The Phase RuleThe Phase Ruleand its applicationand its application

ThermodynamicsThermodynamicsA A system:system:Some portion of the universe that you wish to studySome portion of the universe that you wish to study

The The surroundings:surroundings:The adjacent part of the universe outside the The adjacent part of the universe outside the

systemsystem

Open system: exchange of energies and mass

Closed system: only exchange of mechanical and thermal energy, no mass exchange

A phase: is a physically distinct part of a system that is mechanically separable from other parts in the system;e.g. a melt or a mineral

ThermodynamicsThermodynamics

A A phase diagram:phase diagram:P, T Shows the stability ranges of phases (minerals, melts, solutions) P, T Shows the stability ranges of phases (minerals, melts, solutions)

as functions of composition,, pH and Eh.as functions of composition,, pH and Eh.

ComponentsComponents:: Minimum number of chemical constituents that Minimum number of chemical constituents that

are required to describe the compositions of all phases in the systemare required to describe the compositions of all phases in the system

(there are never be more components than phases in a system)(there are never be more components than phases in a system)

Degrees of freedom: The number of variables to define the position of a mineral assemblage in a phase diagram

Intensive properties: P, T, pH, Eh (Environmental variables)

Extensive properties: V, m, partial pressure

F F = C - P + v= C - P + vF = number of degrees of freedomF = number of degrees of freedomThe number of variables to define the position of a mineral assemblage in a phase diagram

The Gibbs Phase RuleThe Gibbs Phase Rule

F = C – F = C – PP + v + vF = number of degrees of freedomF = number of degrees of freedom

The number of variables to define the position of a mineral assemblage in a phase diagram

P = number of phasesP = number of phasesphases are phases are mechanically separablemechanically separable constituents constituents

The Phase RuleThe Phase Rule

F = F = CC - P + v - P + vF = Number of degrees of freedomF = Number of degrees of freedom

The number of variables to define the position of a mineral assemblage in a phase diagram

P = number of phasesP = number of phases

phases are mechanically separable constituentsphases are mechanically separable constituents

C = minimum number of componentsC = minimum number of components (chemical (chemical constituents that must be specified in order to define all phases)constituents that must be specified in order to define all phases)

The Phase RuleThe Phase Rule

The Phase RuleThe Phase RuleF = C - P + F = C - P + vv

F = # degrees of freedomF = # degrees of freedom The number of variables to define the position of a mineral assemblage in a phase diagram

P = number of phasesP = number of phasesphases are mechanically separable constituentsphases are mechanically separable constituents

C = minimum # of components C = minimum # of components (chemical (chemical

constituents that must be specified in order to define all phases)constituents that must be specified in order to define all phases)

v = intensive properties or environmental variables, v = intensive properties or environmental variables,

in P/T and pH/Eh diagrams = 2in P/T and pH/Eh diagrams = 2

1 - 1 - C SystemsC Systems

1. 1. The system SiOThe system SiO22

AC

B

Point A:F = C – P + 2F = 1 – 1 + 2F = 2

Divariant area = two variables to define a position in the coesite stability field

Two environmental variables: P and T

One component = SiO2

7 different phases

1 - 1 - C SystemsC Systems

1. 1. The system SiOThe system SiO22

AC

B

Point B:F = C – P + 2F = 1 – 2 + 2F = 1Univariant curve =one variable to define a position on the the coesite - α-quartz phase boundary

Two environmental variables: P and T

One component = SiO2

7 different phases

1 - 1 - C SystemsC Systems

1. 1. The system SiOThe system SiO22

AC

B

Point C:F = C – P + 2F = 1 – 3 + 2F = 0invariantinvariant = Triple pointdo not need any variableto define equilibrium between coesite, a- and b-quartz

Two environmental variables: P and T

One component = SiO2

7 different phases

1 - 1 - C C SystemsSystems

2. 2. The system HThe system H22OO

Point C:F = C – P + 2F = 1 – 3 + 2F = 0

Triple point

C

2 - 2 - C SystemsC Systems

1. 1. PlagioclasePlagioclase (Ab-An, NaAlSi(Ab-An, NaAlSi33OO88 - CaAl - CaAl22SiSi22OO88))

A. Systems with A. Systems with Complete Solid SolutionComplete Solid Solution

Solidus = a curve or a surfacealong which compositions of a crystalline phase are in equilibrium with a melt.

Liquidus = a curve or a surfacealong which compositions of amelt are in equilibrium with a crystalline phase.

Bulk composition of melt Bulk composition of melt

aa = An = An60 60 = 60 g An + 40 g Ab= 60 g An + 40 g Ab XXAnAn = 60/(60+40) = 0.60 = 60/(60+40) = 0.60

Point a : C = 2, environmental variable = 1, phases = 1F = C – P + v = 2 (“divariant”)(“divariant”)

Get Get new phasenew phase joining liquid: joining liquid: first crystals of first crystals of plagioclase: plagioclase: = 0.87 (point = 0.87 (point cc))

F = F = at b ?, (C= 2, P=2, v=1)at b ?, (C= 2, P=2, v=1) = 1, = 1, (“univariant”)(“univariant”)

XXAnAnplagplag

At 1450At 1450ooC, liquid C, liquid dd and plagioclase and plagioclase f f coexist at equilibriumcoexist at equilibrium

A A continuous reactioncontinuous reaction of the type:of the type:

liquidliquidAA + solid + solidBB = =

liquidliquidCC + solid + solidDD

fd e

dede efef

The lever principle:The lever principle:

Amount of liquidAmount of liquid

Amount of solidAmount of solid dede

efef==

where where dd = the = the liquidliquid composition, composition, ff = the = the solidsolid composition composition and and ee = the = the bulkbulk composition composition

liquidusliquidus

solidussolidus

When XWhen Xplagplag hh, then X, then Xplagplag = X = Xbulkbulk and, according to the and, according to the

lever principle, the amount of liquid lever principle, the amount of liquid 0 0

Thus Thus gg is the composition of the last liquid to crystallize at is the composition of the last liquid to crystallize at 13401340ooC for bulk X = 0.60C for bulk X = 0.60

Final plagioclase to form is Final plagioclase to form is ii when = 0.60 when = 0.60

Now Now P = 1P = 1 so F = 2 - 1 + 1 = 2 so F = 2 - 1 + 1 = 2

XXAnAnplagplag

Note the following:Note the following:1.1. The melt crystallized over a T range of 135 The melt crystallized over a T range of 135ooC *C *2.2. The composition of the liquid changed from The composition of the liquid changed from bb to to gg3.3. The composition of the solid changed from The composition of the solid changed from cc to to hh

Equilibrium Equilibrium meltingmelting is exactly the opposite is exactly the opposite Heat AnHeat An6060 and the first melt is and the first melt is g at Ang at An20 20 and 1340and 1340ooCC Continue heating: both melt and plagioclase change compositionContinue heating: both melt and plagioclase change composition Last plagioclase to melt is Last plagioclase to melt is c (Anc (An8787) at 1475) at 1475ooCC

Fractional crystallization:Fractional crystallization: Remove crystals as they form so they can’t Remove crystals as they form so they can’t undergo a continuous reaction with the meltundergo a continuous reaction with the meltAt any T At any T XXbulkbulk = X = Xliqliq due to the removal of the crystalsdue to the removal of the crystals

Partial Melting:Partial Melting:Remove first meltRemove first melt as forms as formsMelt XMelt Xbulkbulk = 0.60 first liquid = = 0.60 first liquid = gg

remove and cool bulk = g remove and cool bulk = g final plagioclase = final plagioclase = ii

PlagioclasePlagioclase

Liquid

LiquidLiquid

plus

Plagioclase

Note the difference between the two types of fieldsNote the difference between the two types of fields

The The blueblue fields are fields are oneone phase fields phase fields

Any point in these fields represents a true Any point in these fields represents a true phase compositionphase composition

The blank field is a The blank field is a twotwo phase field phase field

Any point in this fieldAny point in this field represents a bulk represents a bulk

composition composed of two phasescomposition composed of two phases at the at the edge of the blue fields and connected by a edge of the blue fields and connected by a

horizontal tie-linehorizontal tie-line

2-2-C Eutectic SystemsC Eutectic Systems Example: Diopside - AnorthiteExample: Diopside - Anorthite

No solid solutionNo solid solution

Cool composition Cool composition aa::bulk composition = bulk composition = AnAn7070

Cool to 1455Cool to 1455ooC (point C (point bb))

Continue cooling as XContinue cooling as Xliqliq varies along the liquidus varies along the liquidus

Continuous reactionContinuous reaction: liq: liqAA anorthite + liq anorthite + liqBB

at 1274at 1274ooC PC P = 3 = 3 so F = 2 - 3 + 1 = so F = 2 - 3 + 1 = 0 invariant0 invariant (P) T and the composition of all phases is fixed(P) T and the composition of all phases is fixed Must remain at 1274Must remain at 1274ooC as a C as a discontinuous discontinuous

reactionreaction proceeds until a phase is lost proceeds until a phase is lost

Left of the eutecticLeft of the eutectic get a similar situation get a similar situation

Note the following:Note the following:

1.1. The melt crystallizes over a T range up to ~280 The melt crystallizes over a T range up to ~280ooCC

2.2. A sequence of minerals forms over this interval A sequence of minerals forms over this interval

- And the number of minerals increases as T drops- And the number of minerals increases as T drops

3.3. The minerals that crystallize depend upon T The minerals that crystallize depend upon T

- The sequence changes with the bulk composition- The sequence changes with the bulk composition

Augite forms before Augite forms before plagioclaseplagioclase

This forms on the This forms on the leftleft side of the eutectic side of the eutectic

Gabbro of Gabbro of the the Stillwater Stillwater Complex, Complex, MontanaMontana

Plagioclase forms before Plagioclase forms before augiteaugite

This forms on the This forms on the rightright side of the eutectic side of the eutectic

OphiticOphitic texture texture

Diabase dikeDiabase dike

Also note:Also note:• The last melt to crystallize in any binary eutectic The last melt to crystallize in any binary eutectic

mixture is the mixture is the eutecticeutectic composition composition• Equilibrium Equilibrium meltingmelting is the opposite of equilibrium is the opposite of equilibrium

crystallizationcrystallization• Thus the Thus the first meltfirst melt of any mixture of Di and An of any mixture of Di and Anmust be the eutectic composition as wellmust be the eutectic composition as well

Fractional crystallization:Fractional crystallization:

The alkali feldspar phase diagram

The disordered solid solution can only exist at high temperatures.

Below the solvus the solid solution breaks down to 2 phases - one Na-rich, the other K-rich.

This exsolution process results in a 2-phase intergrowth, called perthite

aa

M

M

T

T

Composition0 20 40 60 80 100

K-FeldsparNa-Feldspar

200

400

600

800

1000

HighAb

LowAb

Disordered solidsolution

Na-feldspar + K-feldspar

Intergrowth = Perthite

Al,Si ordering

solvus

Miscibility gap

Phase diagram for the alkali feldspars

Perthite microstructure - an intergrowth of Na-feldspar in K-feldspar

Antiperthite: K-feldspar in Na-Feldspar Na-feldspar

whi

te

Cross-hatched twinningin K-feldspar

Binary Peritectic Binary Peritectic SystemSystem

Peritectic pointPeritectic point - The point on a phase diagram - The point on a phase diagram where a reaction takes place between a previously where a reaction takes place between a previously precipitated phase and the liquid to produce a new precipitated phase and the liquid to produce a new solid phase. When this point is reached, the solid phase. When this point is reached, the temperature must remain constant until the reaction temperature must remain constant until the reaction has run to completion. A peritectic is also an invariant has run to completion. A peritectic is also an invariant point in a T-x section at constant pressure.point in a T-x section at constant pressure.

Intermediate compound - A phase that has a composition intermediate between two other phases. Congruent melting - melting wherein a phase melts to a liquid with the same composition as the solid. Incongruent melting - melting wherein a phase melts to a liquid with a composition different from the solid and produces a solid of different composition to the original solid. 

Some additional terms:

For the case of incongruent melting, we will use the system forsterite (Mg2SiO4) - silica (SiO2), which has an intermediate compound, enstatite (MgSiO3). This system is a prime example of the phenomena of incongruent melting in rocks, and therefore gives insights into many aspects of mineral formation. 

Crystallization of Composition X  Composition X is a mixture of 13 wt. % SiO2 and 87 wt. % Mg2SiO4. Because this composition falls between the compositions of pure forsterite and pure enstatite, it must end its crystallization history containing only crystals of forsterite and enstatite. i.e. no quartz will occur in the final crystalline mixture. If a mixture such as composition X is taken to a temperature above its liquidus (i.e. above 1800oC in Figure 2) it will be in an all liquid state. We now trace the cooling history of composition X.

As a liquid of composition X is cooled, nothing will happen until the temperature is equal to the liquidus temperature at 1800o. At this point crystals of forsterite (Fo) begin to precipitate out of the liquid. As the temperature is further lowered, the composition of the liquid will change along the liquidus toward the peritectic (P), and the crystals forming from the liquid will always be pure Fo until P is reached. At the temperature of the peritectic, about 1580o, note that three phases must be in equilibrium, Fo, liquid, and enstatite (En). At this point some of the crystals of Fo react with the liquid to produce crystals of En. The reaction that takes place can be written as follows:Mg2SiO4 + SiO2 = 2MgSiO3Fo + liq = 2EnAfter all of the liquid is consumed by this reaction, only crystals of Fo and En will remain. The proportions of Fo and En in the final crystalline product can be found by applying the lever rule.%Fo crystals = [d/(c + d)] x 100%En crystals = [c/(c + d)] x 100At any intermediate stage in the process, such as at 1700o the proportion of all phases present (Fo and liquid in this case) can similarly be found by applying the lever rule.at 1700oC%Fo crystals = [b/(a + b)] x 100%liquid = [a/(a + b)] x 100Note that melting of composition X is exactly the reverse of crystallization. Mixture X will begin to melt at the peritectic temperature. At this point En will melt to crystals of Fo plus liquid (incongruent melting). As soon as all of the En crystals have been consumed by this reaction, the temperature can be increased until it reaches 1800o at which point all of the Fo crystals will have been consumed and the only phase left will be liquid with a composition of the starting material.

Crystallization of Composition Y Composition Y is equivalent to pure En. Thus only En may appear in the final crystalline product if perfect equilibrium is maintained.If composition Y is cooled from an all liquid state it first begins to crystallize at about 1650o. At 1650o crystals of Fo will begin to precipitate from the liquid. This will continue with further cooling until the temperature of the peritectic is reached. In this interval, the composition of the liquid must become more enriched in SiO2 and will thus change along the liquidus until it has the composition of the peritectic, P. At the peritectic temperature (1580o) all of the remaining liquid will react with all of the previously precipitated Fo to produce crystals of En. The temperature will remain constant until this reaction has gone to completion, after which the only phase present will be pure En.Thus, it can be seen that enstatite melts incongruently. If pure enstatite is heated to a temperature of 1580o it melts to Fo plus liquid.

Crystallization of Composition ZCrystallization of Composition Z Since composition Z lies between Since composition Z lies between En and SiO2, it must end up with crystals of En and Qz (Quartz). If En and SiO2, it must end up with crystals of En and Qz (Quartz). If such a composition were cooled from some high temperature where such a composition were cooled from some high temperature where it is in the all liquid state, it would remain all liquid until it reached the it is in the all liquid state, it would remain all liquid until it reached the liquidus temperature at about 1600o. At this temperature crystals of liquidus temperature at about 1600o. At this temperature crystals of Fo would begin to precipitate and the composition of the liquid Fo would begin to precipitate and the composition of the liquid would begin to change along the liquidus toward the peritectic, P. At would begin to change along the liquidus toward the peritectic, P. At P, all of the Fo previously precipitated would react with the liquid to P, all of the Fo previously precipitated would react with the liquid to produce crystals of En. After this reaction has run to completion, produce crystals of En. After this reaction has run to completion, and all of the previously precipitated Fo is consumed, there would and all of the previously precipitated Fo is consumed, there would still remain some liquid. With decreasing temperature, more crystals still remain some liquid. With decreasing temperature, more crystals of En would form, and the liquid composition would change along of En would form, and the liquid composition would change along the liquidus toward the eutectic, E. At E crystals of Qz would begin the liquidus toward the eutectic, E. At E crystals of Qz would begin to form, the temperature would remain constant until all of the liquid to form, the temperature would remain constant until all of the liquid was used up, leaving crystals of Qz and En as the final solid. Note was used up, leaving crystals of Qz and En as the final solid. Note that because composition Z lies very close to the composition of that because composition Z lies very close to the composition of pure En, the final crystalline product would consist mostly of En with pure En, the final crystalline product would consist mostly of En with a very small amount of Qz.a very small amount of Qz.For all compositions between P and 100% SiO2 the system would For all compositions between P and 100% SiO2 the system would behave in an identical fashion to the simple Eutectic system behave in an identical fashion to the simple Eutectic system discussed previously.discussed previously.

Fractional Crystallization in the System Up to this point we have always been discussing the case of equilibrium crystallization. That is all solids remain in contact with the liquid until any reaction that takes place has run to completion. As is often the case in natural systems crystals can somehow become separated from the system so that they will not react at reaction points such as P. This is the case of fractional crystallization. Under fractional crystallization conditions the cooling and crystallization histories will be drastically different. In particular, the rule that the final composition must equal the initial composition will not be followed.As an example of this phenomena we will examine the fractional crystallization of composition X. Furthermore, we will look at the case of perfect fractional crystallization. During perfect fractional crystallization of composition X all of the Fo that is precipitated will be somehow removed from the system. (In nature this can occur by crystals sinking to the bottom of the liquid due to the fact that crystals generally tend to be more dense than liquids.) Note that if only some of the crystals are removed from the liquid we will have a case intermediate between perfect fractional crystallization and equilibrium crystallization.

Cooling a liquid of composition X to the liquidus at 1800o will cause Fo to precipitate as before. With further cooling the liquid composition will change along the liquidus and more Fo will be precipitated. In this case, however, all of the Fo will be removed from the system as it crystallizes. Since the Fo is no longer present, the composition of the system will have the composition of the liquid (the Fo removed can no longer contribute to the composition of the system). Therefore, when the temperature reaches the peritectic temperature, 1580o, there will be no Fo available to react with the liquid, and the liquid (and system) will have a composition, P. Thus the liquid will now precipitate crystals of En and continue cooling to the eutectic, E, where crystals of Qz will form. The final crystalline product will consist of Qz and En. Compare this case with the previously discussed case of equilibrium crystallization of composition X. Note that under equilibrium conditions the final crystalline product of composition X contained crystals of Fo and En, while in the fractional crystallization case the final product contains En and Qz. Thus fractional crystallization has allowed an originally Fo rich composition to produce an SiO2 rich liquid and Qz appears in the final crystalline product.

If you go back and look at simple eutectic systems, or look at fractional crystallization of composition Z in the more complex system, you should be able to see that fractional crystallization will have no effect on the phases produced in the final crystalline product, but will only change the proportions of the phases produced. Fractional crystallization is only effective in producing a different final phase assemblage if there is a reaction relationship of one of the phases to the liquid.