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Phase Equilibria in Silicate Systems. Intro. Petrol. EPSC-212, Francis-14. Winter, J.D.; 2001: An Introduction to Igneous and Metamorphic Petrology. Prentice Hall, Chapter 4, 2001. Fundamentals: Units Types of units commonly used:

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Phase Equilibria in Silicate Systems

Intro. Petrol. EPSC-212, Francis-14


Winter, J.D.; 2001: An Introduction to Igneous and Metamorphic Petrology. Prentice Hall, Chapter 4, 2001.

Fundamentals: Units

Types of units commonly used:

Weight units: = gms species/ 100gms (X-Ray analytical techniques determine weight

fraction element)

Molar Units: = gms species / (molecular wt. of species), normalized to 100 (most

chemical phenomena are proportional to molecular proportions)

Atomic units: = (no. moles species) x (no. atoms per species), normalized to 100 atoms

Cation units: = (no. moles species) x (no. cations per species), typically normalized to

some total number of cations (or anions).

Oxygen units: = (no. moles species) x (no. oxygens per species), commonly normalized to

some total number of anions (closest to volume proportions)


Coordinates of Enstatite (MgSiO3) in: Mg2SiO4 - SiO2 space

Weight units: 70 30

Molecular units: 50 50

Cations units: 75 25

Oxygen units 66.7 33.3


Volatile have an importance beyond that predicted simply by their abundance


  • - Volatiles have low molecular weights:
    • H2O = 18
    • CO2 = 44
    • SiO4 = 92
    • NaAlSi3O8 = 262
  • In a melt consisting of NaAlSi3O8 clusters and H2O molecules:
    • 0.5 wt. % H20 ~ 45 mole % H2O
  • Small amounts of water produce large effects because of its low molecular wt. compared to that of a silicate magma. This effect is enhanced by the fact that at XH2O < 0.3, molecular water dissociates into 2 OH’s

fH20 ~ PH2Oα XH2O.2

H2O + Obridging 2 × OH


The Rock Forming Minerals:

A mineralis defined as a naturally occurring crystalline phase or compound that is made up of a 3-D ordered atomic arrangement or structure of different atoms.

The dominant rock forming minerals are silicates, compounds of Si and O, because Si is the most abundant metal and oxygen the most abundant anion in the Earth's crust.

Minerals can be thought of as close packings of oxygens to a first approximation, because of the large size of O, with the smaller Si and other metals occupying the interstices, or sites, between the oxygens.










Mineral Name:

Structural Formula:

Chemical Composition:


Rock Forming Minerals:

The ratio of Si to O determines the type and abundance of the silicate mineral(s) present, and the other metals distribute themselves in sites between the oxygens according to the size and charge of their cations. The amounts of different elements in any given mineral may vary somewhat, but the type and proportion of occupied sites is fixed by the structure of the silicate mineral and can be expressed by a formula of the type:

Ww Xx Yy Tt Aa

Where the capital letters stand for occupied sites of different co-ordination number and the small letters are small whole numbers.

A = anion site

W = 12 co-ordinated site

X = 8 co-ordinated site

Y = 6 co-ordinated site

T = 4 co-ordinated site


K, Rb, Ba, Na

Ca, Mn, Na

Mg, Fe, Mn, Al, Ti

Si, Al


The Mineralogical Phase Rule

In any chemical system at equilibrium, the following relationship holds:

FDegrees of Freedom = Components - Phases + 2

F equals the minimum number of variables that must be specified in order to completely define the state of a system. F is thus the variance of the system, the number of unknowns.

C equals the number of independent chemical components needed to define the composition of the system.

Pequals the number of physical phases present in the system, which include the number of solid minerals, plus liquid and gas phases, if present.

2 represents the variables pressure and temperature.


Single Component Systems:

  • SiO2
  • When a solid consist of 2 coexisting minerals (phases):
    • F = C – P + 2 = 1 - 2 + 2 = 1
  • Such a system is invariant at any given pressure, and thus a single component solid phase will melt at 1 unique temperature at any specified pressure. The boundary between the 2 phases in P - T space will be a univariant line with a slope approximated by:
    • dG = - SdT + VdP = 0
    • dP/dT = S/ V
  • This is also true for solid - liquid phase boundaries because, to a first approximation, Ho and So are constant for small changes in temperature (true for all reactions not involving a relatively compressible vapour phase).
Two Component Systems:

Mg2SiO4 – SiO2

Pure forsterite melts at 2163oC at 1 atm. If extra SiO2 is added to the

system, SiO2 will be present only in the melt phase, while

forsterite will remain a pure phase. The temperature at which

forsterite crystallizes is now an inverse function of SiO2 content:

at equilibrium:

GFoOl = GFoLiqand GFo = 0

GoFo + R × T × Ln(aFoOl) = GoFoLiq + R × T × Ln(aFoLiq)

GoFo = - R × T × Ln(aFoLiq/aFoOl)


GoFo = - R × T × Ln(1-XSiO2),

aFoOl = 1.0, assume activity (a) = mole fraction X)


GoFo = HoFo – T × SoFo = HoFo – T × HoFo / TFo

If Ho and So are insensitive to small changes in T & P, then:

HoFo – T × HoFo / TFo = - R × T × Ln(1-XSiO2)


Ln(1-XSiO2) = HoFo / R × (T - TFo) / (T×TFo)

van't Hoff equation for melting point depression


Two Component Systems:

Binary Phase Diagrams

  • If a system is comprised of 2 components, then where a solid and liquid phase coexist:
    • F = 2 - 2 + 2 = 2
  • Such a system is univariant at any given pressure, and thus the melting point of a solid will depend on the proportions of the two components. In the absence of extensive solid solution, the presence of an additional component will reduce the melting temperature of single component solid phases because the additional component typically dissolves preferentially in the liquid phase.


No solid-solution

Between end-members

The Eutectic point “e” of a two component system is invariant (F = 0, if pressure fixed) and is defined by the intersection of two univariant (F = 1) liquidus curves, originating from the melting temperatures of the two pure end-member phases.


Peritectic versus Eutectic

Invariant points



p - oliv + liq opx

e1 - liq albite + qtz

e2 - liq neph + albite

e - liq opx + qtz

the lever rule
The Lever Rule


T3 : Forst / liquid :b / a; Forst / whole = b /(a+b)

For bulk

Composition X

T2: Forst / liquid :c / a; Forst / whole = c /(a+c)

T1: Forst / Enst : d / a; Forst / whole = d /(a+d)


Cumulate Rocks versus Rocks

that represent liquids








Fractional Crystallization vs Partial Melting

Upon cooling to 1557oC, early crystallized olivine exhibits a reaction relationship with the residual liquid of composition “p” to form orthopyroxene. Either olivine or melt must disappear before cooling can continue. During partial melting, orthopyroxene begins to melt incongruently at 1557oC to form olivine plus a liquid of composition “p”. Orthopyroxene must be consumed before the temperature can increase.


The presence of other components in solid solution at levels that are insufficient to stabilize a separate phase destroys the invariant nature of melting.

The temperature and composition of the first melt are determined by the amount of the additional component. During partial melting, these additional components are typically the first to be refined out into the melt.


Bianary Systems with extensive Solid Solution:

  • Olivine exhibits complete solid solution between the forsterite (Mg2SiO4) and fayalite (Fe2SiO4) end-members. In Fe and Mg bearing systems, neither the olivine solid nor the olivine liquid are pure end-member components:
  • We now have two van't Hoff equations:
  • Ln(XFoLiq / XFoOl) = HoFo / R × (T -TFo) / (T × TFo)
  • Ln(XFaLiq / XFaOl) = HoFa /R × (T -TFa) /(T × TFa)
  • Because:XFaLig = 1-XFoLiq andXFaOl = 1-XFoOl
  • Then:
  • Ln(XFoLiq / XFoOl) = HoFo / R × (T -TFo) / (T × TFo)
  • Ln(1-XFoLiq / (1-XFoOl)) = HoFa /R × (T -TFa) / (T × TFa)
  • The choice of any T betweenTFo and TFa will enable the calculation of the compositions of the coexisting olivine and liquid for that T, and thus the solidus and liquidus at any T. Exactly analogous solid solution relationships can be developed for the plagioclase series feldspars:
    • anorthiteCaAl2Si2O8 - albite NaAlSi3O8

Ternary Systems: Forsterite – Diopside – Anorthite

Liquidus Projection

In order to portray the magmatic phase relations of systems with more than two chemical components, we need to develop specialized projection schemes.

Three component systems can be represented on a two dimensional sheet of paper, if we project only those phase relationships for which a magmatic liquid is present.

P = 1 atm


Ternary Systems: Forsterite – Diopside - Anorthite

A Liquid of bulk composition X cools to the olivine liquidus surface at 1600°C, at which point Forsterite begins to crystalize

P = 1 atm

The liquid composition moves directly away from Fo, producing a dunite cumulate, until it reaches the cotectic, at which point Diopside begins to crystallize with Forsterite.



Forsterite – Diopside - Anorthite

P = 1 atm

Co-precipitation of Forsterite + Diopside causes liquid composition to move down cotectic curve, producing a wehrlite cumulate.



Forsterite – Diopside - Anorthite

P = 1 atm

The liquid composition reaches the ternary eutectic at 1270°C, at which point Anorthite begins to crystallize with Diopside and Forsterite, producing a gabbroic cumulate.

The composition of the liquid remains at the eutectic point until all the liquid is consumed.



Ternary Systems

  • The composition of the first melt of an assemblage ABD is that of invariant eutectic point eABD, while the composition of the first melt of assemblage DBC is that of invariant eutectic point eDBC.
  • The intersection of a univariant curve with the Alkemade line joining the compositions of the coexisting solid phases defines a thermal maximum along the univariant curve.

Ternary Systems


Solid Solution:


Oliv - Cpx - Qtz Liquidus Projection:

The invariant point “p”, at which olivine, clinopyroxene and orthopyroxene coexist with a liquid, is a peritectic point because it lies outside of the compositional volume of the solid phases. It represents the first melt of any assemblage consisting of olivine, opx, and cpx (mantle peridotite) at 1 atm., and is analogous in composition to a quartz-normative basalt.

Similarly, the univariant curve along which olivine and orthopyroxene coexist with a liquid is a reaction curve because the tangent to the curve at any point cuts the olivine - orthopyroxene Alkemade line with a negative olivine intercept.

The invariant point “e” is a eutectic and represents the composition of the first melt of an assemblage of quartz-diopside-orthopyroxene. The composition of “e” approximates that of the Earth’s continental crust.


Mantle Ocean Continent

crust crust

SiO2 45.2 49.4 60.3

TiO2 0.7 1.4 1.0

Al2O3 3.5 15.4 15.6

MgO 37.5 7.6 3.9

FeO 8.5 10.1 7.2

CaO 3.1 12.5 5.8

Na2O 0.6 2.6 3.2

K2O 0.1 0.3 2.5

Total 99.2 99.3 99.5

Cations normalized to 100 cations

Si 38.5 46.1 56.4

Ti 0.5 1.0 0.7

Al 3.6 16.9 17.2

Mg 47.6 10.6 5.4

Fe 6.0 7.9 5.6

Ca 2.8 12.5 5.8

Na 0.9 4.7 5.8

K 0.1 0.5 3.0

O 140.2 153.0 161.3

Mineralogy (oxygen units, XFe3+ = 0.10)

Quartz 0.0 0.0 13.0

Feldspar 13.2 57.3 64.3

Clinopyroxene 6.7 25.7 5.9

Orthopyroxene 18.3 4.1 14.7

Olivine 59.9 9.9 0.0

Oxides 1.8 3.0 2.0

Oceanic crust - MORB basalt p

Continental crust - granite e


Liquidus Projections for haplo-basalts

The Basalt Tetrahedron at 1 atm:

The olivine - clinopyroxene - plagioclaseplane is a thermal divide in the haplo-basalt system at low pressures and separates natural magmas into two fundamentally different magmatic series. Sub-alkaline basaltic magmas with compositions to the Qtz-rich side of the plane fractionate towards Qtz-saturated residual liquids, such as rhyolite. Alkaline basaltic magmas with compositions to the Qtz-poor side of the plane fractionate towards residual liquids saturated in a feldspathoid, such as nepheline phonolite.


Since the dominant mineral in the mantle source of basaltic magmas is olivine, we can achieve a further simplification by projecting the liquidus of basaltic systems from the perspective of olivine:


Alkaline basalts fall to the Foid-side of the olivine-clinopyroxene-plagioclase plane (1 atm thermal divide) and fractionate to foid-saturated residual liquids. Sub-alkaline basalts fall to the Quartz-side and fractionate towards quartz-saturated residual liquids.

Alkaline basaltic lavas are volumetrically insignificant (~1%), but strongly enriched in highly incompatible trace elements profiles compared to sub-alkaline lavas, and low in HREE, Y, & Sc. These characteristics are generally ascribed to small degrees of partial melting at elevated pressures, leaving garnet as a phase in the refractory residue.


The Effect of Pressure

1 atm

Increasing pressure shifts the oliv-cpx-opx peritectic point towards less Si-rich compositions. At approximately 10 kbs this invariant point moves into the oliv - cpx- opx compositional volume, and the first melt of the mantle has an olivine basalt composition. The invariant point is still a peritectic point, however, because of the extensive solid solution of cpx towards opx. At pressures exceeding 15-20 kbs, this invariant point moves outside the simple olivine - cpx - qtz system, into the Neph-normative volume of the basalt tetrahedron. The first melt of mantle peridotite is an alkaline olivine basalt at these high pressures.


Since the dominant mineral in the mantle source of basaltic magmas is olivine, we can achieve a further simplification by projecting the liquidus of basaltic systems from the perspective of olivine:

Movement of the invariant point determining the composition of the first melt with increasing pressure.