6. Phase Transitions and Phase Equilibria 57 6 Phase Transitions and Phase Equilibria Phase transitions are part of everyday experience (e.g. the formation of water clouds) and very important in atmospheric dynamics. But they are also very important inside planets... they’re everywhere, as melting to provide volcanism; abrupt changes in density and seismic properties in mantle (the 410 and 660km discontinuity in Earth’s mantle), the formation of Earth’s inner core, etc. They have a profound influence on planet evolution as well as affecting the static structure (density as a function of depth). We’ve already seen one example of a (hypothesized) phase transition in hydrogen; below we examine phase transitions in more detail. 6.1 Why do Phase Transitions Happen? At a given T, P; a thermodynamic system will seek to minimize the Gibbs energy: G = E + PV - TS. As one increases pressure, this will clearly favor phases of lower volume (more compact packing of constituents). Or as one increases temperature, it will clearly favor phases of higher entropy (more disorder). Figure 6.1 6.2 The Clausius-Clapeyron Relation Consider a single component system. (By this we mean a system that lacks compositional degrees of freedom. Systems that have such degrees of freedom include alloys, salt water, etc. and do not satisfy the simplest form of Clausius -Clapeyron). G = E + PV - TS ⇒ dG = dE + PdV + VdP - TdS - S dT But dE = -PdV + TdS [first law of thermodynamics] ⇒ dG = VdP - S dT; (∂G/∂P)T=V and (∂G/∂T)P= -S. Now consider two phases (1 and 2) that are in equilibrium at P, T and still in equilibrium at P + dP, T+dT. G1 = G2 and G1 +dG1 = G2 + dG2 ⇒ dG1 = dG2 ⇒ V1 dP -S1dT = V2 dP- S2 dT ⇒ (dP/dT)ph = (S2 - S1 )/ (V2 - V1 ) (6.1) or, equivalently,6. Phase Transitions and Phase Equilibria 58 (dnP/ dnT) = TΔS/PΔV (6.2) Note that the RHS is a ratio of energies. The dimensionless derivative on the LHS will be large if the volume change is small and/or the entropy change is large; and small if the volume change is large and/or the entropy change is small. This is a very useful thing to remember because it will enable you to understand phase diagrams better. This equation will tell you things such as: The pressure dependence of the condensation temperature of a gas (solar nebular thermodynamics), the pressure dependence of the melting point of a solid (e.g., why ice I has a melting point that decreases with pressure), the topography on phase boundaries in Earth’s mantle associated with convective upwellings and downwellings (purportedly detected seismologically), etc. 6.3 Multicomponent systems This is a big topic, but the main points are these: When two multicomponent phases are in contact and in equilibrium, the chemical potential (specific Gibbs energy) of each species must be equal. The two phases need not have the same composition and in general will not have the same composition. This is an example of the Gibbs phase rule. (See Appendix to this chapter). For example, multimineralic assemblages do not have a melting point (in the sense of a single T at which you transition from completely solid to completely liquid)..... they have a solidus (a T at which melting begins) and a liquidus (at which the material is completely molten). Simple melting (where solidus = liquidus = “melting point”) is the exception rather than the rule in planetary systems (including icy satellites). 6.4 Kinds of Phase Transitions 6.4.1 Electronic Reorganizations of the electron distribution (nature of the electronic states) can occur. This could be an insulator metal transition or a change in spin states, for example. The major example in our solar system could be Molecular hydrogen → Metallic hydrogen and probably involves a substantial volume change (e.g. 10%) but a modest entropy change (e.g. Boltzmann’s constant per proton) implying that the transition pressure is only weakly T-dependent according to the Clausius-Clapeyron equation. As discussed previously, the hydrogen case is in reality probably more complicated, i.e. Molecular hydrogen → Metallic molecular hydrogen → Monatomic metal 6.4.2 Structural (Solid→ Solid) These generally arise because there exists a lower volume structure that can (by lowering the Gibbs energy) exist at higher P. All major mantle transitions in terrestrial planets are driven in this way. Examples: Olivine → spinel6. Phase Transitions and Phase Equilibria 59 Spinel or post-spinel → Magnesiowustite + Perovskite although the latter also involves disproportionation (breakdown into new mineral phases that have a different chemical formula. The 660km discontinuity in Earth corresponds to spinel →perovskite and magnesiowustite. The phase boundary is shown below as a composition vs. pressure slice of the phase diagram. Notice (by the Gibbs phase rule) that there must be two lines separating the spinel and pv+mw field. But in this case they’re very close together, which means that the phase transition is sharply defined. Figure 6.2 Perovskite is the most common mineral in Earth. However, it may transform at still higher pressure:6. Phase Transitions and Phase Equilibria 60 Figure 6.36. Phase Transitions and Phase Equilibria 61 This figure is taken from Oganov et al, Nature 411,934-937(2004). The top part shows the crossover in enthalpy (same as Gibbs energy at low T) that favors the new phase above a pressure of ~ 1 megabar, according to two theoretical calculations. The bottom half shows the data (X-ray crystallographic determination of structure) and the inferred phase boundary in T-P space. Water ice undergoes many structural phase transitions as a function of pressure, even in the pressure range encountered in icy satellites. Figure 6.4 Ice I has a density of ~0.92g/cm3 (hence the negative slope of the melting curve.) Ice III has a density of ~1.16g/cm3, ice V has a density of ~1.25, Ice VI a density of ~1.31 and ice VII a density of 1.49. Notice the remarkable differences over a range of only a few tens of kilobars (i.e., much more compressible than rock). At still higher pressures, water symmetrizes (i.e. hydrogen bond and covalent bonds have same length and strength). This is ice X (way beyond the pressures of icy satellites). At that stage you should no longer think of “separate” water molecules. Here is a recent phase diagram for ice at higher pressure [Lin JF, Gregoryanz