 |
4.7 |
|
Silver |
14.2 |
17.8 |
18.9 |
20.6 |
23.7 |
27.1 |
– |
|
Tantalum |
4.8 |
6.0 |
6.3 |
6.8 |
7.2 |
7.4 |
7.8 |
|
Thallium* |
25.2 |
28.0 |
29.9 |
34.7 |
– |
– |
– |
|
Tin* |
16.5 |
19.6 |
22.0 |
27.2 |
– |
– |
– |
|
Titanium* |
4.5 |
7.4 |
8.6 |
9.9 |
11.1 |
11.7d |
12.9 |
|
Tungsten |
2.6 |
4.1 |
4.5 |
4.6 |
5.0 |
5.3 |
5.5 |
|
Uranium* |
10.0 |
13.4 |
13.9 |
16.9 |
24.3e |
22.9e |
– |
|
Vanadium |
5.1 |
7.1 |
8.4 |
9.9 |
10.9 |
12.0 |
14.1 |
|
Zinc* |
24.5 |
28.6 |
30.2 |
32.8 |
– |
– |
– |
|
|
|
|
|
|
|
|
|
* Crystallographically
anisotropic. Data are for isotropic polycrystalline bodies. For anisotropic
bodies, data vary.
** Data for isotropic POCO Grade
AXM-5Q isotropic graphite. Most polycrystalline graphites are anisotropic.
a Phase change at 311
K.
b Phase change at 690 K.
c Phase changes at 1 185 K
and 1 667 K.
d Phase
change at 1 156 K.
e
Phase change at 941 K and 1 048 K.
|
Metal alloys
(Approximate compositions in mass
%) |
α/(10−6
K−1) |
|
100 K |
200 K |
293 K |
500 K |
800 K |
1 100 k |
|
|
|
|
|
|
|
|
|
Aluminium bronze (90 Cu + 5 Al + 4.5 Ni) |
12–14 |
– |
15.9 |
18.1 |
20.3 |
– |
|
Brass (67 Cu + 33 Zn) |
– |
– |
17.5 |
20.0 |
22.5 |
– |
|
Bronze (85 Cu + 15 Sn) |
– |
– |
17.3 |
19.3 |
21.9 |
– |
|
Cast iron (Fe + 3 C + 2 Si) |
– |
– |
11.9 |
13.1 |
14.5 |
– |
|
Constantan (65 Cu + 35 Ni) |
11.2 |
– |
15.0 |
17.4 |
19.2 |
– |
|
Cupro-nickel (65 Ni + 30 Cu + 1.5 Fe + 1 Mn) |
9.8 |
– |
12.7 |
15.4 |
18.2 |
– |
|
Dural (94 Al + 4 to 5 Cu) |
13.1 |
– |
21.6 |
27.5 |
30.1 |
– |
|
Inconel |
8.7 |
– |
11.6 |
14.4 |
17.6 |
– |
|
Nickel–iron alloys* |
|
|
|
|
|
|
|
(64 Fe + 36 Ni,
Invar) |
1.4 |
0.53 |
0.13 |
5.1 |
17.1 |
– |
|
(63 Fe + 32 Ni +
4 Co, Super Invar) |
– |
– |
0.0 |
– |
– |
– |
|
(50 Fe + 50 Ni)
|
– |
– |
9.9 |
10.2 |
13.7 |
17.3 |
|
Phosphor bronze |
– |
– |
17.0 |
20.0 |
– |
– |
|
Stainless steel |
|
|
|
|
|
|
|
(ferritic types,
16–19 Cr + 6–14 Ni) |
7.0 |
9.2 |
11.1 |
15.0 |
19.7 |
23.2 |
|
(austenitic types
e.g. 13–17 Cr + 10 Ni) |
6.0 |
7.9 |
9.5 |
12.1 |
13.8 |
13.9 |
|
Steel, carbon (0.7–1.4 C) |
6.9 |
– |
10.7 |
13.7 |
16.2 |
– |
|
Stellite (65 Co + 20–30 Cr + 6–15 W)
|
6.9 |
9.3 |
11.2 |
14.6 |
17.2 |
17.4 |
|
Tungsten carbide cermets (4–11 Co) |
– |
– |
3.7 |
4.3 |
4.8 |
5.5 |
|
|
|
|
|
|
|
|
* Note that Ni–Fe–Co
alloys have low expansivities below the gamma to alpha phase transformation and
high expansions above this temperature. Expansivities and the transition
temperature depend critically on the proportions of the major as well as minor
elements; see for example. Partridge (1949) or ASM Metals Handbook
(1981).
|
Ceramics, glasses, semiconductors |
α/(10−6
K−1) |
|
100 K |
200 K |
293 K |
500 K |
800 K |
1 100 K |
1 500 K |
|
|
|
|
|
|
|
|
|
|
Alumina (Al2O3) |
0.6 |
3.3 |
5.5 |
7.8 |
8.5 |
9.4 |
10.2 |
|
Beryllia (BeO) |
– |
– |
5.5 |
8.0 |
9.5 |
10.6 |
12.4 |
|
Boron nitride: |
|
|
|
|
|
|
|
|
para. hot
pressing |
– |
0–2 |
0–2.3 |
0–3 |
0–4 |
1–8 |
4–9 |
|
perp, hot
pressing |
– |
~1 |
0–1 |
0–1 |
0–1 |
1–2 |
1–2 |
|
Cordierites
(Mg2Al4Si5O18) |
– |
– |
~0 |
~1.0 |
1–2 |
– |
– |
|
Forsterites (Mg2SiO4) |
– |
– |
9.0 |
10.3 |
11.8 |
13.2 |
– |
|
Glasses:* |
|
|
|
|
|
|
|
|
Borosilicate, Pyrex
|
1.5 |
2.7 |
2.8 |
3.3 |
5.0 |
– |
– |
|
Borosilicate,
crown |
– |
– |
7–8 |
– |
– |
– |
– |
|
Dense flint |
– |
– |
8–9 |
– |
– |
– |
– |
|
Fused silica |
−0.53 |
0.13 |
0.49 |
0.63 |
0.47 |
0.35 |
– |
|
Soda-lime
(Float) |
– |
– |
7.5 |
– |
– |
– |
– |
|
Glass–ceramics:* |
|
|
|
|
|
|
|
|
Corning 9606 |
– |
– |
1.9 |
3.6 |
4.1 |
4.6 |
– |
|
Corning 9608 |
– |
– |
1–2 |
1–3 |
1–4 |
2–5 |
– |
|
Macor
machinable |
– |
– |
~8 |
~9 |
~11 |
~14 |
– |
|
Zerodur |
– |
<0.1 |
<0.1 |
– |
– |
– |
– |
|
Magnesia (MgO) |
2.2 |
7.6 |
– |
11.0 |
13 |
15.2 |
16.2 |
|
Magnesium fluoride: |
|
|
|
|
|
|
|
|
para. c-axis |
3.9 |
– |
14.5 |
17.0 |
19.2 |
– |
– |
|
perp. c-axis |
1.4 |
– |
9.5 |
11.5 |
15.8 |
– |
– |
|
polycrystalline |
2.2 |
– |
11.1 |
13.3 |
16.8 |
– |
– |
| Mullites
(Al6Si2O13) |
– |
– |
3.0 |
4.4 |
5.2 |
6.0 |
6.5 |
|
Porcelains: |
|
|
|
|
|
|
|
|
aluminous |
– |
– |
3–6 |
4–7 |
5–8 |
7–10 |
– |
|
chemical |
– |
– |
2–4 |
– |
– |
– |
– |
|
quartz |
– |
– |
3–6 |
4–7 |
5–8 |
7–10 |
– |
|
Pyrophillite, fired 1 250°C** |
– |
– |
2–3 |
3–4 |
3–4 |
– |
– |
|
Quartz single crystal: |
|
|
|
|
|
|
|
|
para. c-axis |
4.0 |
5.2 |
6.8 |
11.4 |
31.4 |
– |
– |
|
perp. c-axis |
9.1 |
10.3 |
12.2 |
19.5 |
37.6 |
– |
– |
|
Sapphire single crystal: |
|
|
|
|
|
|
|
|
para. c-axis |
3.6 |
4.1 |
4.8 |
7.9 |
8.9 |
9.8 |
10.9 |
|
perp. c-axis |
3.3 |
2.2 |
6.6 |
7.4 |
8.3 |
9.1 |
10.0 |
|
Semiconductors: |
|
|
|
|
|
|
|
|
gallium arsenide |
1.9 |
– |
5.7 |
6.5 |
7.1 |
– |
– |
|
| |
