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Exsolution and Phase Diagrams Lecture 11

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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.

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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.

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Inflection Points At 800˚C, ∆G real 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 ∆G real 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. Albite-Orthoclase

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

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

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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 (∆G r =0).

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Thermodynamics of Melting Melting occurs when free energy of melting, ∆G m, is 0 (and only when it is 0). This occurs when: ∆G m = ∆H m –T∆S m Hence: Assuming ∆S and ∆H are independent of T: where T i,m is the freezing point of pure i, T is 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.

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Computing an Approximate Phase Diagram We assume the liquid is an ideal solution (a i = X i ) and compute over the range of X i

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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

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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 liquids0 Divariant ϕ = 2; 1 solid + 1 liquid, 2 solids, 2 liquids0 Trivariant ϕ = 1; 1 solid or 1 liquid1 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. Trivariant System

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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 o (a) Liquid solution plus pure solid o (b) Liquid solution plus solid solution o (c) Two pure solids o (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.

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Univariant Systems One degree of freedom. o We specify only P or T. o Three phases in binary system can coexist along a line (not a plane) in P-T-X space. o only at one T, once we specify P (and visa versa). Compositions of solutions are determined by drawing tangents.

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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.

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Plagioclase Solution Condition for equilibrium: o e.g. Chemical potential is Combining these: o standard states are the pure end member solids and liquids.

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Plagioclase Solution The l.h.s. is simply ∆G m for the pure component: rearranging Since X An = 1 - X Ab error in book: Ab on lhs should be An

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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 ∆G m and T. In a non-ideal case, it would depend on G excess as well. Computing the equation above (and a similar one for the solid), we can compute the phase diagram.

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Lecture 12: Phase diagrams PHYS 430/603 material Laszlo Takacs UMBC Department of Physics.

Lecture 12: Phase diagrams PHYS 430/603 material Laszlo Takacs UMBC Department of Physics.

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