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Exsolution and Phase Diagrams. Lecture 11. Alkali Feldspar Exsolution. ‘Microcline’ - an alkali feldspar in which Na- and K-rich bands have formed perpendicular to the twinning direction. This leads to this cross-hatched or fabric-like texture under crossed polarizers. G-bar–X and Exsolution.

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alkali feldspar exsolution
Alkali Feldspar Exsolution

‘Microcline’ - an alkali feldspar in which Na- and K-rich bands have formed perpendicular to the twinning direction. This leads to this cross-hatched or fabric-like texture under crossed polarizers.

g bar x and exsolution
G-bar–X and Exsolution
  • We can use G-bar–X diagrams to predict when exsolution will occur.
  • Our rule is that the stable configuration is the one with the lowest free energy.
  • A solution is stable so long as its free energy is lower than that of a physical mixture.
  • Gets tricky because the phases in the mixture can be solutions themselves.
inflection points
Inflection Points
  • At 800˚C, ∆Greal defines a continuously concave upward path, while at lower temperatures, such as 600˚C (Figure 4.1), inflections occur and there is a region where ∆Greal is concave downward. All this suggests we can use calculus to predict exsolution.
  • Inflection points occur when curves go from convex to concave (and visa versa).
  • What property does a function have at these points?
  • Second derivative is 0.


inflection points1
Inflection Points
  • Second derivative is:
  • First term on r.h.s. is always positive (concave up).
  • Inflection will occur when

Actual solubility gap can be less than predicted because an increase is free energy is required to begin the exsolution process.

phase diagrams
Phase Diagrams
  • Phase diagrams illustrate stability of phases or assemblages of phases as a function of two or more thermodynamic variables (such as P, T, X, V).
  • Lines mark boundaries where one assemblage reacts to form the other (∆Gr=0).
thermodynamics of melting
Thermodynamics of Melting
  • Melting occurs when free energy of melting, ∆Gm, is 0 (and only when it is 0).
  • This occurs when:

∆Gm = ∆Hm –T∆Sm

  • Hence:
  • Assuming ∆S and ∆H are independent of T:
  • where Ti,m is the freezing point of pure i, Tis the freezing point of the solution, and the activity is the activity of i in the liquid phase.

T-X phase diagram for the system anorthite-diopside.

computing an approximate phase diagram
Computing an Approximate Phase Diagram

We assume the liquid is an ideal solution (ai = Xi)

and compute

over the range of Xi

constructing t x phase diagrams from g bar x diagrams
Constructing T-X phase diagrams from G-bar–X diagrams

We can use thermodynamic data to predict phase stability, in this case as a function of temperature and composition

phase rule and phase diagrams
Phase Rule and Phase Diagrams
  • Phase rule: ƒ = c – ϕ + 2; c = 2 for a binary system.
  • Accordingly, we have ƒ = 4 – ϕ and:

PhasesFree compositional variables

  • Univariant ϕ = 3; 2 solids + liquid, 2 liquids + solid

3 solids or liquids 0

  • Divariantϕ = 2; 1 solid + 1 liquid, 2 solids, 2 liquids 0
  • Trivariantϕ = 1; 1 solid or 1 liquid 1

Trivariant System

G-bar-X diagram for a trivariant, one-phase system exhibiting complete solid solution. Need to specify P, T, and X to completely describe the system.

divariant systems
Divariant Systems
  • We need to specify both T and P (G-bar–X relevant only to that T and P). Two phases coexist on a plane in T–P–X space.
  • G-bar-X diagrams for different divariant systems
    • (a) Liquid solution plus pure solid
    • (b) Liquid solution plus solid solution
    • (c) Two pure solids
    • (d) Limited solid solution (limited liquid solution would be the same)
  • The free energy of the system as a whole is that of a mechanical mixture of phases – described by straight line through or tangent to free energies of individual phases.
  • We deduce compositions of solutions by drawing tangents between curves (or points) for phases.
univariant systems
Univariant Systems
  • One degree of freedom.
    • We specify only P or T.
    • Three phases in binary system can coexist along a line (not a plane) in P-T-X space.
    • only at one T, once we specify P (and visa versa).
  • Compositions of solutions are determined by drawing tangents.
plagioclase solution
Plagioclase Solution
  • Unlike alkali feldspar, Na-Ca feldspar (plagioclase) forms a complete solid (and liquid) solution.
  • Let’s construct the melting phase diagram from thermodynamics.
  • For simplicity, we assume both liquid and solid solutions are ideal.
plagioclase solution1
Plagioclase Solution
  • Condition for equilibrium:
    • e.g.
  • Chemical potential is
  • Combining these:
    • standard states are the pure end member solids and liquids.
plagioclase solution2
Plagioclase Solution
  • The l.h.s. is simply ∆Gm for the pure component:
  • rearranging
  • Since XAn = 1 - XAb

error in book: Ab on lhs should be An

plagioclase solution3
Plagioclase Solution
  • From:
  • Solving for mole fraction of Ab in the liquid:
  • The mole fraction of any component of any phase in this system can be predicted from the thermodynamic properties of the end-members.
  • In the ideal case, as here, it simply depends on ∆Gmand T.
  • In a non-ideal case, it would depend on Gexcessas well.
  • Computing the equation above (and a similar one for the solid), we can compute the phase diagram.