### Phase Diagram

```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
10D110030
UG sophomore year
Dept. of ME & MS
Mentor:
Prof. M. P. Gururajan
Dept. of ME & MS
Shrey Singh
10D110010
UG sophomore year
Dept. of ME & MS
Glossary
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.
Explanation
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.
G0β
Gα
A
G1α
G0α
β
G1β
G1
α1 X0
Figure 1
β1
Gβ
G(α + β)
Gβ
α
Gα
B
A
XBα XB0
XBβ
B
Figure 2
In figure 1, we show the free energies of two different phases called alpha and beta. If Gα
is the free energy of the alpha phase (per mole) and Gβ is the free energy of the beta phase
(per mole) then a mechanical mixture of XBα moles of alpha and XBβ moles of beta will have a
free energy of (XBα Gα + XBβ Gβ ). That is as shown in figure 2 , for any composition between XBα
and XBβ the total free energy of the mechanical mixture lies on the straight line that connects
Gα and Gβ. 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.
Gβ
Gα
G0β
G eα
C
A
αe
G0α
Ge
X0
G eβ
D
βe
B
Taking the argument further it is
clear that for any composition
between αe and βe ,a mechanical
mixture of alpha and beta phases
of equilibrium compositions αe
and βe 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
Introduction
Slide 2
Glossary
Tab 03
Tab 04
Tab 05
Tab 06
Tab 07
Binary Phase Diagrams
Instructions/ Working area
Credits
Action/Description
Show the animation
description and the
learning objectives from
slide 1.
Show the glossary from
slide 2
Audio
narration
Text to be displayed
Slide 3,4 and 5
Introduction
Glossary
Explanation
Tab 04
Tab 05
Tab 06
Tab 07
Binary Phase Diagrams
Instructions/ Working area
Credits
Action/Description
Audio
narration
Text to be displayed
Text on
the slide 3
Text on the slide 3
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
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 Text on slide 4.
4.
Next slide
Introduction
Glossary
Explanation
Animation
Tab 05
Tab 06
Tab 07
Binary Phase Diagrams
Instructions/ Working area
Credits
Master Layout 1
The Animator has to
make diagrams by
Figure
himselfA 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+αFigure
regionb needs to lie
above the point c .So
all the figures used
ahead will the ones
made by the animator
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
C
Figure 1
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 .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
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
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
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
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
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
Click on Next for next
animation
Master Layout 2
The
The Animator
Animator has
has to
to
make
make diagrams
diagrams by
by
himself
himself of
of all
all the
the
figures
figures in
in the
the image
image
alongside
alongside in
in the
the
current
current slide
slide .All
.All the
the
dotted
dotted lines
lines needs
needs to
to
be
be completely
completely
horizontal
horizontal or
or vertical
vertical in
in
the
the figures
by
the
the animator
animator .For
.For
Figure
Figure two
two the
the dotted
dotted
horizontal
horizontal line
line in
in the
the
l+α
l+α region
region needs
needs to
to lie
lie
above
above the
the point
point cc .So
.So
all
all the
the figures
figures used
used
will the
the ones
ones
by the
the animator
animator
Figure 1
Figure 2
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
animator
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
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
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.
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.
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.
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.
Same as
audio
narration
Same as
audio
narration
Action/Description
Audio narration
Text to be
displayed
Figure A will change from 3 to 4.
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
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.
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
Click on Next for next slide.
Next Slide
Introduction
Glossary
Explanation
Animation
Questionnaire
Tab 06
Tab 07
Binary Phase Diagrams
Instructions/ Working area
Credits
Questionnaire
I. What is the no of regions that will exist in
the phase diagram at the temperature at
which the free energy
vs. composition curve is
 5
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)
(b)
Ans.
(a).Peritactic
(b).Eutectic
(c).Eutectoid
(c)
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.
Same Same as
as on on the
the slide
slide
Next Slide
Introduction
Glossary
Explanation
Activity Area
Questionnaire
Summary
Tab 07
Binary Phase Diagrams
Instructions/ Working area
Credits
Summary
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
Show the summary
from previous slide
Same
as on
the
slide
Text to be displayed
Next Slide
Introduction
Glossary
Explanation
Activity Area
Questionnaire
Summary
Binary Phase Diagrams
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
• 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