Complex Phase Diagram construction using
Free Energy vs. Composition Curves
• To construct free energy vs. composition curves given a complex phase diagram and vice versa.
Learning Objectives :
After going through this learning aid ,user will be able to:
 Construct a phase diagram given free energy vs. composition curves at different temperatures.
 Draw the schematic free energy vs. composition curves at different temperatures given the phase diagrams.
 State and interpret the Gibbs Phase Rule.
Author:
Kartikay Agarwal Shrey Singh
10D110030 10D110010
UG sophomore year UG sophomore year
Dept. of ME & MS Dept. of ME & MS
Mentor:
Prof. M. P. Gururajan
Dept. of ME & MS

Phase Diagram - Phase Diagrams are charts which give the information regarding the equilibrium phases in a system under the given thermodynamic constraints.
Gibbs Phase Rule – At constant pressure P+F-C=1 where C=no. of components,
F=no. of Degrees of freedom, P=no of phases.
Eutectic Point - The point at which liquid phase tranforms into two solid phases.
Peritactic Point – The point at which a liquid and a solid phase transforms into another solid phase.
Eutectoid Point - The point at which solid phase tranforms into two solid phases.

Phase Diagrams are a way of cataloguing the information regarding the equilibrium phases in a system under the given thermodynamic constraints.
In this learning object, we will consider a system of a binary alloy at constant pressure and different temperatures . So the phase diagram is a plot of composition vs. temperature with the equilibrium phases marked at different combinations of these two parameters.
Specifically in this animation we consider a system that shows the existence of one ,two and three at any temperature and composition.
We will show that the equilibrium phases are a result of the minimization of the free energy of the system.

G
0
β
G α G
1
α
G
0
α
G
1
G
1
β
G β
G(α + β)
G α
α
β
G β
A α
1
X
0
β
1
B
Figure 1
A X
B
α X
B
0
Figure 2
X
B
β B
In figure 1, we show the free energies of two different phases called alpha and beta. If G α is the free energy of the alpha phase (per mole) and G β is the free energy of the beta phase
(per mole) then a mechanical mixture of X free energy of (X
B
α G α + X
B
β
B
α moles of alpha and X
B
β moles of beta will have a
G β ). That is as shown in figure 2 , for any composition between X
B
α and X
B
β the total free energy of the mechanical mixture lies on the straight line that connects
G α and G β . Hence from figure 1, it is clear that an alpha phase or a beta phase of uniform composition x0 will have higher free energy as compared to a mechanical mixture of compositions alpha1 and beta1 for the alpha and beta phases respectively.

G
G
C
β e
α
A α e
G
G
0
α
0
β
G e
G
X
0 e
β
β
G
D e
α
B
Taking the argument further it is clear that for any composition between α e and β e
,a mechanical mixture of alpha and beta phases of equilibrium compositions α e and β e will have the lowest free energy. Pictorially the equilibrium compositions of the phases that form the mechanical mixture are given by the common tangent constructed to the free energies where they are concave.
Conversely any region of a phase diagram where a mechanical mixture is seen the free energy vs. composition curve will have a concave curvature.
Similarly ,for a binary system with two solid phases ,the eutectic point, the eutectoid point and the peritactic point exist at the temperature at which the three free energy curves of the two solid and one liquid phase will share a common tangent with each other and the composition will be the one corresponding to the contact point of the common tangent with the free energy curve of liquid phase.

Slide 1 Slide 2
Introduction Glossary
Binary Phase Diagrams
Tab 03 Tab 04 Tab 05 Tab 06 Tab 07
Instructions/ Working area
Credits

Action/Description
Audio narration
Text to be displayed

Slide 3,4 and 5
Introduction Glossary
Binary Phase Diagrams
Explanation Tab 04 Tab 05 Tab 06 Tab 07
Instructions/ Working area
Credits

Action/Description Audio narration
Text to be displayed
As the user clicks on the Explanation ,the whole slide 3 appears on the screen at the same time
Now as the user clicks the next button (present at the right bottom corner of the slide)
The figures on slide 4 appears on the screen
Text on the slide 3
Text on the slide 3
Now as the user clicks the next button (present at the right bottom corner of the slide)
The figure on slide 5 appears on the screen
Text on slide
4.
Text on slide 4.

Introduction Glossary
Binary Phase Diagrams
Explanation
Next slide
Animation Tab 05 Tab 06 Tab 07
Instructions/ Working area
Credits

The Animator has to make diagrams by himself of all the figures in the image alongside in the current slide .All the dotted lines needs to be completely horizontal or vertical in the figures made by the animator .For
Figure two the dotted horizontal line in the l+α region needs to lie above the point c .So all the figures used ahead will the ones made by the animator

Figure 1
C
Figure 2
The Animator has to make diagrams by himself of all the figures in the image alongside.All the dotted lines needs to be completely horizontal or vertical in the figures made by the animator .For
Figure two the dotted horizontal line in the l+α region needs to lie above the point c .So all the figures used ahead will the ones made by the animator

Figure 3 Figure 4
The Animator has to make diagrams by himself of all the figures in the image alongside .All the dotted lines needs to be completely horizontal or vertical in the figures made by the animator .So all the figures used ahead will the ones made by the animator

Figure 5
The Animator has to make diagrams by himself of all the figures in the image alongside .All the dotted lines needs to be completely horizontal or vertical in the figures made by the animator .So all the figures used ahead will the ones made by the animator

Action/Description Audio narration
The Slide Master Layout 1 will appear on the screen in this tab but the figure in the layout will be figure 1 instead of what is shown in the master layout slide
The system will exist in liquid phase until melting temperature of pure A is reached.
As soon as melting point of A is reached ,a mixture of liquid and
α phase will exists in the region between the points where common tangent touches the two free energy curves
Text to be displayed
Same as
Audio
Narration
The figure will change from figure 1 to 2(figure 1 and 2 needs to be made by the animator as been described in the previous slide)
As the eutectioid temperature is reached i.e., the temperature at which there will be common tangent to the three free energy curves there will be a mechanical mixture of all the three phases(liquid ,α & β) at the eutectoid composition
Same as
Audio
Narration
Similarly, figure will change from 2 to 3.
.
As the temperature is further decreased there will be two common tangents in the free energy vs composition curves and hence there will be two regions in the phase diagram where a mechanical mixture of two phases will exist i.e., of β &
α and liquid & β
Same as
Audio
Narration

Action/Description Audio narration Text to be displayed
Figure A will change from 3 to 4.
As the temperature is further decreased there will be three common tangents in the free energy vs composition curves and hence there will be three regions in the phase diagram where a mechanical mixture of two phases will exist i.e., of β &
α and liquid & β and liquid + ϒ.
Same as
Audio
Narration
Now the figure will change from figure 4 to figure 5
During transition curve α & β & ϒ will move downward and l will move upward gradually relative to each other.
Next Button Appears on the Right Bottom
As soon as the eutectic temperature is reached, there will be a mechanical mixture of three phases l, beta and gama at the composition where the common tangent to the three free energy curves of the same touches the liquid free energy curve Also there exists a mechanical mixture of alpha and beta in the region of common tangent between alpha and beta free energy curves
Same as
Audio
Narration
Click on Next for next animation

Figure 1 Figure 2
The Animator has to make diagrams by himself of all the figures in the image alongside in the current slide .All the dotted lines needs to be completely horizontal or vertical in the figures made by the animator .For
Figure two the dotted horizontal line in the l+α region needs to lie above the point c .So all the figures used ahead will the ones made by the animator

Figure 3 Figure 4
The Animator has to make diagrams by himself of all the figures in the image alongside in the current slide .All the dotted lines needs to be completely horizontal or vertical in the figures made by the animator .For
Figure two the dotted horizontal line in the l+α region needs to lie above the point c .So all the figures used ahead will the ones made by the animator

Figure 5
The Animator has to make diagrams by himself of all the figures in the image alongside in the current slide .All the dotted lines needs to be completely horizontal or vertical in the figures made by the animator .For
Figure two the dotted horizontal line in the l+α region needs to lie above the point c .So all the figures used ahead will the ones made by the animator

Action/Description Audio narration Text to be displayed
The Slide Master Layout 1 will appear on the screen in this tab but the figure in the layout will be figure 1 instead of what is shown in the master layout slide
For region above melting temperature of A, No of Phases is just 1 i.e., solid and components are 2 so by Gibbs Phase Rule degrees of freedom will be 1 which means that we can obtain a particular composition of liquid phase by varying temperature and composition simultaneously.
Same as audio narration
The figure will change from figure 1 to 2(figure 1 and 2 needs to be made by the animator as been described in the previous slide)
Similarly, figure will change from 2 to 3.
At the peritactic temperature and composition ,three phases exists which gives out degrees of freedom to be 0 from Gibbs Phase
Rule which means that for a system at constant pressure, peritactic point exists at a fixed temperature and composition.
Same as audio narration
The region where exists a mechanical mixture of two phases
(β+α or l+β), F comes out to be 1 which means that in order to obtain a particular composition of a phase we can chose either one of temperature or composition independently but the other one needs to be fixed.

Action/Description
Figure A will change from 3 to 4.
Audio narration Text to be displayed
On going below melting temperature of B, There will three regions with 1 degree of freedom(β+α or l+β or l+ϒ)
Same as audio narration
Now the figure will change from figure 4 to figure 5
During transition curve α & β & ϒ will move downward and l will move upward gradually relative to each other.
At the Eutectic point and composition ,three phases exists which gives out degrees of freedom to be 0 from gibbs phase rule which means that for a system at constant pressure ,eutectic point exists at a fixed temperature and composition. Although there
Will a region where the degree will be two representing a mixture of
β+α .
Same as audio narration.
Next Button Appears on the Right Bottom Click on Next for next slide.

Introduction Glossary
Binary Phase Diagrams
Explanation Animation
Next Slide
Questionnaire Tab 06 Tab 07
Instructions/ Working area
Credits

I.
What is the no of regions that will exist in the phase diagram at the temperature at which the free energy
 5 vs. composition curve is ll.
What will be those regions?
 α, α+β, β, β +l, l This diagram needs to be made by the animator

Identify the points a, b and c.
(a)
Ans.
(a).Peritactic
(b).Eutectic
(c).Eutectoid
(c)
(b)

Action/Description Audio narration
Text to be displayed
As the user clicks on the Questionnaire tab ,the questions written in black color appear on screen along with the figure(to be made by the animator as described in the previous slide) at the same time .Then on selecting the answers tab made at the bottom of screen, the answer, i.e. text written in red color appears in the same format as shown in previous slide.

Introduction Glossary
Binary Phase Diagrams
Explanation
Next Slide
Activity Area Questionnaire Summary Tab 07
Instructions/ Working area
Credits

By the interpretation of free energy vs. composition curves ,we can estimate the phase diagram and also the free energy vs. composition curves can be estimated on the basis of phase diagrams. Free energy vs. composition curves explain the reason behind the existence of multiple phases at different temperatures and compositions.
The phase or a mixture of phases with the lowest free energy will exist at all combination of temperatures and compositions in order to attain equilibrium.

Action/Description
Audio narration
Text to be displayed

Next Slide
Introduction Glossary
Binary Phase Diagrams
Explanation Activity Area Questionnaire Summary Further Reading
Instructions/ Working area
Credits

Books
• Phase Transformations in Metals and Alloys by David A
Porter K.E. Easterling Published by Chapman & Hall, 2-6
Bormdary Row, London
SEl 8HN, UK
Links
• http://www.sv.vt.edu/classes/MSE2094_NoteBook/96Class
Proj/analytic/anmeth.html
• http://www.matter.org.uk/matscicdrom/manual/td.html
• http://en.wikipedia.org/wiki/Phase_diagram

Action/Description
Audio narration
Text to be displayed