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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.

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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

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

• First, highlight the boundary between A and C.

And AB and B.

A and AB

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

C

A BAB

C

A ABB

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

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Congruently Melting Compounds

C

A BAB

C

A ABB

• 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.

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Congruently Melting Compounds

C

A BAB

C

A ABB

This Alkemades line drawn from C to AB is a true binary join because it crosses the boundary line that its end components share.

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Congruently Melting Compounds Compatibility Triangles

C

A BAB

C

A ABB

•Alkemades lines divide a phase diagram into compatibility triangles.

•This phase diagram is divided into two triangles.

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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.

C

A BAB

C

A ABB

m

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Congruently Melting Compounds Temperature Arrows

C

A BAB

C

A ABB

This point is a maximum in temperature on the boundary line so the temperature arrows will point away from the point.

mThis 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.

C

A BAB

C

A ABB

mm

m

m m

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C

A BAB

C

A ABB

mm

m

m m

Congruently Melting Compounds

To finish the diagram the binary invariant points are given their appropriate labels.

And the ternary invariant points are labeled.

E

e e

e e

EThe diagram is finished.

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Incongruently Melting Compounds

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

C

A BAB

C

A ABB

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Incongruently Melting Compounds

C

A BAB

C

A ABB

• 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.

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Incongruently Melting Compounds Compatibility Triangles

•Alkemades lines divide a phase diagram into compatibility triangles.

•This phase diagram is divided into two triangles.

C

A BAB

C

A ABB

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m

Incongruently Melting Compounds Alkemades’ Theorem- Temp Arrows

AB

C

A B

C

A ABB

• Begin by labeling the max/min points.

Not an “m” point, the boundary line does not cross the Alkemades line it is associated with.

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

mm

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C

A B

C

A ABB

AB

mm

m

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

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 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.

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•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.

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

C

A B

C

A ABB

mm

m

Incongruently Melting Compounds Alkemades’ Theorem- Temp Arrows

AB

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

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

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AB

C

A B

C

A ABB

mm

m

• 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

e

p

Incongruently Melting Compounds

• And the ternary invariant points are labeled.

PE

• The diagram is finished.

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Congruently Melting Ternary Compounds

C

A B

AB

C

A

B

• 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

C

A B

AB

C

A

B

•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

A B

AB

C

A

B

mm

m

m

mm

C

• Begin by labeling the max/min points.

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

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Congruently Melting Ternary Compounds

A B

AB

C

A

B

mm

m

m

mm

C

• The binary invariant points are given their appropriate labels.

e

• And the ternary invariant points are labeled.

E• The diagram is

finished. E

E

ee

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Incongruently Melting Ternary Compounds- Single Incongruency

C

A B

ABCC

A

B

ABC

• 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.

To Table of Contents:

C

A B

ABCC

A

B

ABC

•Alkemades lines divide a phase diagram into compatibility triangles.

•This phase diagram is divided into three triangles.

Incongruently Melting Ternary Compounds Compatibility Triangles

To Table of Contents:

C

A B

ABCC

B

ABC

A

mm

m

Incongruently Melting Ternary Compounds

• Begin by labeling the max/min points.

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

To Table of Contents:

C

A B

ABCC

B

ABC

A

mm

m

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.

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

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C

A B

ABCC

B

ABC

A

mm

m

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.

To Table of Contents:

C

A B

ABCC

B

ABC

A

mm

m

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.

To Table of Contents:

C

A B

ABCC

B

ABC

A

mm

m

e e

e

PP

E

Incongruently Melting Ternary Compounds

• The binary invariant points are given their appropriate labels.

• And the ternary invariant points are labeled.

• The diagram is finished.

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

To Table of Contents:

C

A B

C

A

BABC

ABC

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

Incongruently Melting Ternary Compounds-Double Incongruency

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C

A B

C

A

BABC

ABC

•Alkemades lines divide a phase diagram into compatibility triangles.

•This phase diagram is divided into three triangles.

Incongruently Melting Ternary Compounds Compatibility Triangles

To Table of Contents:

C

A B

C

A

BABC

ABC

mm

m

m

Incongruently Melting Ternary Compounds

• Begin by labeling the max/min points.

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

To Table of Contents:

C

A B

C

A

BABC

ABC

mm

m

m

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.

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• The binary invariant points are given their appropriate labels.

C

A B

C

A

BABC

ABC

mm

m

m

Pe

E

Ee

e

Incongruently Melting Ternary Compounds

• And the ternary invariant points are labeled.

• The diagram is finished.

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