Thermodynamic preamble: the context of the binding energy

Formally, the stability of a crystal structure with respect to other possible structures (whether drastically or subtlely different) is determined by theGibbs free energy, G. This is written as

G= U + P V − T S (5.1)

where U is the internal energy, P the pressure, V the volume, T the temperature, and S the entropy. There are two other important thermodynamic

quantities: theenthalpy H is given by

H= U + P V (5.2)

and theHelmholtz free energy, F , given by

F = U − T S (5.3)

The Helmholtz free energy also has the alternative symbol A. The four thermodynamic potentials are also written in differential form:

dU= T dS − P dV dH = T dS + V dP dF = −S dT − P dV dG= −S dT + V dP

The various quantities in the thermodynamic potentials are related to their differentials by standard thermodynamic relationships, called the Maxwell relations: Some of these relations will prove to be of considerable value in later discussions, which is why we briefly review them here. Note that the fact that S and V are positive quantities means that the derivatives of the free energy with respect to temperature and pressure must be negative and positive respectively. Moreover, since entropy increases on heating, there will be a downwards curvature of the free energy with respect to temperature, and the fact that the volume decreases with pressure means that dG/dP will decrease on increasing pressure as will be seen in Fig. 5.2.

The various components of the free energy reflect different aspects of the properties and behaviour of the crystal. The entropy arises from two main sources. One, which we will discuss only fleetingly in places, is due to disorder of atom positions. The second source is the vibrations of the atoms, which we will discuss in some detail in Chapters 8 and 9. The internal energy also has two main sources, namely the potential energy of interactions between atoms, and the energy of atomic vibrations. In this chapter we will focus specifically on the energies associated with interatomic interactions, taking both an empirical perspective, and a more formal view from quantum mechanics.

The concept of the free energy has considerable power. The important point about the free energy is that the equilibrium state of a material is that for

which the free energy is a minimum. Moreover, the free energy is a continuous function of external variables such as T or P , which means that it will not undergo any discontinuous change. Therefore, at a temperature and pressure at which a structure undergoes a phase transition, the free energies of both phases will be equal. This is illustrated by considering a pressure–temperature phase diagram of the form given in Fig. 5.1. This involves two phases, which

Temperature

Pressure

a

b

Fig. 5.1 Phase diagram with phase boundary in P and T with two phases, α and β.

we label α and β. If it is possible to understand the nature of the free energy, whether in general terms or with a specific theoretical model, it is possible to understand why specific crystal structures are stable over any particular range of temperature or pressure.

In Fig. 5.2 we show general representations of the curves of free energy versus temperature for both phases at a fixed pressure, and the curves of free energy versus pressure at a fixed temperature. We consider first the phase transition at a fixed pressure, which we arbitrarily set as zero. The stable phase at low temperature, denoted as α in Fig. 5.1, will have the lower internal energy, and the phase that is stable at high temperature, which is denoted as β, will have the higher entropy. Thus at some temperature Tcthe free energies of the two phases will be identical:

G_α|T=Tc,P=0= Gβ

T=Tc,P=0 ⇒ Uα− TcS_α = Uβ − TcS_β

⇒ Tc=U_α− Uβ

S_α− Sβ = U

S (5.8)

The values of U and S used in this equation should be those at the fixed pressure and T = Tc. However, if the dependence of U and S on pressure and temperature is not significant – and we note that both will change in the same sense so that the ratio U/S will vary even less – measurements or calculations of U and S at some general temperature and pressure can be used to give a reasonable estimate of Tc.

We now consider the case of a high-pressure phase transition at low temperature (which we set arbitrarily to zero). The low-pressure phase, α, will have the lower internal energy at low pressures. However, it will also have the higher volume, and at a high pressure the high-pressure phase, β, will have the lower enthalpy even though it has the higher internal energy. At some pressure,

.

Free energy

Temperature Pressure

a a

b b

Transition

temperature Transition

pressure

Fig. 5.2 Plots of crossing of free energies with variation of T or P for two phases of a material. Note the slopes that give S and V respectively.

which we denote as Pc, the free energies of the two phases will be equal:

where the values of U and V are strictly those at the transition pressure.

However, simply by knowing that the difference between the volumes of the two phases has an opposite sign to the difference between the internal energies is enough to understand that the phase transition will happen at a positive pressure. A rough measure or calculation of U and V will be enough to give an estimate of the transition pressure Pc.

Now we consider a phase diagram across both pressure and temperature, as illustrated in Fig. 5.3. For all points along the phase diagram Gα = Gβ. We Temperature represents the line where the free energies of both phases are the same. The Clausius–

Clapeyron equation is found by considering small incremental changes in T and P along the phase boundary.

take one point along the phase boundary, and envisage moving along the phase boundary by infinitesimal increments of temperature and pressure dT and dP respectively. This will give changes in the free energies of the two phases:

dGα = −SαdT + VαdP

where we have used the result that dGα = dGβ, and assumed that the entropy and volume of each phase do not change with the infinitesimal changes in T and P. This relation is a form of the well-known Clausius-Clapeyron equation, relating the ratio of the changes in entropy and volume to the slope of the phase diagram (often this relation contains the substitution L/T = S, where L is the latent heat of the phase transition). It might be expected that a phase with higher volume is likely to be less stiff than the phase with lower volume. Because of this, the phase with higher volume is likely to be somewhat more flexible, with lower-frequency vibrations, and hence with higher entropy (this statement will be properly quantified in Chapters 8 and 9). Thus we expect the ratio S/V to be positive (although some exceptions are known).

We have gone through the thermodynamic arguments to highlight how much we can understand about structural stability from qualitative reasoning. Our task now is to flesh out this qualitative understanding by considering the forces between atoms. In this chapter we will focus on the bonding contribution to the internal energy. The vibrational contributions to the internal energy and the entropy will be discussed in later chapters, but these will themselves also depend on the interactions between atoms in the crystals.