1. Phase Equilibria Ceramic Engineering 251
Alkemades Lines
Phase Equilibria
Ceramic Engineering 251

2. 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 Lines Table 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…. C A B AB
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. C A B AB
It must be drawn to connect the compositions of A and C.

6. Congruently Melting CompoundsA B AB
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. C A B AB

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”.

9. 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 B AB m

10. 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. C A B AB m
This point is a minimum in temperature on the Alkemades line so the arrows will point toward the point.
Dot and M should show up as closely together as possible.
All arrows on a line should come in together.
Minimum

11. Congruently Melting Compounds Temperature Arrows
We can go through the same process for the other lines on the diagram. C A B AB m m
Arrows look best going toward where they are pointing - in their entrance.
Again...M and dot together m m m

12. Congruently Melting Compounds
To finish the diagram the binary invariant points are given their appropriate labels. C A B AB m e e
And the ternary invariant points are labeled. E E
The diagram is finished. e e

13. Incongruently Melting Compounds
Begin by highlighting each boundary line and the Alkemades line associated with it. C B A AB A B AB

14. Incongruently Melting Compounds
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 A B AB

15. Incongruently Melting Compounds Compatibility Triangles
Alkemades lines divide a phase diagram into compatibility triangles.
This phase diagram is divided into two triangles. C A B AB

16. Incongruently Melting Compounds Alkemades’ Theorem- Temp ArrowsB AB
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.

17. 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. C A B AB m

18. Incongruently Melting Compounds Alkemades’ Theorem- Temp Arrows
First draw an extension of the boundary line so that it will intersect the Alkemades line.
The boundary line for C and AB can be drawn in the same way it was on the previous diagram. C A B AB m
Then use the point where the lines meet as a guide for drawing in the temperature arrows. AB

19. Incongruently Melting Compounds
Then the extension lines are removed, leaving only the Alkemades and boundary lines with their temperature arrows. C A B AB m 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. C A B AB
Then draw in the Alkemades lines for the three remaining boundary lines.

21. Congruently Melting Ternary Compounds Compatibility Triangles
Alkemades lines divide a phase diagram into compatibility triangles.
This phase diagram is divided into three triangles. C A B AB

22. Congruently Melting Ternary CompoundsB AB C 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

23. Congruently Melting Ternary CompoundsB AB C C
The binary invariant points are given their appropriate labels.
And the ternary invariant points are labeled. m e m e
The diagram is finished. E m E m m E m e

24. Incongruently Melting Ternary Compounds - Single Incongruency
Begin by highlighting each boundary line and the Alkemades line associated with it. C A B
ABC
Then draw in the Alkemades lines for the three remaining boundary lines.

25. Incongruently Melting Ternary Compounds Compatibility Triangles
Alkemades lines divide a phase diagram into compatibility triangles.
This phase diagram is divided into three triangles. C A B
ABC

26. Incongruently Melting Ternary CompoundsB
ABC
Begin by labeling the max/min points.
Then draw the temperature arrows on the boundary and Alkemades lines. m m A m

27. Incongruently Melting Ternary CompoundsB
ABC m
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 CompoundsB
ABC m
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 CompoundsB
ABC m
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 CompoundsB
ABC 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.
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
ABC B A A B

32. Incongruently Melting Ternary Compounds Compatibility Triangles
Alkemades lines divide a phase diagram into compatibility triangles.
This phase diagram is divided into three triangles. C A B
ABC

33. Incongruently Melting Ternary Compounds
Begin by labeling the max/min points.
Then draw the temperature arrows on the boundary and Alkemades lines. C m m
ABC
ABC B m A A B m

34. Incongruently Melting Ternary CompoundsB
ABC m
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 CompoundsB
ABC 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 P
And the ternary invariant points are labeled. E m
The diagram is finished. E e