Douce A.P. Thermodynamics of the Earth and Planets

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468 Thermodynamics of planetary volatiles

Note that I wrote two versions of the oxygen equation (ϕ

). Because n

is vanishingly

small it is permissible to ignore its contribution to the function ϕ

, but not to the derivative

∂ϕ

/∂n

=2, which enters into the function ∂Z/∂n

. This will become clear in a second.

The Gibbs free energy of the system fluid + graphite, G

system

, is made up of three contri-

butions: G

fluid

as given by (9.91), G

and G

graphite

. Even if molecular oxygen is present in

vanishingly small quantities there is a finite chemical potential of oxygen that contributes

to the Gibbs free energy of the system and that must therefore be accounted for in the

minimization calculation. We write the contribution of O

to the total Gibbs free energy of

the system as follows:

0,O

1,T

RT lnfO

. (9.96)

Because we are considering a graphite saturated system we must also have an equation for

the Gibbs free energy of graphite. As this is a pure phase it is simply:

graphite

0,graphite

1,T

graphite



graphite

dP . (9.97)

The function Z in this case is as follows:

Z = G

fluid

graphite



j=1

(9.98)

with G

f luid

given by (9.91). This function has seven partial derivatives, all of which must

vanish at the minimum. Five of the partial derivatives are of the form of equation (9.94),

one each for CO

, CO, CH

and H

O. As an example, the CO

equation is:

∂Z

∂n

0,CO

1,T

+1 +ln





+ln





−

+λ

+2λ

=0. (9.99)

In this equation n

= n

+ n

,asn

can be ignored in this

context. The other four partial derivatives, for CO, CH

and H

O, are analogous, and

you should write them out yourself. The partial derivative relative to n

is:

∂Z

∂n

0,O

1,T

+lnf O

+2λ

=0. (9.100)

The variable n

disappears, but we now see why it is necessary to leave it in ϕ

when

forming equation (9.98) and calculating the derivative ∂ϕ

/∂n

: it generates the coefficient

of the Lagrange multiplier. Finally, the graphite partial derivative is:

∂Z

∂n

graphite

0,graphite

1,T



graphite

+λ

=0. (9.101)

469 9.6 Speciation in multicomponent volatile phases

Equations (9.95), (9.99) and its equivalents, (9.100) and (9.101) constitute a system

of 10 non-linear equations in the 10 unknowns: n

, n

graphite

ln(fO

), λ

, λ

and λ

. The input parameters are P, T and the three bulk gram-atom num-

bers: N

, N

and N

. As in the chemical equilibrium method, the solution is numerical

and iterative. Recalling that the fugacity coefficients are functions of fluid composition,

we must initially calculate the standard state fugacity coefficients, use these to obtain an

initial solution set, and then use the fluid composition from this solution set to calculate

fugacity coefficients in the mixture, iterating until consecutive solutions converge within a

desired interval. Implementation in Maple is straightforward and is discussed in Software

Box 9.4.

Software Box 9.4 Calculation of fluid speciation by Gibbs free energy minimization

The Maple worksheet minG_fluid_species.mw contains a procedure that

calculates fluid speciation in a C–O–H fluid saturated in graphite as a func-

tion of bulk O/H ratio, at constant temperature and pressure. The proce-

dure, COH_graphite_saturation, implements the solution described in

general terms in Section 9.6.2 , and in particular solves the problem

described in Worked Example 9.8. The procedure is invoked with the fol-

lowing parameters, in this order: (pressure in bar, temperature in

centigrade, name of output file). It calculates fluid speciation, oxy-

gen fugacity and fluid composition on the graphite saturation boundary, along the

O–H join. The content and organization of the output file are described in the Maple

listing. Thermodynamic properties of the gas species are entered in the spreadsheet

RefStateData, with a Shomate-type heat capacity equation and volumetric prop-

erties for solid phases from Holland and Powell (1998). The properties used in this

example can be imported in tab-delimited format from the file agaspeciesdata,or

they can be copied from a spreadsheet.

Given that we are explicitly studying speciation in graphite-saturated fluids, there is

an additional step that we must take, which is to determine the bulk composition of the

fluid at which graphite saturation takes place. The graphite saturation boundary defines the

boundary of the bulk composition region within which the calculations are valid. For bulk

compositions outside of this region the fluid is not saturated in graphite and equation (9.97),

and hence (9.98) and (9.101), are no longer valid.

Let us call the ratio N

, that we will use as our compositional variable, N

=z.

As long as the system is saturated in graphite, for each value of P , T and z there is one and

only one equilibrium fluid composition, because the thermodynamic state of the system

is fully determined. This means that species mol fractions, and hence the ratios among

them, are fixed and independent of the value of N

. Along the graphite saturation boundary

graphite

vanishes, so that we can write the following two compositional equations valid

along the boundary:

=4n

+2n

(9.102)

470 Thermodynamics of planetary volatiles

Using the fact that species mol fractions are fixed at graphite saturation for given P ,T and

z we can define the following four constant ratios:

, a

, (9.103)

which, substituting in (9.102), yield the ratio N

along the graphite saturation boundary:

+2a

(9.104)

For any P –T combination we can choose values of z = N

from z → 0toz →∞,

solve for the fluid speciation at each z using any value of N

large enough to ensure that

the system is saturated in graphite (which we can easily verify because the calculation

must yield n

graphite

> 0), and then use (9.103) and (9.104) to calculate N

along the

graphite saturation boundary. The combination of N

and N

yields a unique point

on the graphite saturation boundary. The tedious algebra is easily incorporated in the Maple

procedure that calculates fluid speciation (Software Box 9.4).

An example of the results obtained by this procedure is shown in Fig. 9.16. The ternary

diagram shows three graphite saturation boundaries, calculated at the same temperature,

1200

◦

C, and pressures of 1, 5 and 10 kbar. These conditions may bracket core form-

ing conditions in 100–1000 km planetesimals. Fluids above the boundary are saturated in

graphite. Bear in mind that the results depend on the Redlich–Kwong EOS for calcula-

tion of the fugacity coefficients, so the 10 kbar values are unlikely to be very accurate,

and the same may be true of the 5 kbar results. The qualitative trend is however cor-

rect: increasing pressure expands the graphite stability field, as we should expect from

the fact that graphite is the low volume phase relative to fluid. The shaded region in

the diagram, between the CO

–H

O join and the oxygen vertex, represents a range of

bulk compositions that cannot be modeled with the equations given above, because in this

region oxygen mol number is never negligible (you should explain to yourself why this is

the case).

At a given P and T fluids with a constant O/H ratio anywhere within the graphite saturated

region (i.e. anywhere along a tie line joining the graphite saturation boundary and the carbon

vertex, such as the one shown in the diagram) have the same composition, and therefore

also the same oxygen fugacity. Variation of the oxygen fugacity of graphite saturated fluids

with O/H ratio is shown in the bottom diagram of Fig. 9.16. Also shown in the plot are

oxygen fugacities at the QFM and QFI buffers (the bands correspond to the ranges in f (O

)

between 1 and 10 kbar). We see that f (O

) of graphite-saturated C–O–H fluids is some

3–4 orders of magnitude lower than at the QFM buffer, which is generally consistent with

separation of metallic Fe (Exercise 9.14).

Fluid speciation as a function of O/H ratio is shown in Fig. 9.17, which reveals some

interesting trends. At low pressure (1 kbar) the predominant species is CO over a wide range

of O/H ratios, and CH

for fluids very rich in H. Except over a very narrow interval, H

Ois

everywhere subordinate to CO and CH

. With increasing pressure, however, H

O becomes

the chief species over a progressively wider compositional range. Methane is always the

dominant species for H-rich fluids, but carbon monoxide is replaced by carbon dioxide

471 9.6 Speciation in multicomponent volatile phases

0.2 0.4 0.6 0.8

–14

–12

–10

–8

–6

Graphite saturation

at 1200°C

1 kbar

5 kbar

10 kbar

QFM

QFI

Graphite saturation at 1 200°C

log f(O

)bar

Fig. 9.16 Graphite saturation conditions for a C–O–H ﬂuid at 1200

◦

C, with ﬂuid properties calculated with the RK EOS. Top:

graphite saturation boundaries at 1, 5 and 10 kbar pressure. Increasing pressure expands the graphite stability ﬁeld,

as expected from fundamental thermodynamic relationships. Graphite-saturated ﬂuids anywhere along a join

between the graphite saturation boundary and the C vertex, such as the one shown, have the same species

distribution, which is shown in Fig. 9.17. The model discussed in the text breaks down in the shaded region, where

molecular oxygen concentrations are not negligible. Bottom: oxygen fugacity of graphite-saturated ﬂuids as a

function of bulk O/H ratio. Fugacities are in every case several orders of magnitude lower than along the QFM buffer,

making crystallization of metallic Fe possible.

as the dominant species in hydrogen-poor fluids at high pressure. These results suggest

that retention of volatile components during differentiation of planetesimals smaller than

a certain radius may be made very difficult, not only by the low gravitational attraction

but also by the fact that H

O, which could combine with silicates to form hydrous mineral

phases, is not an abundant species in the fluid phase (hence its chemical potential will be

relatively low). Retention of volatiles in the deep interior of planetesimals by formation of

hydrous phases becomes easier with increasing body size, as core formation will take place

at higher pressures.

472 Thermodynamics of planetary volatiles

0.2 0.4 0.6 0.8

–1

–2

–3

–1

–2

–3

–1

–2

–3

0.2 0.6 0.8

0.2

0.4

0.80.4 0.6

Graphite saturation at 1200°C - 10 kbar

Graphite saturation at 1200°C - 5 kbar

Graphite saturation at 1200°C - 1 kbar

Fig. 9.17 Species distribution in C–O–H ﬂuids saturated in graphite at 1200

◦

C and 1, 5 and 10 kbar, as a function of bulk O/H

ratio. Each horizontal coordinate value corresponds to one tie-line such as the one in the ternary diagram of Fig. 9.16.

Carbon monoxide is an important species at 1 kbar, but is replaced by carbon dioxide with increasing pressure. H

Ois

not an abundant species at low pressure, which complicates volatile retention during core formation in small

planetesimals.

473 9.7 Fluids at the conditions of giant planet interiors

9.7 Fluids at the conditions of giant planet interiors

The equations of state that we discussed to this point are generally not applicable to fluids

at the conditions that exist in the interiors of the giant planets (Section 1.15). Fluids at

particle densities of ∼10

−3

, even before they become degenerate (Fig. 1.16), become

as incompressible as solids. When electron degeneracy sets in the behavior can only be

described in terms of quantum mechanical effects. Explicit EOS often do not exist, and

are substituted by tables of thermodynamic properties listed as a function of pressure and

temperature. Thermodynamic properties are generally obtained from numerical simulations

of interatomic and electronic potentials, and the predictions of the simulations are checked

against the results of shock-wave, diamond anvil or other extreme-condition experimental

techniques.

Once an EOS for fluids at ultrahigh pressure is available (even if only as a table of

thermodynamic values), calculation of chemical potentials, speciation, etc., can be carried

out as described in the previous sections, and applied to model processes in the interiors of

giant planets, such as unmixing of molecular helium and fluid metallic hydrogen. The EOS

for fluid planets is also important to understand their thermal evolution. Let us write the EOS

for fluids at ultrahigh pressure as two separate equations, for pressure and internal energy.

In each case we consider zero temperature and thermal components (see Section 8.4):

P = P

γ C

T (9.105)

E =E

T , (9.106)

where γ is the Grüneisen ratio (equations (8.61)) and the variables E, V and C

are all

extensive quantities. We recall that the zero temperature energy is the strain energy of the

material when compressed under P

. We also recall the relationship between pressure and

internal energy valid at zero temperature (equation (8.48)) and, using the chain rule, we find:

=−

∂E

∂V

=−

∂E

∂m

∂V

=−ρ

∂E

∂m

, (9.107)

where m is mass, and ρ density = m/V . We saw in Chapter 2 that in a self-gravitating

body at hydrostatic and thermodynamic equilibrium, the gravitational binding energy and

the body’s thermodynamic state are related to one another by equation (2.42), which we

now write as follows:

=−3





dm +





. (9.108)

From (9.107) we have:



dm =−



=−E

, (9.109)

i.e. the total strain energy of the planet. We re-write the thermal pressure as follows:

=γρc

T , (9.110)

474 Thermodynamics of planetary volatiles

where c

/m is the specific heat capacity. Substituting in (9.108) and assuming some

average values for temperature and heat capacity:



dm =



γC

T dm =γ C

T = γ

(

E −E

)

=γ E

. (9.111)

Substituting (9.109) and (9.111)in(9.108) we arrive at:

(

−γ E

)

. (9.112)

Consider now a planet that contracts by an infinitesimal amount between two states of

hydrostatic and thermodynamic equilibrium. The change in gravitational binding energy is

given by:

(

−γ dE

)

< 0 (9.113)

where the negative sign states that the planet is contracting. The thermal energy radiated by

the planet is (see equation (2.50)):

dQ =dE +dU

=dE

+dE

(

−γ dE

)

=4dE

(

1 −3γ

)

< 0,

(9.114)

where dQ must be a negative quantity because planets do not absorb heat from space. We

see that the strain (zero–temperature) and thermal energies of the planet must be related as

follows:

3γ −1

. (9.115)

This is a general result, that follows from the conditions of thermodynamic and hydrostatic

equilibrium. The actual values of dE

and dE

are given by the specific equation of state,

but they must satisfy (9.115). In the absence of an EOS they remain unknown, but we can

analyze some limiting cases.

The Grüneisen ratio is characteristically a number of order 1, so we rewrite (9.115)as

follows:

. (9.116)

Clearly it must be dE

≥ 0, for the strain energy cannot decrease under compression

(equation (9.107)). Suppose first that dE

=0. We then get:

=−

, dQ =

. (9.117)

This case is unphysical, as it implies a material incapable of storing strain energy, or equiv-

alently, a material with no strength. If a planet behaved in this way then upon contraction it

would store one third of the gravitational energy dissipated as thermal energy, and radiate

the other two thirds to space. Compare this result to the behavior of a monatomic ideal gas

(dQ =−dE, equation (2.52)), which is as close as we can get to a material with no strength.

Consider now the other limiting case dE

. In this case we would have dQ =0,

which would imply a planet in thermal equilibrium with its environment and hence not

contracting. As long as a planet has a temperature higher than ∼2.7 K, however, it will

475 Exercises for Chapter 9

radiate thermal energy to space (Chapter 13) and hence contract. We can think of this case,

then, as one in which the planet is losing thermal energy stored during an earlier stage of

contraction during which dE

, whereas at present dE

. We then have:

=−

, dE

=−

, (9.118)

i.e. two thirds of the gravitational energy would be stored as thermal energy, and the other

third as strain energy. The key point here is that in a planet composed of this type of material

all of the gravitational energy released during contraction is stored as internal energy, much

of it as heat.

The EOS for hydrogen, helium and H

O–NH

–CH

mixtures at conditions of the deep

interiors of the giant planets suggest that their present day behaviors approach this second

limiting case. Today the planets are cooling by releasing either thermal energy stored during

a previous stage of contraction during which the properties of the (less dense) fluids were

significantly different, or heat generated by radioactive decay, or some combination. Cooling

causes contraction, but the gravitational binding energy released in this process is chiefly

stored in the planets’ deep interiors, as in (9.118).

Exercises for Chapter 9

9.1 Write a Maple procedure to calculate f (H

O) along the serpentinization reaction

(9.15). Compare your results to Fig. 9.1.

9.2 The bottom panel of Fig. 9.1shows that dehydration occurs with increasing pressure.

Explain this apparently inconsistent result.

9.3 Write equations for f (CO

) as a function of f (O

) at constant P and T along the (hm)

and (mt) reactions in Fig. 6.14. Plot the two reactions at 1 bar, 25

◦

C and 1 kbar, 400

◦

and confirm that their intersections coincide with the (sd) reaction (see Fig. 9.3). What

is the stable phase assemblage at the Earth’s surface? What can you infer about the

fluid composition during crystallization of a hydrothermal vein with the equilibrium

assemblage siderite + magnetite?

9.4 Prove that for a system at equilibrium the derivative (∂P /∂V )

must be negative.

(Hint: start from the definition of Helmholtz free energy.)

9.5 Write a Maple procedure to solve the system of simultaneous equations (9.1.1 ), (9.1.5

) and (9.1.6 ) for the three non-dimensional variables φ

liq

, φ

vap

, and π, as a function

of non-dimensional temperature, τ . Verify your results against Fig. 9.7, 9.8and 9.9.

9.6 Explain why the critical isochore (φ = 1) is the continuation of the univariant phase

transition (Fig. 9.9).

9.7 Plot the following three fluids in Figs. 9.7, 9.8and 9.9: (i) N

at 1 bar, 300 K; (ii) N

1.5 bar, 100 K; (iii) CO

at 92 bar, 800 K. Comment on the nature of the atmosphere

at the surfaces of Earth, Titan and Venus.

9.8 Compare the behavior of : (i) N

at 1 bar, 300 K; (ii) N

at 1.5 bar, 100 K; (iii) CO

at 92 bar, 800 K, as predicted by the ideal gas EOS and by the Redlich–Kwong EOS.

Which EOS would you use to model the atmospheres of Earth, Titan and Venus?

9.9 Use the Maple worksheet that you wrote for Exercise 9.1 to justify the choice of

f (H

O) in Worked Example 9.7.

476 Thermodynamics of planetary volatiles

9.10 Recalculate the speciation calculation in Worked Example 9.7 ignoring CO, and

compare your results to Fig. 9.16.

9.11 A metamorphic rock contains the equilibrium assemblage: muscovite + sillimanite

+ quartz + microcline + graphite. During metamorphism this assemblage was at

equilibrium with a volatile phase. Construct plots showing the composition of the

volatile phase as a function of temperature at 1, 5 and 10 kbar. Begin by finding the

upper temperature limit for each plot by determining the location of the muscovite

+ sillimanite + quartz + microcline + vapor equilibrium in the carbon-free system.

Then modify the Maple worksheet described in Software Box 9.3to calculate the

diagrams.

9.12 Recalculate the CO–CO

speciation calculation (Exercise 5.5) using Gibbs free energy

minimization. Plot isobaric graphite saturation boundaries in a diagram of T vs. com-

position (the binary C–O join). Use your diagram to discuss preservation of elemental

carbon during metamorphism of organic sediments.

9.13 Plot log f (O

) vs. X

in olivine at quartz saturation, and log f (O

) vs. a

SiO

for pure

fayalite, along the QFI buffer at 1200

◦

C and constant pressures of 1, 5 and 10 kbar.

Discuss the implications for crystallization of metallic Fe during differentiation of

planetesimals (assume that the fluids are C–O–H fluids saturated in graphite).

9.14 Plot fugacities of ethane and formaldehyde in the speciation diagrams in Fig. 9.17and

discuss whether including these species in the Gibbs free energy minimization calcu-

lations would make any significant difference. Necessary thermodynamic properties

can be obtained from NIST’s Chemistry WebBook.

Melting in planetary bodies

The liquid state extends from the melting point to the boiling point. Beyond the critical point

fluids with liquid-like densities transition continuously to fluids with gas-like densities. A

liquid close to its freezing temperature may differ significantly in such properties as vis-

cosity, microscopic structure and chemical behavior from a liquid of the same composition

near its boiling or critical points. For this reason it is convenient to define a melt as a liquid

that is at, or very near, its freezing point. A melt is therefore saturated, or nearly so, in a

solid phase (or assemblage) of broadly similar bulk composition. The exact meaning of

“broadly similar” will remain undefined, but will become clear from the context of this

and the following chapter, in which we will discuss electrolyte solutions. There is a parallel

between this definition of melt and that of vapor, which is a gas that is at equilibrium with

its liquid.

This chapter focuses on the ways in which melts form in planetary interiors. Because

several excellent and up-to-date textbooks on igneous petrology are available (see, Winter,

2001; McBirney, 2006; Philpotts & Ague, 2009), and the research literature in the field is

vibrant, I will not discuss processes of magma evolution and crystallization. There is no

point in repeating here what is explained in much greater detail elsewhere. It is impor-

tant to recall that a magma is an assemblage of melt, suspended solids and dissolved

volatiles. As geologists we are most familiar with the silicate (and minor carbonate)

magmatism characteristic of terrestrial planets, but there is no fundamental thermo-

dynamic distinction between that kind of magmatism, ice magmatism in bodies such

as Titan, Ganymede or Triton, and equilibrium between molten and solid metals in

planetary cores.

10.1 Principles of melting

10.1.1 Melting of simple solids

From the point of view of thermodynamics, melting of a simple substance (e.g. a one-

component system) at constant P and T is simply defined as a process that causes the

Gibbs free energy of a solid and that of its liquid to become identical. But what is melt-

ing at the microscopic scale? Let us begin by looking at elements, which are the simplest

possible thermodynamic systems. A crystal of an element consists of an assemblage of

identical atoms arranged in a lattice. We can consider the lattice as being made up of

lattice points that are occupied by atoms, and interstitial sites which remain vacant. The

latter is the case because, if an atom were to occupy one of these interstitial sites, then it

would be closer to some other atoms and the increased repulsive potential (e.g. Section

8.3) would raise the free energy of the crystal. The lattice points are the equilibrium

477