Temperature and pressure induced polymorphism 1. Introduction

With polymorphism we mean that an element or a compound can ex-ist in several different crystalline forms. (Some authors use the word allotropy in the case of elements. Others use the word polytypism.) We will be interested in the cases when a solid transforms from one thermodynamically stable crystal structure to another, with varying temperature or pressure. A typical example of a temperature-induced transformation is that of tin at 286 K, from the low temperature semi-conducting phase (gray tin) to the metallic /3-phase (white tin). A typical

Point of transition^a

Examples of temperature or pressure-induced polymorphism Material

aData from Young (1991).

pressure induced transformation is that of alkali halides, where at least eight of the twelve compounds which crystallise in the B1 (i.e. NaCl-type) structure at zero pressure transform to the B2 (i.e. CsCl-NaCl-type) structure at pressures below 100 GPa (Kim and Gordon 1974). Table 7.1 gives some additional examples. Data from Young (1991) and the JANAF thermochemical tables (1985).

6.2. Temperature-induced transformations

That structure is the most stable, which has the lowest Gibbs en-ergy G = U — TS + pV'. We first discuss a temperature-induced transformation and then one induced by pressure. Let us describe the temperature-dependent part of G by harmonic lattice vibrations, and specialise to p « 0 (i.e. normal atmospheric pressure) and high temper-atures (T > 9D). Then the difference G2 — G\ between two phases, 1

Temperature and pressure induced polymorphism 131

600

Fig. 7.4. The Gibbs energy G(T) for a-Sn, £-Sn and liquid Sn. The inset shows the heat capacity, normalised as Cp/3R. Data from Dinsdale (1991) and Hultgren et al.

(1973a).

If the difference Aco = o;i(0) - co2(0) is small compared with &>i(0) [^ &>(0)], we get

Aco AH

(o(0) 3NkBTt' (7.70)

Example: gray and white tin. Figure 7.4 shows G{T) for gray, white and liquid tin, as derived from experiments at ambient pressure. Gray tin (a-Sn) is a semiconductor with the diamond-type lattice structure and white tin 06-Sn) is a metal with tetragonal structure. The electronic structure, and hence the bonding, is radically different in the two phases.

This is reflected in the entropy Debye temperatures evaluated at T = T_t = 286 K, where eg (grey )/flg (white) % 1.3.

6.3. Pressure-induced transformations

To illustrate the pressure-induced transformation we consider r = 0K.

The relative lattice stability is determined by the enthalpy H = U+pV.

At the transformation pressure pu

UiiVx) + ptVi = U2(V2) + PtV2. Thus, a transformation occurs at

U2(V2) - Ui(V{)

Pt =

Vi-V₂

(7.71)

(7.72)

30

_ 20 o

3 10

0

"^w 9 10 11 12

v (A³)

Fig. 7.5. The energy U(T) for bcc and hep Fe, as a function of the volume per atom.

From ab initio electron structure calculations (Ekman 1998, unpublished).

The pressure pt has a geometrical interpretation as the slope of the common tangent to the U\(V) and U₂(V) versus V curves (fig. 7.5).

Because our approach only compares the energy difference between two equilibrium structures, we cannot say anything about kinetic as-pects, such as nucleation and growth of a phase (Yamada et al. 1984), but see the discussion of dynamic instabilities in §6.4 below and in Chapter 6 (§13).

Example: pressure-induced transformation in Fe. At ambient temper-ature and pressure, iron has the bcc structure. Under pressure, it trans-forms to an hep structure. Figure 7.5 shows the energy U at T = 0 K for bcc and hep Fe, from ab initio electron structure calculations. The tran-sition pressure pt as given by eq. (7.72) corresponds to the slope of the common tangent to U(V; bcc) and U(V; hep) in fig. 7.5. The calculation yields pt « 10 GPa, in good agreement with experiments, which are uncertain due to hysteresis in p_t. The large difference in atomic volume of Fe between the bcc and hep lattices is due to magnetic effects. There is no magnetic moment in hep Fe at the volumes considered here. (In a model with atoms represented by rigid spheres, the atomic volume of the bcc phase would be 9% larger than for the hep phase, see Chapter 19, §2.)

Example: temperature- and pressure-induced polymorphism in TIL Thallium iodide has some interesting properties (Samara et al. 1967). At ambient pressure and T_t = 429 K, it transforms from a low temperature orthorhombic structure (1) to a more dense cubic CsCl-type structure

J I I I L

Temperature and pressure induced polymorphism 133

(2). The same structural transformation takes place if, at ambient tem-perature, the pressure is increased to p_t = 0.29 GPa (2.9 kbar). The heat of transformation at 0.1 MPa (1 bar) is AH = 1230 ± 160 J/mol. The molar volume is lower by 3.3% in the CsCl structure; V\ — V2 = 1.5 cm³/mol. We shall analyse these transformations in a simple model.

With N = 2NA and T = 429 K, eq. (7.69) gives o;2(0)M(0) = 0.94.

In the pressure-induced transformation, temperature effects cannot be neglected. If the high temperature form of G2 — G\ in eq. (7.68) re-places U2 — U\ in eq. (7.72), and the volume dependence of AH and CL>2(0)/CO\(0) is ignored, eq. (7.72) yields pt = 0.26 GPa, to be compared with the measured p_t = 0.29 GPa.

6.4. Approaching a lattice instability

In the solid-solid phase transformations discussed above, each phase also exists as a metastable phase in a certain temperature or pressure interval beyond the point of equal Gibbs energies. The actual transfor-mation takes place through nucleation and growth of the new phase. We shall now comment on the case when a phase becomes dynamically un-stable due to a change in temperature, pressure or chemical composition (cf. Chapter 4, §3, Chapter 6, §13). Then the concept of a vibrational entropy has no physical meaning, and the Gibbs (or Helmholtz) energy is undefined.

For example, consider the behaviour of G when a change in the pressure p takes the solid from a dynamically stable state to one that has an unstable phonon mode. In a strict Debye model, and at high temperatures, the Gibbs energy can be written (eq. (D.8)),

G % U(V) + pV - 3NkBT[l/3 + ln(77#D)]. (7.73) If we let #D approach 0 in eq. (7.73), G would diverge towards — 00, and

hence this phase would be stabilised relative to other phases just before it becomes unstable at a pressure p_c. However, eq. (7.73) assumes that all phonon frequencies tend to zero at the same rate, and become 0 at pc. In a real material, there will be one particular mode (q, s) that first reaches co(q, s) — 0. The singularity therefore is very weak, and G does not diverge when p —• p_c (Fernandez Guillermet et al. 1995). Hence, on approaching p_c, there is only a small precursor effect in the phase di-agram. Figure 7.6 shows this for Mg, which is stable in the bcc structure

2000

g

I- 1000

0 0 10 20 30 40 50 60 P (GPa)

Fig. 7.6. The temperature-pressure phase diagram of magnesium, as obtained through ab initio electron structure calculations. After Moriarty and Althoff (1995).

10

5 5

CO <>

3 0

u q 0.5

Fig. 7.7. At ambient pressure (volume VQ per atom) the fee lattice of tungsten is dy-namically unstable under shear, but the phonons are stabilised with increasing pressure, as shown here for phonons in the [100] direction when V = 0.44VQ- The figure gives -\co(q, s)\ when co²(q, s) < 0. For each V, the upper curve refers to longitudinal and the lower curve to transverse modes. After Einarsdotter et al. (1997).

at high p but with a transverse [110] mode that becomes unstable at low pressures (Moriarty and Althoff 1995). Analogously, the Pt-W phase diagram does not show any feature revealing that the fee lattice becomes unstable as one goes from Pt towards pure W (Fernandez Guillermet et al. 1995). See also Craievich and Sanchez (1997) for calculations on Ni-Cr alloys, showing that the elastic constant C = (cn — ci2)/2 is

l r

unstable J I i_

Temperature and pressure induced polymorphism 135

negative for fee Cr and for bec Ni (pure Cr has the bee and pure Ni the fee lattice structure). Strong anharmonieity in the soft modes will modify the arguments above, but not change the essential point about the singularity in G. Finally, it should be remarked that the actual transfor-mation may take place before the point of instability, and be of the first-order type discussed in §§6.2 and 6.3 above. For instance, the bec phase of Ti and Zr appears to be dynamically unstable at low temperatures, but the phonons are well defined at the bec «> hep transition (see fig.

6.17).

Example: phonon instability in fee tungsten. Tungsten at ambient con-ditions has the bec lattice structure, and the fee structure is dynamically unstable (Einarsdotter et al. 1997). Under compression, the phonon modes of the fee lattice are gradually stabilised. Figure 7.7, based on ab initio electron structure calculations, shows this for phonons in the [100]

direction. Modes in some other directions are also strongly affected.