Lecture 15: Phase diagrams of three-component systems

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Lecture 15 Phase diagrams of three-component
systems

• PHYS 430/603 material
• Laszlo Takacs
• UMBC Department of Physics

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How to read concentrations from a triangular
phase diagram?cA 100 at point A, 0 along
the BC side, and has a constant value along any
line parallel to BC. If we draw a parallel to BC
through the point of interest (y) we can read the
concentration of A on either AB or AC.
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It is also customary to use rectangular
coordinate axes for two components, especially if
those compositions are low. The diagrams can be
extended to 3-d to show temperature.
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The Fe - Ni - Cr system
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Application of the Gibbs phase rule

• f n - P 2, where
• f number of degrees of freedom (or freely
  selectable parameters)
• Temperature, pressure, composition of the
  constituent phases
• The average composition of a sample is NOT a
  degree of freedom.
• n number of chemical components
• Element or compound
• P number of phases

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Single - component

• f n - P 2
• n 1
• Possible degrees of freedom T, p
• Inside an area, P 1 ? f 2. T and p can be
  chosen freely.
• On a line, 2 phases are in equilibrium, P 2 ? f
  1. Either T or p can be chosen, but it fixes
  the other variable. The relative amounts are not
  determined by the phase diagram.
• On an intersection of lines,
• 3 phases are in equilibrium,
• P 2 ? f 0. This is an invariant point
  where the parameters are fixed. How much of each
  phase is present is undetermined.
• Four phases cannot be in equilibrium.

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Two - component

• f n - P 2
• f n - P 1, if p is fixed.
• n 2
• Possible degrees of freedom
• T, p, c of constituent phases
• In a single-phase area, P 1
• ? f 2. T and c can be chosen freely.
• In a 2-phase area, P 2 ? f 1. If either T,
  c1 or c2 is chosen, the others get fixed. (E.g.
  choose c1, the composition of the Ag-based solid
  solution. That determines T and the horizontal
  line at that T determines the composition of the
  Cu-based solution.) Relative quantities by lever
  rule.
• On the invariant line at the eutectic
  temperature, 3 phases are in equilibrium, (Ag),
  (Cu), L.
• P 3 ? f 0. There is no free parameter here.

The invariant line can be pictured as a flattened
triangle containing a mix of all three phases.
With that in mind, any area is bordered by areas
with one more or one fewer phases.
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Three - component

• f n - P 1
• n 3
• Inside an empty area, P 1
• ? f 3. T and two concentration can be chosen
  freely (the third obviously follows.)
• In a striped area, P 2 ?
• f 2. T and µ can be chosen freely. Instead of
  the chemical potential, µ, the composition of one
  component is usually chosen, The stripes are
  tie lines that fix the composition of the other
  phase.
• In a shaded triangle, 3 phases are in
  equilibrium, P 3 ?
• f 1. Thus T can be chosen freely, but then
  the vertices of the triangle determine the
  concentration of each phase.

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3-d representation and projection of the liquidus
surface of a 3-component eutectic
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The Al2O3 - Fe2O3 - Fe system
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Classification of granite and its relatives