EQUILIBRIUM PHASE DIAGRAMS
Transcript of EQUILIBRIUM PHASE DIAGRAMS
EQUILIBRIUM PHASE DIAGRAMSEQUILIBRIUM PHASE DIAGRAMS
ANANDH SUBRAMANIAMMaterials Science and Engineering
INDIAN INSTITUTE OF TECHNOLOGY KANPURKanpur-
110016Ph: (+91) (512) 259 7215, Lab: (+91) (512) 259 7147
[email protected]://home.iitk.ac.in/~anandh/
19 Oct 2015MATERIALS SCIENCEMATERIALS SCIENCE&&
ENGINEERING ENGINEERING AN INTRODUCTORY EAN INTRODUCTORY E--BOOKBOOK
http://home.iitk.ac.in/~anandh/E-book.htmhttp://home.iitk.ac.in/~anandh/E-book.htmA Learner’s GuideA LearnerA Learner’’s Guides Guide
Alloy Phase Equilibria, A. Prince, Elsevier Publishing Company, Amsterdam (1966)
Phase diagrams are an important tool in the armory of an materials scientist
In the simplest sense a phase diagram demarcates regions of existence of various phases. This is similar to a map which demarcates regions based on political, geographical, ecological etc. criteria.
Phase diagrams are maps*
Thorough understanding of phase diagrams is a must for all materials scientists
Phase diagrams are also referred to as “EQUILIBRIUM PHASE DIAGRAMS”
This usage requires special attention: though the term used is “Equilibrium”, in practical
terms the equilibrium is NOT
GLOBAL EQUILIBRIUM
but MICROSTRUCTURAL LEVEL EQUILIBRIUM (explanation of the same will be considered later)
This implies that any microstructural information overlaid on a phase diagram is for convenience and not implied by the phase diagram.
The fact that Phase Diagrams represent Microstructural Level equilibrium is often not stressed upon.
Phase Diagrams
* there are many other maps that a material scientist will encounter like creep mechanism maps, various kinds of materials selection maps etc.
Broadly two kinds of phase diagrams can be differentiated* → those involving time and those which do not involve time (special care must be taken in understanding the former class-
those involving time).
In this chapter we shall deal with the phase diagrams not involving time.
This type can be further sub-classified into:
Those with composition as a variable (e.g. T
vs
%Cu)
Those without composition as a variable (e.g. P
vs
T)
Temperature-Composition diagrams (i.e. axes are T and composition)
are extensively used in materials science and will be considered in detail in this chapter. Also, we shall restrict ourselves to structural phases (i.e. phases not defined in terms of a physical property)**
Time-Temperature-Transformations (TTT) diagrams and Continuous-Cooling-
Transformation (CCT) diagrams involve time. These diagrams are usually designed to have an overlay of Microstructural information (including microstructural evolution). These diagrams will be considered in the chapter on Phase Transformations.
* this is from a convenience in understanding point of view** we have seen before that phases can be defined based either on a
geometrical entity or a physical property (sometimes phases based on a physical property are overlaid on a structural phase diagram-
e.g. in a Fe-cementite
phase diagram ferromagnetic phase and curie temperatures are overlaid)
DEFINITIONSDEFINITIONS
Components of a system
Independent chemical species which comprise the system: These could be: Elements, Ions, CompoundsE.g.
Au-Cu system
: Components → Au, Cu (elements)
Ice-water system
: Component → H2
O (compound)
Al2
O3
– Cr2
O3
system
: Components → Al2
O3
, Cr2
O3
Let us start with some basic definitions:
This is important to note that components need not be just elements!!
Note that components need not
be only elements
Phase
Physically distinct, chemically homogenous and mechanically separable region of a system (e.g. gas, crystal, amorphous...).
Gases Gaseous state always a single phase
→ mixed at atomic or molecular level
Liquids ►Liquid solution is a single phase
→ e.g. NaCl
in H2
O► Liquid mixture consists of two or more phases
→ e.g. Oil in water
(no mixing at the atomic/molecular level)
Solids
In general due to several compositions and crystals structures many phases are possible
For the same composition different crystal structures represent
different phases.
E.g. Fe (BCC) and Fe (FCC)
are different phases
For the same crystal structure different compositions represent
different phases.
E.g. in Au-Cu alloy
70%Au-30%Cu
& 30%Au-70%Cu
are different phases
This is the typical textbook definition which one would see!!
Three immiscible liquids
What kinds of Phases exist?
Based on state
Gas, Liquid, Solid
Based on atomic order
Amorphous, Quasicrystalline, Crystalline
Based on Band structure
Insulating, Semi-conducting, Semi-metallic, Metallic
Based on Property
Paraelectric, Ferromagnetic, Superconducting, …..
Based on Stability Stable, Metastable, (also-
Neutral, unstable)
Also sometimes-
Based on Size/geometry of an entity
Nanocrystalline, mesoporous,
layered, …
We have already seen the ‘official’
definition of a phase:
Physically distinct, chemically homogenous and mechanically separable region of a system.
However, the term phase is used in diverse contexts and we list below some of these.
Microstructure
Structures requiring magnifications in the region of 100 to 1000
timesOR
The distribution of phases and defects in a material
Grain
The single crystalline part of polycrystalline metal separated by similar entities by a grain boundary
Phase Transformation is the change of one phase into another. E.g.:
► Water → Ice► -
Fe (BCC) →
- Fe (FCC)
-
Fe (FCC) → -
Fe (ferrite) + Cementite
(this involves change in composition)
► Ferromagnetic phase → Paramagnetic phase (based on a property)
Phase transformation
Again this is a typical textbook definition which has been included for…!!
An alternate definition based on magnification
(Phases + defects + residual stress) & their distributions
Phase diagram
Map demarcating regions of stability of various phases. orMap that gives relationship between phases in equilibrium in a system as a function of T, P and composition (the restricted form of the definition sometime considered in materials textbooks)
Variables / Axis of phase diagrams
The axes can be:
Thermodynamic
(T, P, V),
Other possibilities include magnetic field intensity (H), electric field (E) etc.
Kinetic
(t) or
Composition variables
(C, %x) (composition is usually measure in weight%, atom% or mole fraction)
In single component systems (unary systems) the usual variables are T & P
In phase diagrams used in materials science the usual variables are:T & %x
In the study of phase transformation kinetics Time Temperature Transformation (TTT) diagrams or Continuous Cooling Transformation (CCT) diagrams are also used where the axis are T & t
Phase diagrams are also called Equilibrium Phase Diagrams.
Though not explicitly stated the word ‘Equilibrium’
in this context usually means
Microstructural
level equilibrium and NOT
Global Equilibrium.
Microstructural level equilibrium implies that microstructures are ‘allowed to exist’
and the system is not in the global energy minimum state.
This statement also implies that:
Micro-constituents* can be included in phase diagrams
Certain phases (like cementite
in the Fe-C system) maybe included in phase diagrams, which are not strictly equilibrium phases (cementite
will decompose to
graphite and ferrite given sufficient thermal activation and time)
Various defects are ‘tolerated’
in the product obtained. These include defects
like dislocations, excess vacancies, internal interfaces (interphase
boundaries, grain boundaries) etc.
Often cooling ‘lines/paths’
are overlaid on phase diagrams-
strictly speaking this is not
allowed. When this is done, it is implied that the cooling rate is ‘very slow’
and the system is in ‘~equilibrium’
during the entire process. (Sometimes, even fast cooling paths are also overlaid on phase diagrams!)
Important points about phase diagrams (Revision + extra points)
* will be defined later
The GIBBS PHASE RULE
F = C F = C
P + 2P + 2For a system in equilibrium
F F
C +C +
P = 2P = 2or
F –
Degrees of Freedom C –
Number of Components
P –
Number of Phases
The Phase rule is best understood by considering examples from actual phase diagrams as shown in some of the coming slides
Degrees of Freedom: A general definition
In response to a stimulus the ways in which the system can respond corresponds to the degrees of freedom of the system
The phase rule connects the Degrees of Freedom, the number of Components in a system and the number of Phases present in a system via a simple equation.
To understand the phase rule one must understand the variables in the system along with the degrees of freedom.
We start with a general definition of the phrase: “degrees of freedom”
The phase rule
Variables in a Phase Diagram
C
– No. of Components
P
–
No. of Phases
F
–
No. of degrees of Freedom
Variables in the system =
Composition variables + Thermodynamic variables
Composition of a phase specified by (C –
1)
variables
(e.g. If the composition is expressed in %ages then the total is 100% there is one equation connecting the composition variables and we need to specify only
(C1) composition variables)
No. of variables required to specify the composition of all Phases: P(C –
1) (as there are P phases and each phase needs the specification of
(C1) variables)
Thermodynamic variables = P + T
(usually considered) = 2 (at constant Pressure (e.g. atmospheric pressure) the thermodynamic variable becomes 1)
Total no. of variables in the system = P(C –
1) + 2
F < no. of variables F < P(C –
1) + 2
Degrees of Freedom = What you can control What the system controls
F = C + 2 P
Can control the no. of components added and P & T
System decided how many phases to produce given the
conditions
A way of understanding the Gibbs Phase Rule:The degrees of freedom can be thought of as the difference between what you (can)
control
and what the system controls
F = C F = C
P + 2P + 2
Single component phase diagrams (Unary)
Let us start with the simplest system possible: the unary system
wherein there is just one
component.
Though there are many possibilities even in unary phase diagrams
(in terms of the axis and
phases) , we shall only consider a T-P
unary phase diagram.
Let us consider the Fe unary phase diagram as an illustrative example.
Apart from the liquid and gaseous phases many solid phases are possible based on crystal structure. (Diagram on next page).
Note that the units of x-axis are in GPa
(i.e. high pressures are needed in the solid state and
liquid state to see any changes to stability regions of the phases).
The Gibbs phase rule here is: F = C –
P + 2. (2 is for T & P).
Note that how the phase fields of the open structure (BCC-
one in the low T regime () and
one in the high T regime ()) diminish at higher pressures. In fact -
phase field completely vanishes at high pressures.
The variables in the phase diagram are: T & P (no composition variables here!).
Along the 2 phase co-existence lines
the DOF (F) is 1
→ i.e. we can chose either T or P
and the other will be automatically fixed.
The 3
phase co-existence points
are invariant points with F = 0. (Invariant point implies
they are fixed for a given system).
15351410
(BCC)
(HCP)
(FCC)
(BCC)
Liquid
Gas
Pressure (GPa) →
Tem
pera
ture
(ºC
) →
Triple points: 3 phase coexistence
F = 1 –
3 + 2 = 0 triple points are fixed points of a phase diagram (we cannot chose T or P)
Two phase coexistence lines
F = 1 –
2 + 2 = 1 we have only one independent variable (we can chose one of the two variables (T or P) and the other is ‘automatically’
fixed by the phase diagram)
Single phase regions
F = 1 –
1 + 2 = 2
T and P can both be varied while still being in the single phase region
F = C – P + 2
The maximum number of phases which can coexist in a unary P-T phase diagram is 3Note the P is in GPa
“Very High pressures are required for things to happen in the solid state”
Understanding aspects of the iron unary phase diagram
The degrees of freedom for regions, lines and points in the figure are marked in the diagram shown before
The effect of P on the phase stability of various phases is discussed in the diagram below
It also becomes clear that when we say iron is BCC at RT, we mean at atmospheric pressure (as evident from the diagram at higher pressures iron can become
HCP)
(BCC)
(HCP)
(FCC)
(BCC)
Liquid
Gas
Pressure (GPa) →
Tem
pera
ture
(ºC
) →
This line slopes upward as at constant T if we increase the P the gas will liquefy as liquid has lower volume (similarly the reader should draw horizontal lines to understand the effect of pressure on the stability of various phases-
and rationalize the same).
These lines slope downward as: Under higher pressure the phase with higher packing fraction (lower volume) is preferred
Increase P and gas will liquefy on crossing phase boundary
Phase fields of non-close packed structures shrink under higher pressure
Phase fields of close packed structures expand under higher pressure
Usually (P = 1 atm) the high temperature phase is the loose packed structure and the RT structure is close packed.
How come we find BCC phase at RT in iron?
Binary Phase Diagrams
Binary implies that there are two components.
Pressure changes often have little effect on the equilibrium of solid phases (unless ofcourse
we apply ‘huge’
pressures).
Hence, binary phase diagrams are usually drawn at 1 atmosphere pressure.
The Gibbs phase rule is reduced to:
Variables are reduced to: F = C –
P + 1. (1 is for T).
T & Composition
(these are the usual variables in Materials Phase Diagrams)
F = C F = C
P + 1P + 1Phase rule for condensed phases
For T
In the next page we consider the possible binary phase diagrams.
These have been
classified based on:
Complete Solubility in both liquid & solid states
Complete Solubility in both liquid state, but limited solubility in the solid state
Limited Solubility in both liquid & solid states.
Complete Solubility in both liquid state, but limited solubility in the solid state
Overview of Possible Binary Phase diagrams
Isomorphous
Isomorphous
with phase separation
Isomorphous
with ordering
Complete Solubility in both liquid & solid states
Limited Solubility in both liquid & solid states
Eutectic
Peritectic
Liquid StateLiquid State Solid State analogueSolid State analogue
Eutectoid
Peritectoid
Monotectic
Syntectic
Monotectoid
Solid state analogue of Isomorphous
We have already seen that the reduced phase rule at 1Atm pressure is: F = C –
P + 1.
The ‘one’
on RHS above is T.
The other two variables are:
Composition of the liquid (CL
) and composition (CS
) of the solid. In a fully solid state reaction:
Composition of one solid (CS1
) and composition of the other solid (CS2
).
The compositions are defined with respect to one of the components (say B):
CLB, CS
B
The Degrees of Freedom (DOF, F) are defined with respect to these variables.
What are the variables/DOF in a binary phase diagram?
System with complete solid & liquid solubility: ISOMORPHOUS SYSTEM
Let us start with an isomorphous
system with complete liquid and solid solubility Pure
components melt at a single temperature, while alloys in the isomorphous
system melt over a range of temperatures*.
I.e. for a given composition solid and liquid will coexist
over a range of temperatures when heated.
Model Isomorphous
phase diagram
We mention some important points here (may be/have been reiterated elsewhere!):
Such a phase diagram forms when there is complete solid and liquid solubility.
The solid mentioned is crystalline.
The solid + liquid region is not a semi-solid (like partly molten wax or silicate glass). It is a crystal of well defined composition in equilibrium with a liquid
of well defined
composition.
Both the solid and the liquid and the solid (except pure A and pure B) have both A and B components in them.
A and B components could be pure elements (like in the Ag-Au, Au-Pd, Au-Ni, Ge-Si) or compounds (like Al2
O3
-Cr2
O3
).
At low temperatures the picture may not be ‘ideal’
as presented in the diagram below and we may have phase separation (Au-Ni system) or have compound formation (for some compositions) (Au-Pd system). These cases will be considered later.
Each solid, with a different composition is a different phase. The area marked solid in the phase diagram is a phase field.
If heated further the liquid will vaporize, this part of the phase diagram is usually not shown in the diagrams considered.
Liquid
Liquid (solution)
Solid
Solid (solution)
%B →A B
Solid + Liquid
Solid (solution)
Liquid (solution)
T →
Note that between two single phase regions there is a two phase region (for the alloy) (except for special cases)
A B%B →
T →
M.P. of B
M.P. of A
C = 1P = 2F = 0
C = 2P = 1F = 2
C = 2P = 1 (liquid)F = 2
Variables → T, CSB
2
Variables → T, CLB
2
Variables →
T, CL
B, CSB
3
F = 2 –
P F= 3 –
P
C = 2P = 2F = 1
F = 2 –
P
F = C – P + 1Now let us map the variables and degrees of freedom in varions
regions of the isomorphous
phase diagram
T and Composition can both be varied while still being in the single phase region
in the two phase region, if we fix T (and hence exhaust our DOF), the composition of liquid and solid in equilibrium are automatically fixed (i.e. we have no choice over them). Alternately we can use our DOF to chose CL
→ then T and CS
are automatically fixed.
For pure components at any T For alloys
Solid
Solid + Liquid
Liquid
Disordered (substitutional) solid solutions
For pure components all transformation
temperatures (BCC to FCC, etc.) are fixed (i.e.
zero ‘F’)
Gibbs free energy vs
composition plot at various temperatures: Isomorphous
system
As we know at constant T and P the Gibbs free energy determines the stability of a phase. Hence, a phase diagram can be constructed from G-composition (Gmixing
-C) curves at various temperatures. For an isomorphous
system we need to chose 5 sample temperatures: (i) T1
> TA
, (ii) T2
=TA
, (iii) TA
>T3
>TB
, (iv) T4
=TB
, (v) T5
<TB
.
G of L lower than for all compositions and hence L is stable
G of L lower than for all compositions except for pure A.
For compositions between X1
and X2
the common tangent construction gives the free energy of the L+ mixture
5
How to get G versus composition
curves→
Click here to know more
.
How to get G versus composition
curves→
Click here to know more
.
Al2
O3 Cr2
O3%Cr2
O3
→
T (ºC
)→
2000
2100
2200
10 30 50 70 90
L
L + S
Solidus
Liquidus
S
Isomorphous
Phase Diagram: an example
A and B must satisfy Hume-Rothery
rules for the formation of ‘extended’
solid solution.
Examples of systems forming isomorphous
systems: Cu-Ni, Ag-Au, Ge-Si, Al2
O3
-Cr2
O3
.
Note the liquidus
(the line separting
L & L+S regions) and
solidus
(the line separating L+S
and S regions) lines in the figure.
Schematics
Note that the components in this case are compounds
ISOMORPHOUS PHASE DIG.
Points to be noted:
Pure components (A,B) melt at a single temperature. (General) Alloys melt over a range of temperatures (we will see some special cases soon).
Isomorphous
phase diagrams form when there is complete solid and liquid solubility.
Complete solid solubility implies that the crystal structure of the two components have to be same and Hume-Rothery
rules
have to be followed.
In some systems (e.g. Au-Ni system) there might be phase separation in the solid state (i.e.
the complete solid solubility criterion may not be followed) → these will be considered later in this chapter as a variation of the isomorphous
system (with complete solubility in the
solid and the liquid state).
Both the liquid and solid contain the components A and B.
In Binary phase diagrams between two single phase regions there will be a two phase region → In the isomorphous
diagram between the liquid and solid state there is the
(Liquid + Solid) state.
The Liquid + Solid
state is NOT
a ‘semi-solid’
state → it is a solid of fixed composition
and structure, in equilibrium with a liquid of fixed composition.
In the single phase region
the composition of the alloy is ‘the composition’.
In the two phase region the composition of the two phases is different and is NOT
the nominal composition of the alloy (but, is given by the lever rule). Lever rule is considered next.
HUME ROTHERY RULESClick here to know more about
Say the composition C0
is cooled slowly (equilibrium) At T0
there is L + S
equilibrium Solid (crystal) of composition C1
coexists with liquid of composition C2
Tie line and Lever RuleGiven a temperature and composition-
how do we find the fraction of the phases present along with the composition?
We draw a horizontal line (called the Tie Line) at the temperature of interest (say T0
).
Tie line is XY.
Note that tie lines can be drawn only in the two phase coexistence regions (fields). Though they may be extended to mark the temperature.
To find the fractions of solid and liquid we use the lever rule.
%B →A B
T →
L
L + S
SC0C1 C2
T0
Cooling
Fulcrum of the lever
Arm of the lever proportional to
the solid
Arm of the lever proportional to
the liquid
Note: strictly speaking cooling curves cannot be
overlaid on phase diagrams
Tie line
0
0 1
2 1liquid atT
C Cf C C
At T0 The fraction of liquid (fl
) is
(C0
C1
) The fraction of solid (fs
) is
(C2
C0
)
0
2 0
2 1solid atT
C Cf C C
We draw a horizontal line (called the Tie Line) at the temperature of interest (say T0
).
The portion of the horizontal line in the two phase region is akin to a ‘lever’
with the
fulcrum at the nominal composition (C0
).
The opposite arms of the lever are proportional to the fraction of the solid and liquid phases present (this is the lever rule).
Note that tie line is drawn within the two phase region and is horizontal.
Expanded version
Extended tie line
C0C1 C2
T0
Fulcrum of the lever
Arm of the lever proportional to
the solid
Arm of the lever proportional to
the liquid
0 1
2 1liquid
C Cf AB CAC
C
At T0 The fraction of liquid (fl
) is proportional to (C0
C1
) → AC The fraction of solid (fs
) is proportional to
(C2
C0
) → CB
2 0
2 1solid
C Cf ABC
CB
C
BA
C
For a composition C0
At T0
→ Both the liquid and the solid phases contain both the components A and B
To reiterate: The state is NOT
semi-solid but a mixture of a solid of a definite composition (C1
) with a liquid of definite composition (C2
)
If the alloy is slowly cooled (maintaining ~equilibrium) then in
the two phase region (liquid + solid region) the composition of the solid will move along the brown line and the composition of the liquid will move along the blue line.
The composition of the solid and liquid are changing as we cool!
Points to be noted
Isomorphous
Phase Diagrams
Note here that there is solid solubility, but it is not complete
at low temperatures (below the peak of the 1
+ 2
phase field dome)
(we will have to say more about that soon)
Note that
Ag & Au are so ‘similar’
that the phase diagram becomes a thin lens (i.e. any alloy of Au & Ag melts over a small range of temperatures–
as if it were ‘nearly’
a pure metal!!).
Any composition melts above the linear interpolated melting point.
T1Below T1
(820C) for some range of compositions the solid solubility of Au in Ni (and vice-versa) is limited.
Any composition melts above the linearly
interpolated melting point
Extensions of the simple isomorphous
system: Congruently melting alloys
Congruently melting alloys just like a pure metal
Is the DOF 1? No: in requiring that CL
B
= CSB
we have exhausted the degree of freedom. Hence T is automatically fixed → DOF is actually zero! Tie line has shrunk to a point!
Variables → T, CLB, CS
B
3
C = 2P = 2F = 1?? (see below)
We have seen that a pure metal melts at a single temperature (Why?!!).
An alloy typically melts over a range of temperatures. However, there are special compositions which can melt at a single temperature like a pure metal. One of these
is the congruent melting composition-
in a variation of the isomorphous
phase diagram. Some systems show this type of behaviour.
Intermediate compounds also have this feature as we shall see later.
Elevation in MP Depression in MPCase A Case B
AA and BB bonds stronger than AB bonds
Liquid stabilized → Phase separation
in the solid state
AB bonds stronger than AA and BB bonds
Solid stabilized → Ordered solid formation
E.g.
Au-
Ni
Extensions of the simple isomorphous
system: What does this imply w.r.t
the solid state phases?
Elevation in the MP means that the solid state is ‘more stable’
(crudely speaking the
‘ordered state is more stable’) → ordering reaction is seen at low T.
Depression in MP ‘means’
the liquid state (disordered) is more stable → phase separation
is seen at low T. (Phase separation can be ‘thought of’
as the ‘opposite’
of ordering. Ordering (compound formation) occurs for ve
values of Hmix
, while phase separation is favoured
by +ve
values of Hmix
.
Case A Case B
Examples of isomorphous
systems with phase separation and compound formation
Au-Pd system with 3
compounds
Au-Pt system with phase separation at low temperatures
Au-Ni: model system to understand phase separation
Phase separation in a AlCrFeNi
alloy (with composition Al28.5
Cr27.3
Fe24.9
Ni19.3
) into two BCC phases
Congruent transformations
We have seen two congruent transformations (transformations which occur without change in composition). The list is as below.
Melting point minimum
Melting point maximum
Order –
disorder transformation
Formation of an intermediate phase
Melting point maximum
Order disorder transformation
Formation of an intermediate phase
Very few systems exhibit an isomorphous
phase diagram (usually the solid solubility of one
component in another is limited).
Often the solid solubility is severely limited-
though the solid solubility is never zero (due to
entropic reasons).
In a simple eutectic system (binary), there is one composition at which the liquid freezes to two solids at a single temperature. This is in some sense similar to a pure solid which freezes at a single temperature (unlike a pure substance the freezing produces a two solid phases-
both of which contain both the components).
The term EUTECTIC
means Easy Melting
→ The alloy of eutectic composition freezes at a
lower temperature than the melting points of the constituent components.
This has important implications→ e.g. the Pb-Sn*
eutectic alloy melts at 183C, which is lower than the melting points of both Pb
(327C) and Sn
(232C) can be used for
soldering purposes (as we want to input least amount of heat to solder two materials).
In the next page we consider the Pb-Sn
eutectic phase diagram.
As noted before the components need not be only elements. E.g. in the A-Cu system a eutectic reaction is seen between
(Solid solution of Cu in Al) and
(Al2
Cu-
a compound).
Eutectic Phase Diagram
* Actually - eutectic alloy (or (Pb)-(Sn) eutectic alloy)
Pb Sn%Sn
→
T (ºC
)→
100
200
300
10 30 50 70 90
L
+
L +
+ L
Liquidus
Solvus
Solidus
18% 62%97%
183C
232C
327C
18362% 18% 97%
Cool
C
LSn Sn Sn
Eutectic reaction (the proper way of writing the reaction)
Eutectic reaction
L →
+
Ceutectic
= CE
Teutectic
= TE
ECEC
ECEC
EC
E FD
ET
Note that Pb
is CCP, while Sn
at RT is Tetragonal (tI4, I41
amd) → therefore complete solid solubility across compositions is ruled out!!
Note the following points:
and
are terminal solid solutions (usually terminal solid solutions are given symbols ( and )); i.e.
is a solid solution of B (Sn) in A (Pb). (In some systems the terminal solid solubility may be very limited: e.g. the Bi-Cd
system).
has the same crystal structure as that of A (Pb
in the example below) and
has the same crystal structure as B (Sn
in the example below).
Typically, in eutectic systems the solid solubility increases with temperature till the eutectic point (i.e. we have a ‘sloping solvus
line’). In many situations the solubility of component B in A (and vice-
versa) may be very small.
The Liquidus, Solidus
and Solvus
lines are as marked in the figure below.
Pb Sn%Sn →
T (ºC
)→
100
200
300
10 30 50 70 90
L
+
L + + L
Liquidus
Solvus
Solidus
18% 62% 97%
183C
232C
327C
Eutectic reactionL → +
A B%B →
T (ºC
)
100
200
300
10 30 50 70 90
L
+
L +
+ L
Eutectic reaction
L →
+
Increasing solubility of B in A with ↑T
C = 2P = 3F = 0
At the eutectic point E
(fig. below)→ 3 phases co-exist: L,
&
The number of components in a binary phase diagram is 2 the number of degrees of freedom
F = 0.
This implies that the Eutectic point is an Invariant Point
for a given system it occurs at a fixed composition and temperature.
For a binary system the line DF
is a horizontal line.
Any composition lying between D and F will show eutectic solidification at least in part (for composition E the whole liquid will solidify by the eutectic reaction as shown later).
The percentage of
and
produced by eutectic solidification at E is found by considering DF*
as a lever with fulcrum at E.
EFD
EFD
*
Actually just below DF as tie lines are drawn in a two phase region
2 μm2 μm2 μm
Examples of Eutectic microstructures As pointed out before microstructural information is often overlaid on phase diagrams. These represent microstructures which evolve on slow cooling.
Al-Al2
Cu lamellar eutectic
Sn
Pb
Composition plot across lamellae
Pb-Sn
lamellar eutectic
Though we label the microstructure as Pb-Sn
lamellar eutectic it is actually a - eutectic.
Al2
Cu(note that one of the
components is a compound!)
(Al)
C1
C2
C3
C4
The solidification sequence of C4
will be similar to C2
except that the proeutectic
phase will be
Pb-Sn
eutectic
What is meant by ‘microstructural level equilibrium’?Funda
Check
let us understand the concept using an example considered before.During the eutectic reaction (during slow cooling) a lamellar micro constituent is obtained.
2 μm2 μm2 μm
This results in a huge amount of interfacial area between the two phases (Al, Al2
Cu), which will result in a high value of interfacial energy.
Fig.1: Al-Al2
Cu eutectic
The equilibrium state would correspond to the schematic as shown
below.
Since we ‘tolerate’
the microstructure as in Fig.1 (and do not take the system to the global energy minimum state), the equilibrium considered in typical phase diagrams are microstructural level equilibrium.
Polyhedral crystals
Peritectic
Phase Diagram
Like the eutectic system, the peritectic
reaction is found in systems with complete liquid
solubility but limited solid solubility.
In the peritectic
reaction the liquid (L) reacts with one solid () to produce another solid ().
L + .
Since the solid
forms at the interface between the L and the , further reaction is
dependent on solid state diffusion. Needless to say this becomes
the rate limiting step and hence it is difficult to ‘equilibrate’
Peritectic
reactions (as compared to say eutectic
reactions). Figure below.
In some peritectic
reactions (e.g. the Pt-Ag system-
next page), the (pure)
phase is not
stable below the peritectic
temperature (TP
= 1186C for Pt-Ag system) and splits into a mixture of (
+ ) just below TP
.
Pt-Ag Peritectic
system
Peritectic
reaction
L +
→
Melting points of the components vastly different.
Pt-Ag is perhaps not a good example of a peritectic
system–
obvious looking at the
phase field (not stable below the peritectic
composition).
118666.3% 10.5% 42.4%
Cool
C
LAg Ag Ag
PCPC L
PC
PCPC L
PC
PT
Note that below TP
pure is not stable and splits into ( + )
Formal way of writing the peritectic
reaction
Funda
Check
Components need not be only elements-
they can be compounds like Al2
O3
, Cr2
O3
.
Phase diagrams usually do not correspond to the global energy minimum-
hence often
microstructures are ‘tolerated’
in phase diagrams.
Phase diagrams give information on stable phases expected for a given set of thermodynamic parameters (like T, P). E.g. for a given composition, T and P the phase diagram will indicate the stable phase(s) (and their fractions).
Phase diagrams do not contain microstructural information-
they are often ‘overlaid’
on
phase diagrams for convenience.
Metastable phases like cementite
are often included in phase diagrams. This is to extend
the practical utility of phase diagrams.
Strictly speaking ‘cooling curves’
(curves where T changes) should not be overlaid on
phase diagrams.
(Again this is done to extend the practical utility of phase diagrams
assuming that the cooling is ‘slow’).
Precipitation
The presence of dislocations weakens the crystal → leading to easy plastic deformation.
Putting ‘hindrance’
to dislocation motion increases the strength of the crystal.
Fine precipitates dispersed in the matrix provide such an impediment.
Strength of Al → 100 MPa
Strength of Duralumin with proper heat treatment (Al + 4% Cu + other alloying elements)
→ 500 MPa.
Precipitation Hardening
If a high temperature solid solution is slowly cooled, then coarse (large sized) equilibrium precipitates are produced. These precipitates have a large distance between them. These precipitates have incoherent boundaries with the matrix (incoherent precipitates).
Such (coarse) precipitates, which have a large inter-precipitate distance, are ‘not the best’
in terms of the increase in the hardness.
Hence, we device a 3 step process to obtain a fine distribution of precipitates, which have a low inter-precipitate distance, to obtain a good increase in hardness.
Philosophy behind the process steps in Precipitation Hardening
Coarse incoherent precipitates, with large
inter-precipitate distance
Multi-step process used to obtain a fine distribution of
precipitates (with small inter-precipitate distance)
Not good Better
Al-Cu phase diagram: the sloping solvus
line and the design of heat treatments
The Al-Cu system is a model system to understand precipitation hardening (typical composition chosen is Al-4 wt.% Cu).
Primary requirement (for precipitation hardening) is the presence of a sloping solvus
line
(i.e. high solubility at high temperatures and decreasing solubility with decreasing temperature). In the Al rich end, compositions marked with a shaded box can only be used for precipitation hardening.
Sloping Solvus
line: high T → high solubility
low T → low solubility of Cu in Al
AlCu
4 % Cu
+
→
+
Slow equilibrium cooling gives rise to coarse
precipitates which is not good in impeding dislocation motion.*
RTCu
TetragonalCuAl
RTCu
FCC
CCu
FCCcoolslow
o
% 52)(
% 5.0)(
550 % 4
)( 2
*Also refer section on Double Ended
Frank-Read Source in the chapter on plasticity: max
= Gb/L
C
A
B
Heat (to 550oC) → solid solution
Quench (to RT) →
Age (reheat to 200oC)
→ fine precipitates
4 % Cu
+
CA
B
To obtain a fine distribution of precipitates the cycle A
→ B
→ C is used
Note:
Treatments
A,
B,
C are for the same composition
Supersaturated solution
Increased vacancy concentration
Heat treatment steps to obtain a fine distribution of precipitates
Assume that we start with a material having coarse equilibrium precipitates (which has been obtained by prior slow cooling of the
sample)
.
A: We heat the sample to the single phase region () in the phase diagram (550C).
B: We quench (fast cooling) the sample in water to obtain a metastable supersaturated solid solution (the amount of Cu in the sample is more than that allowed at room temperature according to the
phase diagram).
C: We reheat the sample to relatively low temperature (~180C/200C) get a fine distribution of precipitates. We have noted before that at ‘low’
temperatures nucleation is dominant over growth.
Log(t) →
Har
dnes
s → 180oC
100oC
20oC
Higher temperature
less time of aging to obtain peak hardness
Lower temperature
increased peak hardness optimization between time and hardness required
Schematic curves →
Real experimental curves are in later slides
Note: Schematic curves shown-
real curves considered later
Log(t) →
Har
dnes
s →
180oC
OveragedUnderaged
Peak-aged
Region of solid solution strengthening→ Hardness is higher than that of Al
(no precipitation hardening)
Region of precipitation hardening
(but little/some solid solution strengthening)
Dispersion of fine precipitates
(closely spaced)
Coarsening of precipitates
with increased
inter-precipitate spacing
Not zero of hardness scale
Log(t) →
Har
dnes
s →
180oC Peak-aged
Particle radius (r)
→
CR
SS In
crea
se→
21
r r1
Particle shearing
Particle By-pass
)(tfr
Cohe
rent
(GP
zone
s) In-coherent (precipitates)
Section of GP zone parallel to (200) plane
Cu rich zones fully coherent with the matrix → low interfacial energy
(Equilibrium
phase has a complex tetragonal crystal structure which has incoherent
interfaces)
Zones minimize their strain energy by choosing disc-shape
to the elastically soft <100>
directions in the FCC matrix
The driving force (Gv
Gs
) is less but the barrier to nucleation is much less (G*)
2 atomic layers thick, 10nm in diameter with a spacing of ~10nm
The zones seem to be homogenously nucleated (excess vacancies seem to play an important role in their nucleation)
GP Zones
Atomic image of Cu layers in Al matrix
Bright field TEM micrograph of an Al-4% Cu alloy (solutionized and aged) GP zones.
5nm
5nm
Selected area diffraction (SAD) pattern, showing streaks arising from the zones.