Specific heat capacity

6.3.2 Specific heat capacity

The specific heat is another thermal property important

normal mode of vibration.

in the processing operations of casting or heat According to the quantum theory, the mean energy treatment, since it determines the amount of heat

of a normal mode of the crystal is required in the process. Thus, the specific heat (denoted

2 h v by C Cf p , when dealing with the specific heat at constant pressure) controls the increase in temperature, dT,

where 1 2 represents the energy a vibrator will have produced by the addition of a given quantity of heat,

at the absolute zero of temperature, i.e. the zero-point dQ, to one gram of matter so that dQ D C p dT.

energy. Using the assumption made by Einstein (1907) The specific heat of a metal is due almost entirely

that all vibrations have the same frequency (i.e. all to the vibrational motion of the ions. However, a

atoms vibrate independently), the heat capacity is small part of the specific heat is due to the motion of the free electrons, which becomes important at

C v D ⊲ dE/dT⊳ v

high temperatures, especially in transition metals with

electrons in incomplete shells. The classical theory of specific heat assumes that

2 ] an atom can oscillate in any one of three directions,

and hence a crystal of N atoms can vibrate in 3N This equation is rarely written in such a form because independent normal modes, each with its characteristic

by  E D , where  E is known as the Einstein characteristic temperature. Consequently, when C v thermal energy of the metal is E D 3NkT. In solid is

frequency. Furthermore, the mean energy of each nor- mal mode will be kT, so that the total vibrational

and liquid metals, the volume changes on heating are plotted against T/ E , the specific heat curves of all very small and, consequently, it is customary to con-

pure metals coincide and the value approaches zero at sider the specific heat at constant volume. If N, the

very low temperatures and rises to the classical value number of atoms in the crystal, is equal to the number

of 3Nk D 3R ' 25.2 J/g at high temperatures. of atoms in a gram-atom (i.e. Avogadro number), the

Einstein’s formula for the specific heat is in good heat capacity per gram-atom, i.e. the atomic heat, at  agreement with experiment for T E , but is poor for low temperatures where the practical curve falls off

constant volume is given by less rapidly than that given by the Einstein relationship. dQ

dE However, the discrepancy can be accounted for, as

C v dT

D 3Nk D 24.95 J K atomic vibrations are not independent of each other.

D shown by Debye, by taking account of the fact that the

dT

In practice, of course, when the specific heat is exper- This modification to the theory gives rise to a Debye imentally determined, it is the specific heat at constant

characteristic temperature  D , which is defined by pressure, C p , which is measured, not C v , and this is

k D D D

The physical properties of materials 171

D is Debye’s maximum frequency. Figure 6.3b without a rise in temperature, so that the specific heat shows the atomic heat curves of Figure 6.3a plotted

⊲ dQ/dT⊳ at the transformation temperature is infinite. against T/ D ; in most metals for low temperatures

In some cases, known as transformations of the sec- ⊲T/ D − 1⊳ a T 3 law is obeyed, but at high temper-

ond order, the phase transition occurs over a range atures the free electrons make a contribution to the

of temperature (e.g. the order–disorder transformation atomic heat which is proportional to T and this causes

in alloys), and is associated with a specific heat peak

a rise of C above the classical value. of the form shown in Figure 6.4b. Obviously the nar- rower the temperature range T 1 T c , the sharper is

6.3.3 The specific heat curve and

the specific heat peak, and in the limit when the total

change occurs at a single temperature, i.e. T 1 D T c , the specific heat becomes infinite and equal to the latent The specific heat of a metal varies smoothly with tem-

transformations

heat of transformation. A second-order transformation perature, as shown in Figure 6.3a, provided that no

also occurs in iron (see Figure 6.4a), and in this case phase change occurs. On the other hand, if the metal

is due to a change in ferromagnetic properties with undergoes a structural transformation the specific heat

temperature.

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-

6.3.4 Free energy of transformation

formation; for example, the ˛ to , to υ and υ to liq- In Section 3.2.3.2 it was shown that any structural uid phase changes in iron shown in Figure 6.4a. At the

changes of a phase could be accounted for in terms transformation temperature the latent heat is absorbed

of the variation of free energy with temperature. The

Figure 6.3 The variation of atomic heat with temperature .

Figure 6.4 The effect of solid state transformations on the specific heat–temperature curve .

172 Modern Physical Metallurgy and Materials Engineering relative magnitude of the free energy value governs the

phase changes occur the more close-packed structure stability of any phase, and from Figure 3.9a it can be

usually exists at the low temperatures and the more seen that the free energy G at any temperature is in turn

open structures at the high temperatures. From this governed by two factors: (1) the value of G at 0 K,

viewpoint a liquid, which possesses no long-range

G 0 , and (2) the slope of the G versus T curve, i.e. the structure, has a higher entropy than any solid phase temperature-dependence of free energy. Both of these

so that ultimately all metals must melt at a sufficiently terms are influenced by the vibrational frequency, and

high temperature, i.e. when the TS term outweighs the consequently the specific heat of the atoms, as can be

H term in the free energy equation. shown mathematically. For example, if the temperature

The sequence of phase changes in such metals as of the system is raised from T to T C dT the change

titanium, zirconium, etc. is in agreement with this pre- in free energy of the system dG is

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 (˛-iron) even at low temperatures and fcc ( -iron) at

S dT high temperatures, is an exception to this rule. In this case, the stability of the bcc structure is thought to be

so that the free energy of the system at a temperature T

associated with its ferromagnetic properties. By hav- is

ing a bcc structure the interatomic distances are of the

correct value for the exchange interaction to allow the GDG 0 S dT

0 electrons to adopt parallel spins (this is a condition for magnetism). While this state is one of low entropy it is At the absolute zero of temperature, the free energy

also one of minimum internal energy, and in the lower

G 0 is equal to H 0 , and then temperature ranges this is the factor which governs the

phase stability, so that the bcc structure is preferred. GDH 0 S dT

Iron is also of interest because the bcc structure,

0 which is replaced by the fcc structure at temperatures

above 910 °

C, reappears as the υ-phase above 1400 ° C.

which if S is replaced by 0 ⊲C p /T⊳ dT becomes

This behaviour is attributed to the large electronic spe-

cific heat of iron which is a characteristic feature of GDH 0 ⊲C p /T⊳ dT dT

TT

most transition metals. Thus, the Debye characteristic

0 0 temperature of -iron is lower than that of ˛-iron and this is mainly responsible for the ˛ to transformation.

Equation (6.1) indicates that the free energy of a given However, the electronic specific heat of the ˛-phase phase decreases more rapidly with rise in tempera-

becomes greater than that of the -phase above about ture the larger its specific heat. The intersection of the

C and eventually at higher temperatures becomes free energy–temperature curves, shown in Figure 3.9a,

sufficient to bring about the return to the bcc structure therefore takes place because the low-temperature

at 1400 ° C.

phase has a smaller specific heat than the higher- temperature phase.

At low temperatures the second term in equation

6.4 Diffusion

(6.1) is relatively unimportant, and the phase that is stable is the one which has the lowest value

6.4.1 Diffusion laws

of H 0 , i.e. the most close-packed phase which is Some knowledge of diffusion is essential in associated with a strong bonding of the atoms.

understanding the behaviour of materials, particularly However, the more strongly bound the phase, the

at elevated temperatures. A few examples include higher is its elastic constant, the higher the vibrational

such commercially important processes as annealing, frequency, and consequently the smaller the specific

heat-treatment, the age-hardening of alloys, sintering, heat (see Figure 6.3a). Thus, the more weakly bound

surface-hardening, oxidation and creep. Apart from

the specialized diffusion processes, such as grain temperature, is likely to appear as the stable phase

structure, i.e. the phase with the higher H 0 at low

boundary diffusion and diffusion down dislocation at higher temperatures. This is because the second

channels, a distinction is frequently drawn between term in equation (6.1) now becomes important and G

diffusion in pure metals, homogeneous alloys and decreases more rapidly with increasing temperature,

inhomogeneous alloys. In a pure material self-diffusion for the phase with the largest value of ⊲C p /T⊳ dT.

can be observed by using radioactive tracer atoms. From Figure 6.3b it is clear that a large ⊲C p /T⊳ dT

In a homogeneous alloy diffusion of each component is associated with a low characteristic temperature

can also be measured by a tracer method, but in an and hence, with a low vibrational frequency such as

inhomogeneous alloy, diffusion can be determined by is displayed by a metal with a more open structure

chemical analysis merely from the broadening of the and small elastic strength. In general, therefore, when

interface between the two metals as a function of time.

The physical properties of materials 173

Figure 6.5 Effect of diffusion on the distribution of solute in an alloy .

Inhomogeneous alloys are common in metallurgical practice (e.g. cored solid solutions) and in such cases diffusion always occurs in such a way as to produce a macroscopic flow of solute atoms down the concentration gradient. Thus, if a bar of an alloy, along which there is a concentration gradient (Figure 6.5) is heated for a few hours at a temperature where atomic migration is fast, i.e. near the melting point, the solute

Figure 6.6 Diffusion of atoms down a concentration atoms are redistributed until the bar becomes uniform

gradient .

in composition. This occurs even though the individual atomic movements are random, simply because there

with J x the flux of diffusing atoms. Setting c 1 c are more solute atoms to move down the concentration D 2 gradient than there are to move up. This fact forms the b⊲ dc/dx⊳ this flux becomes

basis of Fick’s law of diffusion, which is

p x v v b 2 1 2 v b ⊲ dc/dx⊳

⊲ 6.3⊳ In cubic lattices, diffusion is isotropic and hence all six

D⊲ dc/dx⊳

Here the number of atoms diffusing in unit time across unit area through a unit concentration gradient

orthogonal directions are equally likely so that p x D 1 6 .

is known as the diffusivity or diffusion coefficient, 1 D .

For simple cubic structures b D a and thus

D x D D y D D z D 1 6 v a 2 D D (6.4) depends on the concentration and temperature of the

It is usually expressed as units of cm 2 s 1 or m 2 s 1 and

p alloy.

1 2 and D D 2 v a , To illustrate, we may consider the flow of atoms

whereas in fcc structures b D a/

and in bcc structures D D 24 v a 2 . in one direction x, by taking two atomic planes A

Fick’s first law only applies if a steady state exists and B of unit area separated by a distance b, as

in which the concentration at every point is invariant,

shown in Figure 6.6. If c 1 and c 2 are the concentrations

i.e. ⊲dc/dt⊳ D 0 for all x. To deal with nonstationary

flow in which the concentration at a point changes corresponding number of such atoms in the respective

of diffusing atoms in these two planes ⊲c 1 >c 2 ⊳ the

with time, we take two planes A and B, as before,

separated by unit distance and consider the rate of that any one jump in the Cx direction is p x , then

planes is n 1 D c 1 b and n 2 D c 2 b . If the probability

increase of the number of atoms ⊲dc/dt⊳ in a unit the number of jumps per unit time made by one atom

volume of the specimen; this is equal to the difference is p x

between the flux into and that out of the volume an atom leaves a site irrespective of directions. The

element. The flux across one plane is J x and across the number of diffusing atoms leaving A and arriving at

other ⊲J x C

B in unit time is ⊲p x 1 b⊳ and the number making the We thus obtain Fick’s second law of diffusion

reverse transition is ⊲p x 2 b⊳ so that the net gain of

dc dJ x

D d D x dc (6.5)

atoms at B is

1 c 2 ⊳DJ x When D is independent of concentration this reduces to

1 The conduction of heat in a still medium also follows the

dc x

same laws as diffusion.

D D x 2 (6.6)

dt

dx

174 Modern Physical Metallurgy and Materials Engineering and in three dimensions becomes

dc d dc d dc d dc or

(6.11) is the behaviour of a diffusion couple, where there

ln⊲r 1 /r 0 ⊳ 1 0 ⊳

An illustration of the use of the diffusion equations

Diffusion equations are of importance in many diverse is a sharp interface between pure metal and an alloy.

Figure 6.5 can be used for this example and as the problems and in Chapter 4 are applied to the diffusion solute moves from alloy to the pure metal the way in

of vacancies from dislocation loops and the sintering which the concentration varies is shown by the dotted

of voids.

lines. The solution to Fick’s second law is given by