Thermal expansion

5.3.1 Thermal expansion

If we consider a crystal at absolute zero temperature, the ions sit in a potential well of depth E r 0 below the energy of a free atom (Figure 1.3). The effect of raising the temperature of the crystal is to cause the ions to oscillate in this asymmetrical potential well about their mean positions. As a consequence, this motion causes the energy of the system to rise, increasing with increasing amplitude of vibration. The increasing amplitude of vibration also causes an expansion of the crystal, since as

a result of the sharp rise in energy below r 0 the ions as they vibrate to and fro do not approach much

Physical properties 241 closer than the equilibrium separation, r 0 , but separate more widely when moving apart. When the

distance r is such that the atoms are no longer interacting, the material is transformed to the gaseous phase, and the energy to bring this about is the energy of evaporation.

The change in dimensions with temperature is usually expressed in terms of the linear coefficient of expansion α, given by α = (1/l)(dl/dT ), where l is the original length of the specimen and T is the absolute temperature. Because of the anisotropic nature of crystals, the value of α usually varies with the direction of measurement and even in a particular crystallographic direction the dimensional change with temperature may not always be uniform.

Phase changes in the solid state are usually studied by dilatometry. The change in dimensions of a specimen can be transmitted to a sensitive dial gauge or electrical transducer by means of

a fused silica rod. When a phase transformation takes place, because the new phase usually occupies

a different volume to the old phase, discontinuities are observed in the coefficient of thermal expansion α –T curve. Some of the ‘nuclear metals’ which exist in many allotropic forms, such as uranium and plutonium, show a negative coefficient along one of the crystallographic axes in certain of their allotropic modifications.

The change in volume with temperature is important in many metallurgical operations such as casting, welding and heat treatment. Of particular importance is the volume change associated with the melting or, alternatively, the freezing phenomenon, since this is responsible for many of the defects, both of a macroscopic and microscopic size, which exist in crystals. Most metals increase their volume by about 3% on melting, although those metals which have crystal structures of lower coordination, such as bismuth, antimony or gallium, contract on melting. This volume change is quite small, and while the liquid structure is more open than the solid structure, it is clear that the liquid state resembles the solid state more closely than it does the gaseous phase. For the simple metals the latent heat of melting, which is merely the work done in separating the atoms from the close-packed structure of the solid to the more open liquid structure, is only about one-thirtieth of the latent heat of evaporation, while the electrical and thermal conductivities are reduced only to three-quarters to one-half of the solid-state values.

Worked example

In copper, what percentage of the volume change which occurs as the specimen is heated from room temperature to its melting point is due to the increased vacancy concentration, assuming that the vacancy concentration at the melting point (1083 ◦

∼10 C) is −4 ? (Linear thermal expansion coefficient of copper α is 16.5 × 10 −6 K −1 .)

Solution

At the melting point (1083 ◦

C) vacancy concentration = 10 −4 , i.e. one vacancy every 10 4 atom sites.

= 3 × 16.5 × 10 −6 × (1083 − rT ) = 49.5 × 10 −6 × 1060 = 5.25 × 10 −2 .

Fractional change due to vacancies =

242 Physical Metallurgy and Advanced Materials