The Phase Rule and its application

Thermodynamics A system: Some portion of the universe that you wish to study The surroundings: The adjacent part of the universe outside the system 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

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

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 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 P = number of phases P = number of phases phases are mechanically separable constituents 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) The Phase Rule

F = C - P + v F = # 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 # 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

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

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

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

1 - C Systems 2. The system H 2 O Point C: F = C – P + 2 F = 1 – F = 0 Triple point C

2 - C Systems 1. Plagioclase (Ab-An, NaAlSi 3 O 8 - CaAl 2 Si 2 O 8 ) A. Systems with Complete Solid Solution Solidus = a curve or a surface along which compositions of a crystalline phase are in equilibrium with a melt. Liquidus = a curve or a surface along which compositions of a melt are in equilibrium with a crystalline phase.

Bulk composition of melt a = An 60 = 60 g An + 40 g Ab X An = 60/(60+40) = 0.60 X An = 60/(60+40) = 0.60

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

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 Anplag

At 1450 o C, liquid d and plagioclase f coexist at equilibrium A continuous reaction of the type: liquid A + solid B = liquid C + solid D liquid C + solid D

 f d e deef The lever principle: Amount of liquid Amount of solid de ef = where d = the liquid composition, f = the solid composition and e = the bulk composition and e = the bulk composition liquidus solidus

When X plag  h, then X plag = X bulk and, according to the lever principle, the amount of liquid  0 Thus g is the composition of the last liquid to crystallize at 1340 o C for bulk X = 0.60

Final plagioclase to form is i when = 0.60 Now P = 1 so F = = 2 X Anplag

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

Equilibrium melting is exactly the opposite l Heat An 60 and the first melt is g at An 20 and 1340 o C l Continue heating: both melt and plagioclase change composition l Last plagioclase to melt is c (An 87 ) at 1475 o C

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 melt undergo a continuous reaction with the melt At any T X bulk = X liq due to the removal of the crystals

Partial Melting: Remove first melt as forms Melt X bulk = 0.60 first liquid = g remove and cool bulk = g  final plagioclase = i

Plagioclase Liquid Liquid plus Plagioclase 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

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

Cool composition a: bulk composition = An 70

Cool to 1455 o C (point b)

 Continue cooling as X liq varies along the liquidus  Continuous reaction: liq A  anorthite + liq B

at 1274 o C P = 3 so F = = 0 invariant  (P) T and the composition of all phases is fixed  Must remain at 1274 o C as a discontinuous reaction proceeds until a phase is lost

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

Note the following: 1. The melt crystallizes over a T range up to ~280 o C 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

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

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

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

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 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 white Cross-hatched twinning in K-feldspar