1. The Phase Rule and its application

2. Thermodynamics A system: The surroundings:
Some portion of the universe that you wish to study
The surroundings:
The adjacent part of the universe outside the system
Closed system: only exchange of mechanical and
thermal energy, no mass exchange
Open system: exchange of energies and mass
a system:
Some portion of the universe that you wish to study
a glass of water
plagioclase
the mantle
Changes in a system are associated with the transfer of energy
lift an object: stored chemical  potential
drop an object: potential  kinetic
pump up a bicycle tire: chemical  mechanical  heat (friction + compression)
add an acid to a base: chemical  heat
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

3. Thermodynamics A phase diagram:
P, T Shows the stability ranges of phases (minerals, melts, solutions)
as functions of composition,, pH and Eh.
Components: Minimum number of chemical constituents that
are required to describe the compositions of all phases in the 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

4. The Gibbs Phase Rule F = C - P + v F = number of degrees of freedom
The number of variables to define
the position of a mineral assemblage in a phase diagram

5. The Phase Rule F = C – P + v F = number of degrees of freedom
The number of variables to define
the position of a mineral assemblage in a phase diagram
P = number of phases
phases are mechanically separable constituents

6. The Phase Rule F = C - P + v F = Number of degrees of freedom
The number of variables to define
the position of a mineral assemblage in a phase diagram
P = number of phases
phases are mechanically separable constituents
C = minimum number of components (chemical constituents that must be specified in order to define all phases)

7. The Phase Rule F = # degrees of freedom F = C - P + v
The number of variables to define
the position of a mineral assemblage in a phase diagram
P = number of phases
phases are mechanically separable constituents
C = minimum # of components (chemical
constituents that must be specified in order to define all phases)
v = intensive properties or environmental variables,
in P/T and pH/Eh diagrams = 2

8. 1 - C Systems 1. The system SiO2 Two environmental variables: P and T
One component = SiO2
7 different phases
Point A:
F = C – P + 2
F = 1 – 1 + 2
F = 2
Divariant area
= two variables to
define a position in
the coesite stability field A C B

9. 1 - C Systems 1. The system SiO2 Two environmental variables: P and T
One component = SiO2
7 different phases
Point B:
F = C – P + 2
F = 1 – 2 + 2
F = 1
Univariant curve =
one variable to
define a position on the
the coesite - α-quartz
phase boundary A C B

10. 1 - C Systems 1. The system SiO2 Two environmental variables: P and T
One component = SiO2
7 different phases
Point C:
F = C – P + 2
F = 1 – 3 + 2
F = 0
invariant = Triple point
do not need any variable
to define equilibrium
between coesite,
a- and b-quartz A C B

11. 1 - C Systems 2. The system H2O Point C: F = C – P + 2 F = 1 – 3 + 2
Triple point C

12. 2 - C Systems A. Systems with Complete Solid Solution
1. Plagioclase (Ab-An, NaAlSi3O8 - CaAl2Si2O8)
Liquidus = a curve or a surface
along which compositions of a
melt are in equilibrium with a
crystalline phase.
Solidus = a curve or a surface
along which compositions of
a crystalline phase are in
equilibrium with a melt.
T - X diagram at constant P = 0.1 MPa
F = C - phi since P is constant: lose 1 F

13. Bulk composition of melt a = An60 = 60 g An + 40 g Ab
XAn = 60/(60+40) = 0.60
Example of cooling with a specific bulk composition
Point a: at 1560oC
phi = ? (1)
F = ? = C - phi + 1 = = 2

14. Point a : C = 2, environmental variable = 1, phases = 1
F = C – P + v = 2
(“divariant”)
Intensive variables can be any ones (P, T, X, density, molar G-V-S, etc. as far as the Phase Rule is concerned
Geologically, it’s best to choose T and X (P is constant)
Thus F = T and X(An)Liq is a good choice
Because X(Ab)Liq = 1 - X(An)Liq they are not independent, so may pick either but not both

15. X Get new phase joining liquid:
first crystals of plagioclase: = 0.87 (point c)
F = at b ?, (C= 2, P=2, v=1) = 1, (“univariant”) X An
plag
F now = 1

16. At 1450oC, liquid d and plagioclase f coexist at equilibrium
A continuous reaction of the type:
liquidA + solidB =
liquidC + solidD
As cool, Xliq follows the liquidus and Xsolid follows the solidus, in accordance with F = 1
At any T, X(An)Liq and X(An)Plag are dependent upon T
We can use the lever principle to determine the proportions of liquid and solid at any T

17. The lever principle: = D Amount of liquid Amount of solidef =
Amount of solid de
where d = the liquid composition, f = the solid composition
and e = the bulk composition d e f D
liquidus de ef
solidus

18. When Xplag ® h, then Xplag = Xbulk and, according to the lever principle, the amount of liquid ® 0
Thus g is the composition of the last liquid to crystallize at 1340oC for bulk X = 0.60

19. X Final plagioclase to form is i when = 0.60An
plag
Final plagioclase to form is i when = 0.60
Now P = 1 so F = = 2

20. Note the following:
1. The melt crystallized over a T range of 135oC *
2. The composition of the liquid changed from b to g
3. The composition of the solid changed from c to h

21. Equilibrium melting is exactly the opposite
Heat An60 and the first melt is g at An20 and 1340oC
Continue heating: both melt and plagioclase change composition
Last plagioclase to melt is c (An87) at 1475oC

22. Fractional crystallization: Remove crystals as they form so they can’t
undergo a continuous reaction with the melt
At any T Xbulk = Xliq due to the removal of the crystals
Thus liquid and solid continue to follow liquidus and solidus toward low-T (albite) end of the system
Get quite different final liquid (and hence mineral) composition

23. Remove first melt as forms Melt Xbulk = 0.60 first liquid = g
Partial Melting:
Remove first melt as forms
Melt Xbulk = 0.60 first liquid = g
remove and cool bulk = g ® final plagioclase = i
Fractional crystallization and partial melting are important processes in that they can cause significant changes in the final rock that crystallizes beginning with the same source rock.

24. Note the difference between the two types of fields
The blue fields are one phase fields
Any point in these fields represents a true phase composition
The blank field is a two phase field
Any point in this field represents a bulk composition composed of two phases at the edge of the blue fields and connected by a horizontal tie-line
Plagioclase
Liquid
plus

25. 2-C Eutectic Systems
Example: Diopside - Anorthite
No solid solution

26. Cool composition a:
bulk composition = An70
At a: phi = 1
F = C - phi + 1 = = 2
Good choice is T and X(An)Liq

27. What happens at b? Pure anorthite forms (point c) phi = 2
Cool to 1455oC (point b)
What happens at b? Pure anorthite forms (point c)
phi = 2
F = = 1
composition of all phases determined by T
Xliq is really the only compositional variable in this particular system

28. Continue cooling as Xliq varies along the liquidus
Continuous reaction: liqA ® anorthite + liqB
Lever rule shows less liquid and more anorthite as cooling progresses

29. at 1274oC P = 3 so F = 2 - 3 + 1 = 0 invariant
(P) T and the composition of all phases is fixed
Must remain at 1274oC as a discontinuous reaction proceeds until a phase is lost
What happens at 1274oC?
get new phase
Pure diopside g joins liquid d and anorthite h
d is called the eutectic point: MUST reach it in all cases
d between g and h, so reaction must be:
diopside + anorthite = liquid
must run right -> left as cool (toward low entropy)
therefore first phase lost is liquid
Below 1274oC have diopside + anorthite
phi = 2 and F = = 1

30. Left of the eutectic get a similar situation
Cool An oC (e) and diopside forms first at f
Continue cooling: continuous reaction with F = 1 as liqA -> liqB + diopside
At 1274oC anorthite (h) joins diopside (g) + liquid (d)
Stay at 1274oC until liquid consumed by discontinuous reaction: liquid -> diopside + anorthite

31. Note the following:
1. The melt crystallizes over a T range up to ~280oC
2. A sequence of minerals forms over this interval
- And the number of minerals increases as T drops
3. The minerals that crystallize depend upon T
- The sequence changes with the bulk composition

32. Augite forms before plagioclase
Gabbro of the Stillwater Complex, Montana
This forms on the left side of the eutectic

33. Plagioclase forms before augite
Ophitic texture
Diabase dike
This forms on the right side of the eutectic

34. Also note:
The last melt to crystallize in any binary eutectic mixture is the eutectic composition
Equilibrium melting is the opposite of equilibrium crystallization
Thus the first melt of any mixture of Di and An must be the eutectic composition as well

35. Fractional crystallization:
Since solids are not reactants in eutectic-type continuous reactions, the liquid path is not changed
Only the final rock will differ: = eutectic X, and not bulk X

36. The alkali feldspar phase diagram
Na-feldspar + K-feldspar
Intergrowth = Perthite
Al,Si ordering
solvus
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
Miscibility gap

37. Phase diagram for the alkali feldspars

38. Cross-hatched twinning
Perthite microstructure - an intergrowth of Na-feldspar in K-feldspar
Antiperthite: K-feldspar in Na-Feldspar
Cross-hatched twinning
in K-feldspar
Na-feldspar
white

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

40. Some additional terms:
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.  
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.  

42. 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.

43. 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 = 2MgSiO3
Fo + liq = 2En
After 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 100
At 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 100
Note 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.

44. 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.

45. Crystallization of Composition Z Since composition Z lies between 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 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 Fo would begin to precipitate and the composition of the liquid 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 produce crystals of En. After this reaction has run to completion, and all of the previously precipitated Fo is consumed, there would still remain some liquid. With decreasing temperature, more crystals of En would form, and the liquid composition would change along 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 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 pure En, the final crystalline product would consist mostly of En with a very small amount of Qz.
For all compositions between P and 100% SiO2 the system would behave in an identical fashion to the simple Eutectic system discussed previously.

46. 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.

47. 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.

48. 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.