Fowler A. Mathematical Geoscience

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820 D Melting, Dissolution, and Phase Changes

is the strain rate. We can write

˙ε

=−p∇.u +τ

˙ε

, (D.16)

where τ

is the deviatoric stress tensor, and using the conservation of mass equation

(D.12)

, we ﬁnd





=τ

˙ε

−∇.q ≡R. (D.17)

The right hand side R of this equation consists of the viscous dissipation and the

heat transport. Using (D.11), this leads to

ρT

=R. (D.18)

Using the relation in (D.11), the energy equation can also be written in the form

ρc

−βT

=R, (D.19)

and using the deﬁnition of (speciﬁc) enthalpy, it takes the form

−

=R. (D.20)

These different forms are variously of use depending on the material properties.

In particular, for a perfect gas one can show (see Question D.12) that

dh =c

dT, de =c

dT. (D.21)

The second of these also applies to an incompressible ﬂuid.

D.3 Phase Change: Clapeyron Equation

The use of the free energies G (Gibbs free energy) and F (Helmholtz free energy)

is that they describe thermodynamic equilibrium conditions. Speciﬁcally, they take

minimum (and thus stationary) values at equilibrium. The difference between them

resides in the external conditions. At constant temperature and pressure, the Gibbs

free energy is a minimum, while at constant temperature and volume, the Helmholtz

free energy is a minimum. Of course, we are never really interested in systems

which are at equilibrium. Implicitly, thermodynamics is useful because we typically

assume that in systems away from equilibrium (pretty much everything), there is

a rapid relaxation of some parts of the system towards equilibrium. For example,

it is common to assume that in melting or freezing, the solid–liquid interface is at

the melting point. This is often a good assumption, but not always. One needs to be

aware that in practice, we assume thermodynamic relations in a quasi-equilibrium

D.4 Phase Change in Multi-component Materials 821

manner. If there is a gradient in the Gibbs free energy, then transport will occur to

try to minimise the free energy. A gradient in temperature causes heat transport; a

gradient in pressure causes ﬂuid ﬂow. A gradient in chemical potential (discussed in

Sect. D.4) causes Fickian diffusion.

A simple use of the Gibbs free energy is in determining the Clapeyron relation,

which relates melting temperature (or any phase change temperature) to pressure.

The Gibbs free energy is G = H − TS, and using (D.2), we ﬁnd (for intensive

variables)

dg =vdp−sdT. (D.22)

Suppose now that we have a phase boundary between, say, solid and liquid (of the

same material), denoted by subscripts s and l. At the phase boundary, equilibrium

dictates that g

, where these are the free energies in the solid and liquid phase.

Inequality would cause transport, as we have said. Suppose the melting temperature

is T

, and the system moves to a different temperature and pressure. At the new

equilibrium, the perturbations to the free energies must be equal, thus g

= g

and thus

p −s

T =v

p −s

T , (D.23)

whence

T

p

v

s

, (D.24)

where

v =v

−v

(D.25)

is the change of speciﬁc volume on melting, and

s =s

−s

(D.26)

is the change of speciﬁc entropy on melting. We deﬁne the latent heat to be

L =T

s, (D.27)

so that (D.24) takes the form of the Clapeyron equation,

LT



−



p . (D.28)

This relation, or its differential equivalent, describes the form of the phase transition

curves which, for ice-water-water vapour, have been drawn in Fig. 2.7.

D.4 Phase Change in Multi-component Materials

Now we consider materials, such as alloys or aqueous solutions, which contain more

than one substance. In a sense, we have already introduced this by considering two

822 D Melting, Dissolution, and Phase Changes

different phases of a pure material. If we suppose that we have n

moles of substance

i (these are thus extensive variables), then each substance has its own Gibbs free

energy, and these contribute additively to the total free energy. The free energy of

each phase is called its chemical potential, and the chemical potential μ

of phase i

is deﬁned more precisely by asserting that the total Gibbs free energy satisﬁes

dG =Vdp−SdT +



, (D.29)

thus

∂G

∂n

, (D.30)

where the derivative is evidently at constant temperature and pressure. The chemical

potential is thus an intensive variable. Suppose we have a solid in equilibrium with

a liquid. Since the differential increments in (D.29) are all independent, we can

imagine a change of solid i to liquid i, such that dn

=−dn

. The consequent

change in Gibbs free energy is (μ

− μ

)dn

, and in equilibrium this must be

zero. Thus we must have

=μ

(D.31)

at equilibrium, in each component. Just as heat ﬂows down a temperature gradient,

so substance is transported down a chemical potential gradient.

For a perfect gas, the speciﬁc Gibbs free energy g(T,p) satisﬁes

∂g

∂p



=v =

(D.32)

(since G =ng and pV =nRT ,forn moles of the gas), and thus

g =g

+RT lnp. (D.33)

In a mixture of gases, the partial pressure of each component gas is that pressure

it would have if the other gases were removed. Dalton’s law says that the partial

pressures are additive, so that their sum is the total pressure of the gas mixture. If

we suppose in a mixture that the analogue of (D.33) holds for partial energies and

pressures, i.e.,

+RT ln p

, (D.34)

then since p

V =n

RT and g

is the chemical potential of gas i, we can write

=μ

+RT lnc

, (D.35)

where c

is the molar fraction of phase i (=



). This relation more generally

characterises an ideal mixture, whether it be of gases, liquids or solids.

Now let us consider an interface (we will think of it as a solid-liquid interface)

between the melt and solid of a two component mixture containing substances A

D.4 Phase Change in Multi-component Materials 823

Fig. D.1 The double tangent

construction for c

and c

The curves are the graphs of

the functions g

and g

deﬁned by (D.38), in which

we deﬁne (the units are

arbitrary)

(L) =μ

(L) =RT ,

(S) =1, μ

(S) =4. The

ﬁgure shows the construction

at RT =2.5. c denotes the

concentration as mole

fraction of A

Fig. D.2 Typical phase

equilibrium for an ideal

solution. The same formulae

are used as in constructing

Fig. D.1, with the range

corresponding to 1 ≤RT ≤4

and B. We will suppose the mixture is ideal. At the interface, the chemical potentials

of each component must be equal, thus

=μ

,μ

=μ

, (D.36)

and these will determine the interfacial concentrations as functions of temperature.

To be speciﬁc, let c denote the molar fraction of component A, so that 1 − c is the

molar fraction of B. Then the bulk Gibbs free energies (one in each phase) are

g =μ

c +μ

(1 −c), (D.37)

and for an ideal solution, we have

g =μ

(1 −c) +μ

c +RT



c ln c +(1 −c) ln(1 −c)



. (D.38)

The two functions g

and g

are thus convex upwards functions, and the criterion

for equilibrium as in (D.36) is obtained by drawing a common tangent to g

and g

as indicated in Fig. D.1, and done in Question D.3; this gives the solid and liquid

concentrations in equilibrium for a particular temperature; as the temperature varies,

we obtain the typical phase diagram shown in Fig. D.2.

Although our discussion is motivated by gases, the concept of an ideal solution

applies equally to liquids and solids. Indeed, Fig. 9.4 shows a phase diagram essen-

tially the same as that in Fig. D.2, for the solid solution of albite and anorthite. As

824 D Melting, Dissolution, and Phase Changes

Fig. D.3 A typical phase

diagram for a mixture

(pyroxene–plagioclase) with

a eutectic point. Such

diagrams are common for

aqueous solutions

for liquids and gases, ideal solutions occur when there is no penalty for introducing

molecules of different substance. In the case of solids, this means replacing atoms

in the crystal lattice.

For non-ideal solutions, the logarithmic terms such as lnc in the free energy

are replaced by corresponding quantities ln a, where a is a function of c called the

activity. One typical effect is to make the free energy curves g

and g

have multiple

minima, and this allows for more than one pair of liquidus and solidus values at a

given temperature. A typical such consequent phase diagram is shown in Fig. D.3,

which is actually that for pyroxene and plagioclase shown in Fig. 9.12. Here there

are two liquidus curves, which meet at the eutectic point. The solidus curves in

this diagram are vertical, thus on freezing, one forms either pure pyroxene or pure

plagioclase, depending on which side of the eutectic the liquid composition lies.

Below the eutectic point only solid can exist in equilibrium.

D.5 Melting and Freezing

In discussing phase change, we have mostly referred to melting and freezing. In

terms of pure materials, there is no distinction to be made between this, boiling and

condensation (of liquid and gas), and sublimation and condensation (of solid and

gas). A point we will now make is that there is similarly no distinction between the

different corresponding situations which refer to multi-component phase change.

The melting and freezing of an alloy is familiar in industrial contexts (in forming

solid castings) as well as the environment. The simplest example is the case of an

iceberg, consisting of fresh water ice in equilibrium with a slightly salty ocean. Ice-

bergs are of course not formed by freezing the ocean (but sea ice is), but the principle

will serve. Freezing of salty sea water occurs on a diagram similar to Fig. D.3;fora

sufﬁciently dilute solution, freezing forms more or less pure water ice, with the salt

being rejected into the water. We routinely refer to this as freezing.

D.6 Precipitation and Dissolution

Suppose, however, that we take a salty solution at high temperature. Better, think

of sugar dissolved in water (or tea) at high temperatures. The solubility is greater at

D.7 Evaporation and Boiling 825

higher temperatures, and if we cool the tea (a lot), eventually the sugar will come

out of solution; it precipitates, while at high temperature it dissolves. We do not nor-

mally think of this as melting and freezing, but the process is exactly the same. The

only difference to the iceberg is that we are on the other side of the eutectic. Now,

when we take our saline solution at high salt concentration and high temperature,

and lower the temperature, we reach a liquidus on the other side of the eutectic to

that of the iceberg; solid salt is frozen (but we say it is precipitated), and the rem-

nant water becomes purer. Or, if we pour salt into water when we cook, it dissolves

as we heat the water; we aid the dissolution by stirring, which increases the avail-

able surface area for dissolution. We do not think that the salt is melting; but it is.

There is no distinction between the processes of melting and freezing of alloys and

precipitation and dissolution of solutes.

D.7 Evaporation and Boiling

Surely, however, evaporation and boiling are not the same at all? Evaporation occurs

continually at temperatures below the boiling point: we sweat; boiling occurs at a

ﬁxed temperature. For water, boiling occurs at 100°C at sea level. But evaporation

occurs from oceans at their much lower temperatures. Certainly, on the top of Mount

Everest, boiling temperature is reduced, but this is because the pressure is lower, and

occurs through the Clapeyron effect.

So then, what is evaporation? The saturation vapour pressure of water vapour,

, is a function of temperature, given by the solution of (2.56), and it increases

to a pressure of one bar (sea level atmospheric pressure) at a temperature of 100°C,

where boiling occurs spontaneously.

It is all, in fact, the same story. The ocean, let us say, is pure water (ignore salt).

The atmosphere is a two component mixture (let us say) of water vapour and air;

it is an alloy. If we take a hot atmosphere and reduce its temperature, condensa-

tion occurs at a temperature which depends on atmospheric composition. The molar

fraction of water vapour in the atmosphere is just p

, the vapour pressure di-

vided by the atmospheric pressure. On what would be the liquidus (but now must

be the vaporus

), the vapour pressure has its saturation value, the molar fraction of

water vapour is p

, and the saturation temperature T

is a function of c

What has boiling to do with this? Not much! Evaporation is boiling. What we nor-

mally call boiling refers to the position of the vaporus when c

=1, i.e. p

For given atmospheric pressure, we cannot raise the liquid temperature beyond the

vaporus temperature at vapour concentration of one. If we change atmospheric pres-

sure, then this temperature will change. Yes, because of Clapeyron, but also because

pressure dictates concentration. Gases are different because the amount of gas de-

pends on pressure. For liquids and solids, this is mostly not the case.

Solidus is a perfectly good Latin word, but liquidus is not; vaporus is invented here.

826 D Melting, Dissolution, and Phase Changes

D.8 Chemical Reactions

Surely chemical reactions are different? So it would appear. If we pour vinegar

(acetic acid) into a kettle furred up with limescale (calcium carbonate), the limescale

will dissolve, or react, forming carbon dioxide in the process. In a coal ﬁre, the car-

bon in the coal reacts with oxygen, forming carbon dioxide. There is no equilibrium

surface or phase diagram here, surely?

But in fact the difference is only one of degree. When a salt M dissolves in water

to the point of saturation, the equilibrium that results is a consequence of a simple

reversible reaction



, (D.39)

where k

is the rate of dissolution and k

is the rate of precipitation. The fact that

there is an equilibrium is a consequence of the reversibility. The only effective dif-

ference between this and a chemical reaction is that the examples cited above are

almost irreversible. If we burn coal in a sealed environment, the carbon reacts with

the oxygen to form a mixed atmosphere of O

with CO

, just as when we evap-

orate water vapour in air. If the reaction is reversible, then an equilibrium will be

obtained. In practice (in this example) the backward reaction rate is negligible, and

so the equilibrium which obtains occurs when the coal is (almost) entirely used up.

Chemical reaction is thus the process describing the evolution towards thermody-

namic equilibrium.

D.9 Surface Energy

Interfaces between two materials, be they both ﬂuids, ﬂuid and solid, or any other

such combination, carry a surface energy per unit area, denoted γ . The existence of

a surface energy causes a pressure jump across the interface, and the requirement of

force balance (Newton’s third law) on the massless interface means that the interface

appears to carry a tension, the surface tension. To see how the surface energy induces

this pressure jump, we consider equilibrium of a system containing an interface. For

example, we may think of a box containing ﬂuid with a gas bubble in it. To change

the surface area of the interface, we may alter the external pressure, and thus the

equilibrium is that associated with constant volume and temperature, for which the

relevant minimum is obtained by the Helmholtz free energy F . The basic recipe for

an increment of F for each phase follows from (D.1) and (D.2), and is

dF =−pdV −SdT; (D.40)

when the surface area of a phase interface has a surface energy per unit area γ , then

a change in surface area dA causes an additional contribution γdA, which must

also be included. Suppose the two sides of the interface are denoted by subscripts −

and +, and have corresponding pressures p

−

and p

. For an isothermal change at

D.10 Pre-melting 827

constant total volume, dV

−

=−dV

, and thus the increment of the total Helmholtz

free energy of the system is

dF =−p

−

−p

+γdA=−(p

−

−p

)dV

−

+γdA=0, (D.41)

and thus

−

−p

=γ

∂A

∂V

−

. (D.42)

This determines the pressure jump at the interface. It is a result of differential geom-

etry that

∂A

∂V

−

=2κ, where κ is the mean curvature of the surface (the average of the

two principal curvatures); for example the mean curvature κ of a spherical surface

measured from the side on which the centre of the sphere lies is just 1/R, where R

is the sphere radius.

D.9.1 The Gibbs–Thomson Effect

The curvature of an interface also has an effect on the melting temperature, and this

is known as the Gibbs–Thomson effect. For this we may go back to the Clapeyron

type argument and speciﬁc Gibbs free energy of each phase (i.e., their chemical

potentials). Denoting these as before as g

and g

, but now allowing solid and liquid

pressures to change independently, we have

p

−s

T =v

p

−s

T , (D.43)

and with L =T

s being the latent heat, we have the generalised Clapeyron rela-

tion

LT



−



p

−

−p

)

, (D.44)

in which the ﬁrst term on the right is the Clapeyron effect of changing pressure, and

the second is the Gibbs–Thomson effect, which describes change of melting tem-

perature with surface curvature, since p

−p

=2γκ, with the curvature measured

from the solid side of the interface.

D.10 Pre-melting

It is commonly the case that a solid will maintain a thin liquid ﬁlm of its melt at an

interface with, for example, a quartz grain, even at temperatures below the freezing

point. This phenomenon is known as ‘pre-melting’ (Dash et al. 2006; Wettlaufer and

Worster 2006), and is associated with an excess free energy manifested by very thin

ﬁlms due to a variety of intermolecular forces, for example Van der Waals forces.

The scale on which these forces act is measured in molecular diameters, and so the

828 D Melting, Dissolution, and Phase Changes

ﬁlm thicknesses over which these free energy effects are important are of the order

of nanometres. Just as for surface energy, pre-melting causes an excess pressure,

called the disjoining pressure, to occur in the ﬁlm, and it causes a displacement of

the freezing temperature. A particular geophysical problem in which this disjoining

pressure is important is in the phenomenon of frost heave (Rempel et al. 2004),

wherein freezing soil is uplifted, causing the heave which can be very damaging to

roads and structures. The force generated in frost heave can be very large, of the

order of bars, and this force is due to the disjoining pressure in the thin water ﬁlms

which separate the ice from the soil grains.

To understand the dynamic effects, we consider a thin ﬁlm of thickness h sepa-

rating an ice surface from a foreign solid surface. In the absence of the ﬁlm, the ice-

solid interface has a surface energy which we denote by γ

, while the interposition

of a liquid ﬁlm creates two new surfaces, of interfacial energies γ

(solid-water)

and γ

(ice-water). In addition, the liquid ﬁlm has a Gibbs free energy per unit area

of the form

G =ρ

h +Φ(h), (D.45)

where μ

is the chemical potential energy of the bulk liquid, and Φ is the free energy

associated with intermolecular forces. In particular, we suppose

Φ(0) =γ

,Φ(∞) =γ

+γ

; (D.46)

the liquid ﬁlm is energetically preferred if γ < 0, where

γ =γ

+γ

−γ

, (D.47)

and it is in this case that a positive disjoining pressure occurs. We write

Φ =γ

+γ φ(h), (D.48)

where φ increases monotonically from zero at h =0 to one at h =∞. For example,

Van der Waals forces lead to a form for φ of

φ =



1 −



, (D.49)

where the constant σ is of the order of a molecular diameter. Clearly, if γ < 0,

then Φ is a monotonically decreasing function of h, while the bulk free energy is an

increasing function, and thus a minimum of G in (D.45) will be obtained when h is

ﬁnite, if |γ | is sufﬁciently large. This causes the wetting ﬁlm.

This is perhaps an inverted way of looking at it. Heaving requires the maintenance of the ﬁlm

between ice and soil grains; as long as the ﬁlm is maintained, heave will occur. The presence of a

large overburden pressure will eventually suppress heave, but the necessary pressures are large.

D.11 Liesegang Rings 829

D.10.1 Disjoining Pressure

We consider the Helmholtz free energy of a ﬁlm of thickness h. Following a small

perturbation to the ﬁlm thickness,

dF =−p

−p

−SdT +AdΦ, (D.50)

where A is surface area. We have dV

=Adh; for an isothermal change at constant

volume dV

=−dV

=Adh, and therefore

−p

=−Φ



(h) =−γ φ



(h); (D.51)

this is the disjoining pressure. For (D.49), this leads to

−p

=−

6πh

, (D.52)

where A is the Hamaker constant

A= 12πσ

γ. (D.53)

D.10.2 Freezing Point Depression

Finally we consider the effect of a thin ﬁlm on the freezing point. This simply fol-

lows from (D.44), which we write in the form

LT

=(v

−v

)p

−v

(p

−p

), (D.54)

and thus, from (D.51), (ignoring liquid pressure variations)

L(T −T

)

≈

γ φ



(h)

. (D.55)

For γ < 0, this represents the freezing point depression due to pre-melting only;

the Clapeyron and Gibbs–Thomson effects can be added to the right hand side.

Because φ



∝

, these thin ﬁlms can be maintained to temperatures quite a way

below the normal freezing point.

D.11 Liesegang Rings

As discussed in Chap. 9, Liesegang rings can form when crystals are precipitated

in a dilute solution. Liesegang himself put some silver nitrate on a gel containing

potassium dichromate, and the resulting silver dichromate crystals precipitate in