Free energy of transformation

5.3.4 Free energy of transformation

In Section 2.2.3.2 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 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, G 0 , and (2) the slope of the G–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

dG = dH − T dS − SdT =C p dT − T (C p dT /T ) − SdT = −SdT ,

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

G =G 0 −

SdT .

At the absolute zero of temperature, the free energy G 0 is equal to H 0 , and then

G =H 0 −

SdT ,

Physical properties 245

which if S is replaced by T 0 (C p / T )dT becomes T T

Equation (5.1) indicates that the free energy of a given phase decreases more rapidly with rise in temperature the larger its specific heat. The intersection of the free energy–temperature curves, shown in Figure 2.8a, 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 (5.1) is relatively unimportant, and the phase that is stable is the one which has the lowest value of H 0 , 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 5.2a). Thus, the more weakly bound structure, i.e. the phase with the higher H 0 at low temperature, is likely to appear as the stable phase at higher temperatures. This is because the second term in equation (5.1) now becomes important and G decreases more rapidly with increasing temperature, for the phase with the largest value of (C p / T )dT . From Figure 5.2b it is clear that

a large (C p / T )dT 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 TS 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 prediction and, moreover, the alkali metals, lithium and sodium, which are normally bcc at ordinary temperatures, can be transformed to fcc at sub-zero temperatures. It is interesting to note that iron, being bcc (α-iron) even at low temperatures and fcc (γ-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 having 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 temper- atures above 910 ◦

C. This behavior is attributed to the large electronic specific heat of iron, which is a characteristic feature of most transition metals. Thus, the Debye characteristic temperature of γ-iron is lower than that of α-iron and this is mainly responsible for the α to γ transformation. However, the electronic specific heat of the α-phase becomes greater than that of the γ-phase above about 300 ◦

C, reappears as the δ-phase above 1400 ◦

C and eventually at higher temperatures becomes sufficient to bring about the return to the bcc structure at 1400 ◦ C.