E nergy vs Composition C urves To construct free energy vs composition curves given a eutectic phase diagram and vice versa Learning Objectives ID: 719553
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Slide1
Eutectic Phase Diagram construction using Free Energy vs. Composition Curves
To construct free energy vs. composition curves given a eutectic 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 & MSSlide2
GlossaryPhase Diagram - Phase Diagrams are charts which give the information regarding the equilibrium phases in a system under the given thermodynamic constraints.
Binary alloy – A system with two components.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 a liquid phase tranforms into two solid phases.Slide3
ExplanationPhase 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. Slide4
G
0βG0αG1αG1βG1A α1 X0 β1 B
A XBα XB0 XBβ BG
α
G
β
α
β
G(
α + β
)
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
B
α
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
Figure 1
Figure 2Slide5
A
αe X0 βe BGβGαG0βG0αGeGeβ
GeαCDTaking 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 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.Slide6
Instructions/ Working area
Credits
Binary Phase Diagrams
Glossary
Tab 03
Tab 04
Tab 05
Tab 06
Tab 07
Introduction
Slide
1
Slide 2
Slide7
Action/DescriptionAudio narration
Text to be displayedShow the animation description and the learning objectives from slide 1.Show the glossary from slide 2Slide8
Instructions/ Working area
Credits
Binary Phase Diagrams
Glossary
Explanation
Tab 04
Tab 05
Tab 06
Tab 07
Introduction
Slide
3,4 and 5 Slide9
Action/DescriptionAudio narration
Text to be displayedAs the user clicks on the Explanation ,the whole slide 3 appears on the screen at the same timeNow 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 the slide 3
Text on slide 4.
Text on the slide 3
Text on slide 4.Slide10
Instructions/ Working area
Credits
Binary Phase Diagrams
Glossary
Explanation
Animation
Tab 05
Tab 06
Tab 07
Introduction
Next slideSlide11
Master Layout 1
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 animatorSlide12
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 animatorCFigure 1Figure 2TE
TATBTET
A
T
BSlide13
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 .So all the figures used ahead will the ones made by the animator
Figure 3Figure 4TETATBT
ETATBSlide14
Action/DescriptionAudio narration
Text to be displayedThe 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) Now figure will change from 2 to 3A mixture of liquid and α phase exists in the region between the points where common tangent touches the two curves in figure
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 liquid &
α
and liquid &
β
As the eutectic temperature
is reached i.e., the temperature at which there will be common tangent to all the three free energy curves there will be a mechanical mixture of all the three phases(liquid ,
α
&
β
) at the eutectic composition
Same as Audio Narration
Same as Audio NarrationSlide15
Action/DescriptionAudio narration
Text to be displayedSimilarly, figure A will change from 3 to 4During transition curve
α & β will move downward and l will move upward gradually relative to each other.Next Button Appears on the Right BottomAs the temperature is further decreased the free energy curve of liquid lies completely above to that of alpha and beta so the system will exist only in alpha or beta or alpha +beta
Click on Next for next animation
Same as Audio Narration
Same as Audio NarrationSlide16
A B
Master Layout 2The 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 animatorSlide17
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 animatorFigure 1Figure 2TETATB
TETATBSlide18
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 .So all the figures used ahead will the ones made by the animator
Figure 3Figure 4TETATBT
ETATBSlide19
Action/DescriptionAudio narration
Text to be displayedThe Slide Master Layout 2 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) Now figure will change from 2 to 3The region where exists a mechanical mixture of two phases (l+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.
A
t
the Eutectic 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 ,eutectic point exists at a fixed temperature and composition.
Same as audio narration
Same
as audio narrationSlide20
Action/DescriptionAudio narration
Text to be displayedSimilarly, figure A will change from 3 to 4During transition curve
α & β will move downward and l will move upward gradually relative to each other.Next Button Appears on the Right BottomFor region below TE ,degrees of freedom will be two as per Gibbs Phase Rule which means that we can vary temperature and composition independently.
Click on Next for next slide.
Same as audio narrationSlide21
Instructions/ Working area
Credits
Binary Phase Diagrams
Glossary
Explanation
Activity Area
Questionnaire
Tab 06
Tab 07
Introduction
Next SlideSlide22
QuestionnaireIf the free energy vs. composition curve at some temperature is as shown , then will the system follow a eutectic phase diagram?
YESWhat could be a rational free energy curve at the eutectic point This diagram needs to be made by the animator Slide23
What will be a rational phase diagram for such a system
L+α1 L+α2 α1 +α2 α1 α2 Slide24
What would be the phase diagram of a system with phases having two different curvatures of their free energy
vs
composition curves ,i.e., what the phase diagram would be for a free energy curve as shown at some temperatureSol.Slide25
Action/DescriptionAudio narration
Text to be displayedAs 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 as on the slideSame as on the slideSlide26
Instructions/ Working area
Credits
Binary Phase Diagrams
Glossary
Explanation
Activity Area
Questionnaire
Summary
Tab 07
Introduction
Next SlideSlide27
SummaryBy 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.Slide28
Action/DescriptionAudio narration
Text to be displayedShow the summary from previous slideSame as on the slideSlide29
Instructions/ Working area
Credits
Binary Phase Diagrams
Glossary
Explanation
Activity Area
Questionnaire
Summary
Further Reading
Introduction
Next SlideSlide30
Further Reading BooksPhase Transformations in Metals and Alloys by
David A Porter K.E. Easterling Published by Chapman & Hall, 2-6 Bormdary Row, London SEl 8HN, UK Linkshttp://www.sv.vt.edu/classes/MSE2094_NoteBook/96ClassProj/analytic/anmeth.htmlhttp://www.matter.org.uk/matscicdrom/manual/td.htmlhttp://en.wikipedia.org/wiki/Phase_diagramSlide31
Action/DescriptionAudio narration
Text to be displayedShow further reading from previous slideSame as on the slide