Phase Diagrams II Notes

Week 7a – Phase Diagrams II

Phase Rule in Binary T-X Sections

The phase rule is defined as:
$F + P = C + n$
Where:

$C$ = components
$P$ = phases
$F$ = degrees of freedom
$n$ or integer = intensive variables (Pressure) = 1

Therefore:
$F = C + 1 - P$
If $F = 2$ , then $F = 1$

Tools for Phase Diagrams

Phase proportions = lever rule
Phase compositions = read from axis

Given a diagram with temperature (°C) on the y-axis and composition (A and B) on the x-axis, and regions for A, B, L (liquid), and AB+L (A and B + liquid):

To calculate phase proportions:

% solid = $\frac{x{L}}{x{AB} + x_{L}} * 100$
% liquid = $\frac{x{AB}}{x{AB} + x_{L}} * 100$

Where:

$x_{melt}$ = composition of the melt
$x_{solid}$ = composition of the solid
$x_{AB}$ = composition of the solid phase
$x_{L}$ = composition of the liquid phase

Binary Example: Equilibrium Crystallization

A phase diagram is presented with temperature (°C) on the y-axis and composition on the x-axis ranging from Mg2SiO4 (Fo) to SiO2 (Qz), with MgSiO3 (En) in between.

Regions include:

Fo + L (Forsterite + Liquid)
En + L (Enstatite + Liquid)
Qz + L (Quartz + Liquid)
Fo + En (Forsterite + Enstatite)
En + Qz (Enstatite + Quartz)
L (Liquid)

Binary Example: Fractional Crystallization

A phase diagram is presented with temperature (°C) on the y-axis and composition on the x-axis ranging from Mg2SiO4 (Fo) to SiO2 (Qz), with MgSiO3 (En) in between.

Regions include:

Fo + L (Forsterite + Liquid)
En + L (Enstatite + Liquid)
Qz + L (Quartz + Liquid)
Fo + En (Forsterite + Enstatite)
En + Qz (Enstatite + Quartz)
L (Liquid)

Compositions on Ternary Diagrams

Pure A plots at A.
Compositions with 50% A.
Same for other components, e.g. A20B50C30. A, B, and C are components of the ternary diagram.

Liquidus Projections

Key features:

Liquidus surface
Ternary eutectic
Binary eutectic
Ternary cotectic

Diagram shows regions:

X + Y
Y + Z
X + m
Y + m
Z + m
m + Y

Where X, Y, and Z are end-member components, and 'm' refers to the melt phase.

Equilibrium Crystallization

Crystallization path of A:

What phase crystallizes 1st?
Melt moves directly away from crystals.
Divariant curve → fixed bulk means follows it.
@ peritectic → which 3 phase triangle is the bulk in?
As in Fo-En-An cannot produce Qz

$\frac{x{L}}{x{fo} + x_{L}} * 100 = % liquid remaining$

Where:

$x_L$ = Composition of liquid.
$x_{fo}$ = Composition of forsterite.

$F = C + 1 - P$

Equilibrium Crystallization (Continued)

Crystallization path of A:

What phase crystallizes 1st?
Melt moves directly away from crystals.
Divariant curve → fixed bulk means follows it.
Melt cannot produce any more fo, skips P.
Continues to eutectic.
Bulk in En-An-Qz triangle

$x_{fo+en}$

Equilibrium Melting

Reverse of crystallization:

What 3 phase triangle?
First melt at relevant invariant (P).
Now proceeds along divariant until.
Co-aligned with Fo and bulk (as bulk in Fo field).

Fractional Crystallization

Imagine that the “effective bulk” keeps changing as crystals are removed.
The melt basically just keeps going downhill (it will always end at eutectic if only one is present).
But also - crystals are removed! So it doesn’t (always) follow divariants.

Fractional Crystallization & Lever Rule

The lever rule tells you about the cumulates

1 fo

$\frac{x{L}}{x{fo}}$

2, 3, 4 en. When 2 phases are crystallizing, take the tangent to divariant.
$\frac{x{L}}{x{en}}$

en + an.

$X{an}$ $X{en}$

$x_{solid}$

en + an
$x_{L}$

Reading Resources

Chapter 2 Essentials of Igneous & Metamorphic Petrology (essential)
Chapter 8 Open Petrology (optional, a good resource!!)
Chapter 11 Principles of Igneous and Metamorphic Petrology (optional, more detail than required)

Practical 3: Ternary Phase Diagrams

Practice interpreting ternary phase diagrams [< 2 hours, finish in your own time].
Understand the difference between batch vs. fractional paths.
Apply the lever rule in a ternary system.
Ensure you know how to do this.