The physical properties of materials
6.3 Thermal properties .1 Thermal expansion
The physical properties of materials 169
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Figure 6.1 Strength tr, plotted against density, p (yield strength for metals and polymers, compressive strength for ceramics, tear strength for elastomers and tensile strength for composites). The guide lines of constant a/p, cr 2/3/p and tr 1/2/p are used in minimum weight, yield-limited, design (Ashby, 1989, pp. 1273-93, with permission of Elsevier Science Ltd.).
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Figure 6.2 Variation in potential energy with interatomic distance.
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 or, given by t~ = ( 1 / l ) ( d l / d T ) , 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 c~ 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 ct versus 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.
170 Modern Physical Metallurgy and Materials Engineering 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.
6.3.2 Specific heat capacity
The specific h e a t is another thermal property important in the processing operations of casting or heat treatment, since it determines the amount of heat required in the process. Thus, the specific heat (denoted by Cp, when dealing with the specific heat at constant pressure) controls the increase in temperature, dT, produced by the addition of a given quantity of heat, dQ, to one gram of matter so that dQ = CpdT.
The specific heat of a metal is due almost entirely to the vibrational motion of the ions. However, a small part of the specific heat is due to the motion of the free electrons, which becomes important at high temperatures, especially in transition metals with electrons in incomplete shells.
The classical theory of specific heat assumes that an atom can oscillate in any one of three directions, and hence a crystal of N atoms can vibrate in 3N independent normal modes, each with its characteristic frequency. Furthermore, the mean energy of each nor- mal mode will be kT, so that the total vibrational thermal energy of the metal is E = 3 N k T . In solid and liquid metals, the volume changes on heating are very small and, consequently, it is customary to con- sider the specific heat at constant volume. If N, the number of atoms in the crystal, is equal to the number of atoms in a gram-atom (i.e. Avogadro number), the heat capacity per gram-atom, i.e. the atomic heat, at constant volume is given by
C , , ( d Q ' ~ _ - - - d E _ - - 3 N k = 2 4 . 9 5 J K -i
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v dTIn practice, of course, when the specific heat is exper- imentally determined, it is the specific heat at constant pressure, C p, which is measured, not C,,, and this is
given by
d E + P d V ) _ dH
C p d T p d T
where H - E + P V is known as the heat content or enthalpy, C p is greater than C,, by a few per cent because some work is done against interatomic forces when the crystal expands, and it can be shown that
Cp - C,, = 9 o t 2 V T / f l
where a is the coefficient of linear thermal expansion, V is the volume per gram-atom and fl is the compressibility.
Dulong and Petit were the first to point out that the specific heat of most materials, when determined at sufficiently high temperatures and corrected to apply to constant volume, is approximately equal to 3R, where R is the gas constant. However, deviations from the 'classical' value of the atomic heat occur at low temperatures, as shown in Figure 6.3a. This deviation is readily accounted for by the quantum theory, since the vibrational energy must then be quantized in multiples of hv, where h is Planck's constant and v is the characteristic frequency of the normal mode of vibration.
According to the quantum theory, the mean energy of a normal mode of the crystal is
I h v + { h v / e x p ( h v / k T ) - 1 } E ( v ) = 5
where i h v represents the energy a vibrator will have !
at the absolute zero of temperature, i.e. the zero-point energy. Using the assumption made by Einstein (1907) that all vibrations have the same frequency (i.e. all atoms vibrate independently), the heat capacity is
C.,, = ( d E / d T ) , ,
= 3 N k ( h v / k T ) 2
[exp ( h v / k T ) / { e x p ( h v / k T ) - 112]
This equation is rarely written in such a form because most materials have different values of v. It is more usual to express v as an equivalent temperature defined by | = h v / k , where | is known as the Einstein characteristic temperature. Consequently, when C,, is plotted against T / O E , the specific heat curves of all pure metals coincide and the value approaches zero at very low temperatures and rises to the classical value of 3Nk = 3R _~ 25.2 J/g at high temperatures.
Einstein's formula for the specific heat is in good agreement with experiment for T~>| but is poor for low temperatures where the practical curve falls off less rapidly than that given by the Einstein relationship.
However, the discrepancy can be accounted for, as shown by Debye, by taking account of the fact that the atomic vibrations are not independent of each other.
This modification to the theory gives rise to a Debye characteristic temperature | which is defined by
kOD = hVD
where VD is Debye's maximum frequency. Figure 6.3b shows the atomic heat curves of Figure 6.3a plotted against T/| in most metals for low temperatures (T/| << 1) a T 3 law is obeyed, but at high temper- atures the free electrons make a contribution to the atomic heat which is proportional to T and this causes a rise of C above the classical value.
6.3.3 T h e specific h e a t c u r v e a n d transformations
The specific heat of a metal varies smoothly with tem- perature, as shown in Figure 6.3a, provided that no phase change occurs. On the other hand, if the metal undergoes a structural transformation the specific heat curve exhibits a discontinuity, as shown in Figure 6.4.
If the phase change occurs at a fixed temperature, the metal undergoes what is known as a first-order trans- formation; for example, the a to y, y to ~ and ~ to liq- uid phase changes in iron shown in Figure 6.4a. At the transformation temperature the latent heat is absorbed
The physical properties of materials 171 without a rise in temperature, so that the specific heat
(dQ/dT)
at the transformation temperature is infinite.In some cases, known as transformations of the sec- ond order, the phase transition occurs over a range of temperature (e.g. the order-disorder transformation in alloys), and is associated with a specific heat peak of the form shown in Figure 6.4b. Obviously the nar- rower the temperature range
T ~ - Tr
the sharper is the specific heat peak, and in the limit when the total change occurs at a single temperature, i.e. T~ -- Tr the specific heat becomes infinite and equal to the latent heat of transformation. A second-order transformation also occurs in iron (see Figure 6.4a), and in this case is due to a change in ferromagnetic properties with temperature.6.3.4 Free energy of transformation
In Section 3.2.3.2 it was shown that any structural changes of a phase could be accounted for in terms of the variation of free energy with temperature. The
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172 Modern Physical Metallurgy and Materials Engineering relative magnitude of the free energy value governs the stability of any phase, and from Figure 3.9a it can be seen that the free energy G at any temperature is in turn governed by two factors: (1) the value of G at 0 K, Go, and (2) the slope of the G versus T curve, i.e. the temperature-dependence of free energy. Both of these terms are influenced by the vibrational frequency, and consequently the specific heat of the atoms, as can be shown mathematically. For example, if the temperature of the system is raised from T to T + dT the change in free energy of the system dG is
d G = d H - T d S - S d T
= C p d T - T ( C p d T / T ) - S d T
= - S d T
so that the free energy of the system at a temperature T is
f0 T
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At the absolute zero of temperature, the free energy Go is equal to Ho, and then
f0 T
G = H o - S d T
which if S is replaced by f o r ( C p / T ) d T becomes
E/o J
G = H o - ( C p / T ) d T d T (6.1)
Equation (6.1) indicates that the free energy of a given phase decreases more rapidly with rise in tempera- ture the larger its specific heat. The intersection of the free energy-temperature curves, shown in Figure 3.9a, therefore takes place because the low-temperature phase has a smaller specific heat than the higher- temperature phase.
At low temperatures the second term in equation (6.1) is relatively unimportant, and the phase that is stable is the one which has the lowest value of H0, i.e. the most close-packed phase which is associated with a strong bonding of the atoms.
However, the more strongly bound the phase, the higher is its elastic constant, the higher the vibrational frequency, and consequently the smaller the specific heat (see Figure 6.3a). Thus, the more weakly bound structure, i.e. the phase with the higher H0 at low temperature, is likely to appear as the stable phase at higher temperatures. This is because the second term in equation (6.1) now becomes important and G decreases more rapidly with increasing temperature, for the phase with the largest value of f ( C l , / T ) d T . From Figure 6.3b it is clear that a large f ( C p / T ) d T is associated with a low characteristic temperature and hence, with a low vibrational frequency such as is displayed by a metal with a more open structure and small elastic strength. In general, therefore, when
phase changes occur the more close-packed structure usually exists at the low temperatures and the more open structures at the high temperatures. From this viewpoint a liquid, which possesses no long-range structure, has a higher entropy than any solid phase so that ultimately all metals must melt at a sufficiently high temperature, i.e. when the T S term outweighs the H term in the free energy equation.
The sequence of phase changes in such metals as titanium, zirconium, etc. is in agreement with this pre- diction and, moreover, the alkali metals, lithium and sodium, which are normally bcc at ordinary temper- atures, can be transformed to fcc at sub-zero temper- atures. It is interesting to note that iron, being bcc (c~-iron) even at low temperatures and fcc (y-iron) at high temperatures, is an exception to this rule. In this case, the stability of the bcc structure is thought to be associated with its ferromagnetic properties. By hav- ing a bcc structure the interatomic distances are of the correct value for the exchange interaction to allow the electrons to adopt parallel spins (this is a condition for magnetism). While this state is one of low entropy it is also one of minimum internal energy, and in the lower temperature ranges this is the factor which governs the phase stability, so that the bcc structure is preferred.
Iron is also of interest because the bcc structure, which is replaced by the fcc structure at temperatures above 910~ reappears as the 8-phase above 1400~
This behaviour is attributed to the large electronic spe- cific heat of iron which is a characteristic feature of most transition metals. Thus, the Debye characteristic temperature of y-iron is lower than that of a-iron and this is mainly responsible for the c~ to y transformation.
However, the electronic specific heat of the c~-phase becomes greater than that of the y-phase above about 300~ and eventually at higher temperatures becomes sufficient to bring about the return to the bcc structure at 1400~