Alkemades Lines

1
Alkemades Lines

• Phase Equilibria
• Ceramic Engineering 251

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Alkemade Lines

• Alkemades lines are a subset of Binary Join
  Lines, that connect the primary compositions of
  two phase fields that share a common boundary.
• Alkemades lines divide a phase diagram into
  compatibility triangles.
• Alkemades Theorem can be used to determine the
  direction of decreasing temperature on a phase
  diagram.

3
Alkemade LinesTable of Contents

• Congruently Melting Compounds
• Drawing Alkemades Lines
• Compatibility Triangles
• Alkemades Theorem - Drawing Temperature Arrows
• Incongruently Melting Compounds
• Congruently Melting Ternary Compounds
• Incongruently Melting Ternary Compounds - Single
  Incongruency
• Incongruently Melting Ternary Compounds - Double
  Incongruency
• The End

4
Congruently Melting Compounds
Each boundary line on the diagram has an
Alkemades line associated with it. To determine
which Alkemades line goes with each of the
boundary lines.

• First, highlight the boundary between A and C.

• Then, highlight the Alkemades line that connects
  the compositions A and C, in the same color.

• This can also be done for the boundary lines
  connecting C and B,

A and AB
And AB and B.
5
Congruently Melting Compounds

• The Alkemades line that is associated with the
  boundary line between C and AB is not as obvious.

• It must be drawn to connect the compositions of A
  and C.

6
Congruently Melting Compounds
This Alkemades line drawn from C to AB is a true
binary join because it crosses the boundary line
that its end components share.
7
Congruently Melting Compounds Compatibility
Triangles

• Alkemades lines divide a phase diagram into
  compatibility triangles.
• This phase diagram is divided into two triangles.

8
Congruently Melting Compounds Alkemades Theorem
The intersection of a boundary line (or extended
boundary line) with its corresponding Alkemades
line (or extended Alkemades line) is a maximum in
temperature on the boundary line and a minimum in
temperature on the Alkemades line.

• This theorem can be used to draw arrows on the
  diagram showing the direction of decreasing
  temperature.
• The point where a corresponding Alkemades and
  boundary line meet is a maximum and a minimum, it
  is labeled m.

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Congruently Melting Compounds Temperature Arrows
To draw the temperature arrows on this diagram...
begin by labeling point where the boundary
between A and C crosses the Alkemades line
between B and C as a m point.
m
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Congruently Melting Compounds Temperature Arrows
This point is a maximum in temperature on the
boundary line so the temperature arrows will
point away from the point.
m
This point is a minimum in temperature on the
Alkemades line so the arrows will point toward
the point.
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Congruently Melting Compounds Temperature Arrows
We can go through the same process for the other
lines on the diagram.
m
m
m
m
12
Congruently Melting Compounds
To finish the diagram the binary invariant
points are given their appropriate labels.
e
e
And the ternary invariant points are labeled.
E
E
The diagram is finished.
e
e
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Incongruently Melting Compounds
C

• Begin by highlighting each boundary line and the
  Alkemades line associated with it.

C
B
A
AB
B
A
AB
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Incongruently Melting Compounds
C

• The Alkemades line between C and AB, in this
  diagram, is a Non-binary Join because it does not
  cross the boundary line that its end components
  share.

C
B
A
AB
B
A
AB
15
Incongruently Melting Compounds Compatibility
Triangles

• Alkemades lines divide a phase diagram into
  compatibility triangles.
• This phase diagram is divided into two triangles.

16
Incongruently Melting Compounds Alkemades
Theorem- Temp Arrows

• Begin by labeling the max/min points.

• Then draw the temperature arrows on the boundary
  and Alkemades lines.

m
m
m
AB
Not an m point, the boundary line does not
cross the Alkemades line it is associated with.
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Incongruently Melting Compounds Alkemades
Theorem- Temp Arrows

• The point where the two lines intersect can be
  used the same way the other intersection points
  are. The temperature arrows on the Alkemades
  line will go towards the point. The arrows on
  the boundary line will go away from the point.

• The AB and B boundary line and alkemade line do
  not meet. To correct for this, draw a temporary
  extention of the alkemade line.

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Incongruently Melting Compounds Alkemades
Theorem- Temp Arrows

• The boundary line for C and AB can be drawn in
  the same way it was on the previous diagram.

• First draw an extension of the boundary line so
  that it will intersect the Alkemades line.

• Then use the point where the lines meet as a
  guide for drawing in the temperature arrows.

AB
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Incongruently Melting Compounds

• Then the extension lines are removed, leaving
  only the Alkemades and boundary lines with their
  temperature arrows.

e

• The binary invariant points are given their
  appropriate labels.

e
P

• And the ternary invariant points are labeled.

E

• The diagram is finished.

AB
e
p
20
Congruently Melting Ternary Compounds

• Begin by highlighting each boundary line and the
  Alkemades line associated with it.

• Then draw in the Alkemades lines for the three
  remaining boundary lines.

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Congruently Melting Ternary Compounds
Compatibility Triangles

• Alkemades lines divide a phase diagram into
  compatibility triangles.
• This phase diagram is divided into three
  triangles.

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Congruently Melting Ternary Compounds
C

• Begin by labeling the max/min points.

• Then draw the temperature arrows on the boundary
  and Alkemades lines.

m
m
m
m
m
m
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Congruently Melting Ternary Compounds
C

• The binary invariant points are given their
  appropriate labels.

• And the ternary invariant points are labeled.

m
m
e
e

• The diagram is finished.

m
E
E
m
m
E
m
e
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Incongruently Melting Ternary Compounds- Single
Incongruency

• Begin by highlighting each boundary line and the
  Alkemades line associated with it.

• Then draw in the Alkemades lines for the three
  remaining boundary lines.

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Incongruently Melting Ternary Compounds
Compatibility Triangles

• Alkemades lines divide a phase diagram into
  compatibility triangles.
• This phase diagram is divided into three
  triangles.

26
Incongruently Melting Ternary Compounds

• Begin by labeling the max/min points.

• Then draw the temperature arrows on the boundary
  and Alkemades lines.

m
m
A
m
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Incongruently Melting Ternary Compounds

• The ABC and C boundary line and Alkemades line do
  not cross. To correct for this, draw a temporary
  extension of the Alkemades line.

m
28
Incongruently Melting Ternary Compounds

• The point where the two lines intersect can be
  used the same way the other intersection points
  are. The temperature arrows on the Alkemades
  line will go towards the point. The arrows on
  the boundary line will go away from the point.

m
29
Incongruently Melting Ternary Compounds

• We can go through the same process for the A and
  ABC boundary line and the B and ABC boundary line.

m
30
Incongruently Melting Ternary Compounds

• Then the extension lines are removed, leaving
  only the Alkemades and boundary lines with their
  temperature arrows.

• The binary invariant points are given their
  appropriate labels.

• And the ternary invariant points are labeled.

e
e
P
P

• The diagram is finished.

E
m
e
31
Incongruently Melting Ternary Compounds-Double
Incongruency

• Begin by highlighting each boundary line and the
  Alkemades line associated with it.

C
C
ABC
B
ABC
A
B
A
32
Incongruently Melting Ternary Compounds
Compatibility Triangles

• Alkemades lines divide a phase diagram into
  compatibility triangles.
• This phase diagram is divided into three
  triangles.

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Incongruently Melting Ternary Compounds
C

• Begin by labeling the max/min points.

• Then draw the temperature arrows on the boundary
  and Alkemades lines.

C
m
m
ABC
B
ABC
m
A
B
A
m
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Incongruently Melting Ternary Compounds

• Draw an extension of the ABC and C boundary line
  to use when putting on the temperature arrows.

• An extension of the ABC and A boundary can be
  also be drawn to use when putting on the
  temperature arrows.

m
35
Incongruently Melting Ternary Compounds

• Then the extension lines are removed, leaving
  only the Alkemades and boundary lines with their
  temperature arrows.

• The binary invariant points are given their
  appropriate labels.

e
e
P

• And the ternary invariant points are labeled.

E
m

• The diagram is finished.

E
e