Douce A.P. Thermodynamics of the Earth and Planets

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578 Non-equilibrium thermodynamics

We define the external contribution as the entropy change that arises from exchange of heat

with the environment, i.e.:

S ≡

dE +PdV

. (12.2)

This component can be positive, negative or, in the case of adiabatic or isolated systems,

zero. The internal contribution to entropy arises from processes that occur inside the sys-

tem. These could be, for example, heat transfer, chemical diffusion, chemical reactions,

viscous dissipation, or dissipation of electric currents. An important part of the study of

non-equilibrium thermodynamics consists of finding rigorous mathematical expressions

for these and other entropy production processes. At this point we note that d

S must obey

the following relationship:

S ≥ 0. (12.3)

The equality holds for a system at equilibrium, i.e. static. If a system is not static then d

S>0.

Of course, there is nothing new here: if we apply (12.1) to an isolated system (d

S = 0)

then (12.3) recovers the property of entropy given by equation (4.7). The importance of

separating external and internal entropy contributions as in (12.1) is that internal entropy

production (d

S)isalways positive in a non-static system, regardless of how the system

interacts with its environment. Moreover, it is positive not only for the system as a whole

but also for any (non-static) part of the system that we may wish to analyze independently.

If a system is not at internal equilibrium then gradients in intensive variables must exist

within it. If entropy gradients exist then the concept of molar entropy for the system as

a whole loses meaning. If the system is not far from equilibrium, however, it is possible

to define a lengthscale within which local equilibrium holds (i.e. such that gradients in

intensive variables can be neglected over distances of this magnitude). It is then convenient

to work with entropy per unit volume, so that the entropy of sufficiently small volume

elements is well-defined. More precisely, because we are interested in non-static conditions,

we consider the rate of entropy production per unit volume, σ , defined as:

σ ≡

. (12.4)

We note that the units of σ are entropy per unit volume per unit time, for instance: J K

−1

−3

−1

Entropy production is the result of flows that occur as long as the system is not at

equilibrium, and that cease when the system reaches equilibrium. We shall call these ther-

modynamic flows. They could be, for instance, flow of heat, matter, electric charge or

momentum, or the change in the number of molecules of a given species during a chemical

reaction (a special case of mass flow). The thermodynamic flows are driven by potential

gradients, which are also called thermodynamic forces. For a general system in which

several different entropy production processes operate simultaneously we have:

σ = T

, (12.5)

where the J

are flows (vectors), the F

are potential gradients (one-forms) and (12.5)is

an inner product that produces the scalar quantity σ (Box 1.1). Consider heat flow as an

example. In this case J

is the heat flux vector (Section 3.1), which has units of J s

−1

−2

. On dimensional grounds we see that the units of the potential gradient that drives this

579 12.1 Non-equilibrium thermodynamics

flow must be K

−1

. Restricting our discussion to flow in one spatial dimension, z,we

conclude that the thermodynamic gradient that drives heat flow is:

∂

∂z





. (12.6)

We shall now make the assumption that, if the potential gradients are small, then the flows

are linear functions of the potential gradients. This linear relationship is the formal definition

of a system close to equilibrium. If the linear relationship between flows and forces does

not hold then the system is far from equilibrium and the discussion in this and subsequent

sections does not hold. We write the linear relationship as follows:

, (12.7)

where the L

are constants called phenomenological coefficients. Note that this equation

says that a flow J

may be driven not only by the gradient F

but also by all other poten-

tial gradients that may exist in the system. For instance, if gradients in temperature, F

(equation (12.6)) and chemical potential, F

, exist in a system, then according to (12.7),

both heat flow, J

, and mass flow, J

, each includes two separate contributions, one driven

by F

and the other by F

. Expanding (12.7) for this case we would have:

(12.8)

Each phenomenological coefficient is a scalar, and the matrix composed of all phenomeno-

logical coefficients arranged as in equations (12.8) is a geometric object called a tensor.

The two diagonal coefficients are easily interpreted: L

relates temperature gradient to heat

flow, so it must be somehow related to the heat conductivity, k (Chapter 3), whereas L

links the gradient in chemical potential to mass flow, so it must be related to chemical

diffusivity (Section 12.2.2).

The other two coefficients are more obscure and perhaps unexpected. L

=0 implies that

a gradient in chemical potential drives heat flow, a phenomenon know as the Dufour effect,

whereas L

= 0 means that a gradient in temperature causes chemical diffusion, which

is known as the Soret effect. These and other “cross-flow” phenomena have been known

since the nineteenth century. For example, a temperature engenders an electric current (the

Seebeck effect) and a gradient in electrical potential gives rise to heat flow at constant

temperature (the Peltier effect). Cross chemical diffusion terms arise in systems in which

there are gradients in the chemical potentials of more than one component (Section 12.2.3).

Lars Onsager (1931a, and b) demonstrated that these effects are not random, but rather

a fundamental property of non-equilibrium systems. His work is one of the cornerstones

of non-equilibrium thermodynamics. We shall not discuss it here, but we will mention a

fundamental result that is due to Onsager.This is the fact that the matrixof phenomenological

coefficients is symmetric. In other words, all cross coefficients obey the identity:

. (12.9)

This is known as Onsager’s reciprocal relation.

580 Non-equilibrium thermodynamics

12.1.2 Heat diffusion revisited and the principle of minimum

entropy production rate

Consider a one-component system, in which compositional gradients are by definition

impossible. If we impose a thermal gradient on this system then because F

= J

= 0it

must be L

=0, and (12.8) collapses to:

. (12.10)

The thermodynamic flow is in this case the heat flux, given by equation (3.5):

=q =−k

. (12.11)

The potential gradient F

is given by equation (12.6), which we can also write as:

∂

∂z





=−

. (12.12)

We then obtain a relationship between the phenomenological coefficient L

and the thermal

conductivity, k:

k =

(12.13)

and, using (12.5), the rate of entropy production per unit volume is:

σ =





. (12.14)

As expected, σ is a non-negative quantity, and vanishes only if temperature is uniform and

hence there is no heat flow.

We recall from Chapter 3 that the steady state for heat diffusion, i.e. ∂T /∂t = 0, is

attained, in a system with no heat generation, when the thermal gradient is uniform, i.e.

∂T /∂z =constant (e.g. equation (3.15)). It can be proved beginning from equation (12.14)

that, for any non-zero temperature gradient, σ is minimized if T(z)is a linear function, so

that ∂T /∂z = constant. The formal demonstration of this will not be presented here (see,

for example, Kondepudi & Prigogine, 1998, p. 399). The result is, however, general, and is

known as the theorem of minimum entropy production rate, originally due to Prigogine. In

words, it states that any non-equilibrium system in which at least some of the thermodynamic

forces do not vanish, and in which the linear phenomenological law (12.70) and Onsager

reciprocal relations are valid, evolves to a non-equilibrium steady state in which the rate of

entropy production is minimum. One way to think of this, suggested by Onsager, is that the

rate of entropy production behaves as a potential, that is minimized when a non-equilibrium

system reaches a dynamic state that remains stationary with time.Although the reality of this

result is not in question, different opinions exist on whether minimum entropy production

is a principle (i.e. non-demonstrable from simple statements) or a theorem, as envisioned

by Prigogine (see, for example, Jaynes, 1980).

581 12.2 Chemical diffusion

12.2 Chemical diffusion

12.2.1 Fundamental relationships

Transport of chemical species down a chemical potential gradient is a non-equilibrium

process. It therefore generates entropy. Mass transfer can take different forms. One of them

is chemical diffusion without chemical reaction. We seek an equation that relates this type

of matter flow to entropy production. We begin with the fundamental equation (4.101),

which we re-write as follows:

dS =

dE +PdV

−



. (12.15)

Comparing to (12.1) and (12.2) we find that internal entropy production is given by:

S =−



. (12.16)

Consider diffusive mass transfer of a single component, which requires that there be a

gradient in the chemical potential of only that component. Physically this could be possible,

for example, in a one-component system that is not in equilibrium in a gravitational field

(Chapter 13), or in a two-component system in which the solute is dilute enough that the

concentration of the solvent can be considered to be constant even if the solute concentration

varies (Chapter 11). The latter case is known as tracer diffusion. For simplicity we will

consider diffusion in a single spatial dimension, z, but the equations are easily extended

to diffusion in three dimensions (see Kondepudi & Prigogine, 1998; Zhang, 2008; Borg &

Dienes, 1988).

Consider two parallel surfaces of cross-sectional area a, separated by a small distance

δz, and such that the chemical potentials of the diffusing component at each surface are µ

and µ

, with µ

>µ

(Fig. 12.1). Define δµ = µ

−µ

. Matter flows from 1 to 2, so that

if we call dn = dn

=−dn

> 0, we have:



=µ

+µ

=δµdn. (12.17)

Substituting in (12.16):

S =−

δµdn, (12.18)

which is always positive, as δµ and dn always have opposite signs. Passing to the limit

and noting that this amount of entropy is produced inside a volume of size aδz, we get, by

using (12.4):

σ =



−

dµ



. (12.19)

Comparing (12.19)to(12.5), and allowing for the possibility that temperature gradients may

also exist (Section 12.2.4), we identify the thermodynamic flow and the thermodynamic

582 Non-equilibrium thermodynamics

Fig. 12.1

Matter ﬂow, J

, between two parallel surfaces of area a on which the chemical potentials of the diffusing species are

>µ

potential gradient as follows:

=−





(12.20)

For isothermal diffusion of a single component we write the phenomenological relationship

(12.7) as follows:

=−

dµ

. (12.21)

12.2.2 Fick’s laws of chemical diffusion

Equation (12.21) describes isothermal chemical diffusion of a single chemical species, but

it is not convenient because chemical potential is not a directly measurable quantity. We

seek to recast this equation in terms of concentration of the diffusing species. If c

is the

molar concentration of the diffusing species per unit volume and V is the molar volume

of the system, then the mol fraction X

is X

= Vc

. Taking the standard state as the pure

substance at the temperature and pressure of interest:

=µ

0,i

+RT ln

(

)

. (12.22)

583 12.2 Chemical diffusion

By the chain rule, and assuming that the specis concentration is low enough that it follows

Henry’s law (γ

approximately constant):

dµ

. (12.23)

Substituting in (12.21):

=−

(12.24)

and defining:

D ≡

(12.25)

we arrive at:

=−D

, (12.26)

which is known as Fick’s first law of diffusion. Equation (12.26) is identical to Fourier’s

law of heat conduction (equation (3.5)) and it is no more of a “law” than the latter. Rather,

it is another constitutive equation that is a special case of the general transport relation

expressed by equation (3.4).

We now recall that c

has units of mols per unit volume, and that the matter flux J

has

units of mols per unit area per unit time. It follows that the dimension of D is area per unit

time, e.g. m

−1

, which are units of diffusivity (Section 3.2.3). The parameter D is called the

chemical diffusivity or diffusion coefficient. In particular, for the dilute one-component case

that we are considering here it is called the tracer diffusion coefficient. Diffusion coefficients

are a strong function of temperature (Section 12.4.1) and equation (12.25) shows that they

are also a function of composition. This latter point can cause considerable complications

in the mathematics of diffusion, but the compositional dependency can usually be ignored

in tracer diffusion problems.

Equation (12.26) still has the disadvantage that it includes a matter flux term that is

generally not easily measured, especially if D is small. A better alternative would be

to measure the change in concentration with time. Consider a volume element of unit

cross-sectional area and width δz (Fig. 12.2), and let the matter fluxes across its two faces

be J

and J

z+δz

. The change per unit time in the number of mols of solute contained

in the volume is J

−J

z+δz

, so that the rate of change of concentration (mols per unit

volume) is:

δz

(

−J

z+δz

)

. (12.27)

Using (12.26):

∂c

∂t

δz



∂c

∂z



z+δz

−

∂c

∂z





δz

∂

∂z

δz (12.28)

584 Non-equilibrium thermodynamics

a=1

z + δz

a=1

Fig. 12.2

Geometry of the chemical diffusion equation in one dimension. The volume element has unit cross-sectional area

perpendicular to the matter ﬂow direction and thickness δz along this direction.

or:

∂c

∂t

∂

∂z

. (12.29)

Equation (12.29), which is identical to the heat diffusion equation (3.15) without a source

term, is known as Fick’s second law of diffusion. You may also recognize in equations

(12.27) and (12.28) a compact version of the derivation of equation (3.15) in Section 3.2.2.

As I mentioned there, equation (12.29) is a differential equation that shows up in many

branches of physics and is simply known as the diffusion equation. The mathematics of

diffusion are the same regardless of what is the physical entity that is transported.

Worked Example 12.1 Chemical diffusion on planetary time and lengthscales

We can get a feeling for the relevance of chemical diffusion in planetary processes by

focusing on a few examples. In particular, we will look at diffusion in the atmosphere, the

oceans, magmatic systems and minerals at metamorphic conditions. As for heat diffusion,

we have the relationship:

∼D, (12.30)

where λ is the characteristic diffusive lengthscale, τ the characteristic time scale, and D the

diffusion coefficient. Diffusivities of trace gas components in air at 298 K and 1 bar are of

the order of 10

−5

−1

, whereas for molecular and ionic species in aqueous solution at

298 K typical values are ∼10

−9

−1

(see Kondepudi & Prigogine, 1998; Zhang, 2008).

Diffusivities in silicate melts are somewhat more variable; we will use a typical value for

585 12.2 Chemical diffusion

–10

–8

–6

–4

–2

Diffusion time (years)

Diffusion length (meters)

-5

-9

-12

–22

-32

atmosphere

ocean

magma chamber

metamorphism

Sr in feldspar

Pb in monazite

magma

atmosphere

ocean

Fig. 12.3 Some examples of chemical diffusion in planetary environments. The lines labeled with diffusion coefﬁcients (in

−1

) are plots of equation (12.30). For the atmosphere and ocean we start from their characteristic lengthscales

and infer unrealistically long chemical diffusion time scales, implying that other homogenization mechanisms

operate, such as eddy diffusion. For igneous and metamorphic processes we start with estimates of characteristic time

scales, e.g. 10

years for crystallization of a magma chamber and 20 ×10

years for a metamorphic event, and infer

lengthscales for various processes (see text).

O in rhyolite melts at 900

◦

C, which is ∼10

−12

−1

(Zhang et al., 2007). Tracer

diffusion coefficients in minerals are much more variable. For example, Pb in monazite

at 700

◦

C has a diffusivity of ∼10

−32

−1

(Cherniak et al., 2004), whereas the value

for Sr in feldspars at the same temperature is ∼10

−22

−1

(Cherniak & Watson, 1994).

Figure 12.3 shows plots of equation (12.30) with each of these diffusivity values.

The terrestrial atmosphere and oceans have lengthscales of ∼10 km and 2 km, respec-

tively. We see from Fig. 12.3 that diffusive homogenization of the atmosphere would take

about 300 000 years. The corresponding value for the ocean would be about 130 million

years. Clearly, some other mass transfer mechanism must operate in both of these systems.

For instance, changes in CO

concentration with a periodicity of less than one year are seen

in the atmosphere. A diffusive time for the ocean of 130 million years would imply that

it should not be possible to detect differences between Cretaceous and present-day ocean

chemistry, yet marine sediments record changes on much shorter time scales. If we assume

that the atmosphere mixes over times of order 1 year, and the ocean over times of ∼100 years

(rates are probably faster than these), then we can calculate with equation (12.30) that the

586 Non-equilibrium thermodynamics

effective diffusivities are ∼3 and 10

−3

−1

, respectively. These values are 5 to 6 orders

of magnitude greater than the corresponding chemical diffusivities. The explanation for the

discrepancy is that mass transfer in the atmosphere and ocean is chiefly controlled by a

process known as eddy diffusivity. The idea is that, owing to their relatively low viscosities,

neither air nor water are ever perfectly still. Rather, turbulent motions, i.e. eddies, occur

at all lengthscales, driven by causes such as temperature gradients and motion of animate

or inanimate bodies. Eddies cause local stirring and homogenization, and coupling among

eddies effectively diffuses matter at a rate that renders chemical diffusion in oceans and

atmospheres inconsequential.

A reasonable time scale for crystallization of a large igneous system may be 1 million

years. Over this time diffusive homogenization of compositional gradients would extend

over a distance of about 6 m.At least some magmatic systems are known to be homogeneous

over greater distances, as suggested for instance by the composition of km-size plutons and

of large volcanic eruptions. Again, some other mechanism for chemical mixing appears to

be required, but eddy diffusivity is unlikely to be the answer, given the very high viscosity

of silicate melts. Convective stirring in magma chambers is a possible explanation.

Let us now assume that a typical high-grade metamorphic recrystallization event lasts

20 million years. Over this time Sr in feldspar would diffuse a distance of perhaps 0.5 mm,

whereas Pb in monazite would homogenize over 10

−9

–10

−8

m. Radiogenic Pb formed

by decay of U and Th is essentially immobile relative to the size of a monazite crystal

(say, 10

−5

to 10

−4

m). Monazite is a refractory mineral that commonly survives high-

grade metamorphism. Therefore, monazite crystal cores can be used to date events that

preceded metamorphic recrystallization, even if rims of neoformed monazite grow during

metamorphism. In contrast, feldspar may break down and regrow in response to changes in

metamorphic conditions. Chemical reactions such as these are mass transfer mechanisms,

which may be much faster than simple chemical diffusion. However, even if feldspar growth

did not take place, Sr would homogenize over a length not too different from the size of

feldspar crystals. Rb–Sr dating may then yield the age of metamorphism.

An important generalization that follows from these examples is that, whereas chemical

diffusion is not an important mass transfer mechanism at planetary lengthscales, it pro-

vides the physical underpinnings and constraints for many powerful techniques used to

study planetary processes, such as radiometric dating and estimation of rates of geological

processes. A comprehensive and up to date treatment of these topics is given by Zhang

(2008).

12.2.3 Interdiffusion

Consider now the case of a binary system in which two components are present in com-

parable concentrations. If there is a compositional gradient then the two components will

diffuse in opposite directions. This is known as interdiffusion or, also, binary diffusion.It

is the process by which crystals that grow during metamorphism or igneous crystallization

tend to homogenize.

Let the matter fluxes be J

and J

. As always we consider diffusion in one dimension.

In some systems the following relationship is valid:

=0. (12.31)

587 12.2 Chemical diffusion

Recall that we consider flux in units of mols, or particles, per unit of surface per unit of time.

Interdiffusion in a mixture of ideal gases at constant temperature and pressure obeys (12.31).

The same can be expected to be at least approximately true in a crystal, in which particles

exchange places among fixed sites in a crystalline lattice. But there may be instances in

which (12.31) is not valid, for example if the partial molar volumes of the two components

are significantly different. This requires only a slight modification to the equations, as we

shall see.

Calling the mol fractions of the two components X

and X

we have, from Gibbs–

Duhem’s equation (6.7) at constant temperature and pressure:

dµ

=0 (12.32)

or:

dµ

=0. (12.33)

We can combine (12.31) and (12.33) as follows:

dµ

/dz

dµ

/dz

(12.34)

and define a phenomenological coefficient, L, as follows:

−

≡

dµ

/dz

dµ

/dz

. (12.35)

Calling the molar concentrations per unit volume c

and c

, and using (12.23), we find:

=−

(12.36)

so that the diffusion coefficients are (see equation (12.26)):

(12.37)

For X

→ 1 the coefficient D

becomes the tracer diffusion coefficient given by

equation (12.25), and as long as X

= 0, D

remains bound and the flux J

vanishes

as dc

/dz → 0. In other words, we recover the tracer diffusion case. We also note that, if

the molar volume of the mixture is V , then for all values of X

and X

we have X

=Vc

and X

=Vc

. Substituting these identities in (12.37) we find:

, (12.38)