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Good morning, we continue our lecture on solidification
of binary alloys. Now we have, so far looked

at isomorphous system, it is cooling curve;
we also looked at the lever rule, which can

be applied to find out the percentage phases,
which are present in a particular case, I

mean wherever you have two phase is coexisting.
We also looked at the difference between an

ideal and real solid solutions. We talked
about free energy composition diagram in the

case of isomorphous system. Then, we moved
over to cases, where there is solubility limit,

that means, two metals, they are not soluble
in solid state in all proportion; in those

cases, there will be a situation, where three
phases can coexist. And we looked at two particular

cases; eutectic and peritectic system. And
and we looked at how this structure evolves

in detail in eutectic and peritectic alloy.
Now we will continue for here onwards

And basically, what we have seen that the
transformation that takes place during solidification

of a binary alloy is represented in the form
of a diagram, which is called binary phase

diagram. This is a graphical representation
of phase compositions and and their amount

at a given temperature. And often we do it
at constant pressure, and usually the pressure

is one atmosphere. And one important condition
here if we assume that the cooling is very

slow, such that at every stage, there exists
an equilibrium between liquid and solid.

And these binary diagrams, they are well useful
tool for quantitative evaluation of the micro

structure of the alloy; and hence, it can
also in for some thing about its mechanical

and other physical behavior of the alloy.
Now, in the case of an isomorphous system,

where there is unlimited solubility, and both
liquid and solid state, the phasing takes

place over a range of temperature, we looked
at this. We looked at eutectic system, where

there is a partial solid solubility, and here
there will be a some stage, a liquid, which

dissociates into a mixture of two solids - alpha
and beta; in phase diagram we always represent

liquid using an this roman alpha that L represents
liquid, and Greek alphabets are used to be

represent phases. Now, we also looked at a
peritectic system, where… And this is also

a case, where there is a partial solid solubility;
there is a limited solubility in the solid

state, but unlimited solubility in liquid
state.

And here, on the second conditions over a
range of competition, the liquid reacts with

solid, which has precipitated out to give
a different solid; like in this particular

case, liquid reacts with alpha to give beta,
where beta has entirely different physical

characteristics and structures from that of
alpha. Now these two cases - eutectic and

peritectic, these are examples of invariant
reactions; and in fact, at 1 atmosphere pressure,

if three phases are coexisting in a binary
system, that in binary system, where number

of component is two; in that case, if you
applied this phase rule, you will find the

degree of freedom is 0. So, that means this
type of equilibrium can coexist at a fixed

temperature and between a fixed composition
of liquid and to solids. So, there is no variable

that can be altered. So, in fact, in principle
from the thermodynamic characteristics, it

is possible to calculate these temperatures.
And we will see today, one such example.

Now, there will be several order invariant
reactions as well, which are possible apart

from this, but basically similar to that,
that one phase is separating into two phase

or two phases reacting to give a third phase.
So, this type there are certain other possibilities

as well, and we will look at them in subsequent
part of the lecture. And also we will look

at some different variants of isomorphous
system.

First start with this isomorphous system in
the phase diagram, you know just to recollect

that, this is that phase diagram of an isomorphous
system. And in the two phase region, one can

apply phase rule, that lever rule; one can
apply the lever rule to find out proportion

of alpha and liquid, which are coexisting;
like over here, the alpha of this composition

given by that point p, that composition is
X 1. So, this can coexist with a liquid of

this composition X 2; and in this case, in
using lever rule you can say that alpha, amount

of alpha is proportional to q r q r, and amount
of liquid is proportional to p q. And there

are certain alloys, which exceed it this kind
of phase diagram, which are listed here, like

one of the common alloy, is copper nickel.
And here, if you look at both copper and nickel,

they have identical crystal structure, they
have nearly similar lattice parameter, difference

is less. So therefore, atomic size’s difference
is less, so there exhibit unlimited solubility.

And same, I think will be valid, and for you
to check up what are the crystal structures,

and lattice parameters of these cases germanium-silicon,
antimony-bismuth, they also exhibit a similar

isomorphous behavior; this kind of table will
have at least in the half; until solidification

is complete, they will exhibit this type of
phase diagram

Now, there are cases where this type of isomorphous
system can exhibit maxima or minima; so that

means, in between an alloy, there can be an
alloy, which has in this particular case,

higher melting point than both A and B; and
which is shown in this diagram. And we will

look at it here, say suppose this is actually
T over A, this is B, you have percentage of

B with percentage plotted along this axis,
this is melting point of A, here it is liquid,

and here it is alpha. Now, here you have a
maxima, at so that means, you can look at

it as if it is two isomorphous system, one
this side, another this side.

You can also have a case something like this,
exactly similar; here you have liquid, here

you have alpha, and here this is T over A,
this is T over B, this is melting point of

A, melting point of B; at in between, you
have an alloy, which melts in exactly same

way as that of pure metal. If you try and
plot, it is cooling curve of this alloy, it

will actually be a temperature and time; its
plot will be more or less exactly similar

to that of pure metal. So, this is the melting
point of this law, let us say, this is the

melting point; and it will exactly, I mean,
its solidification behavior will be similar

to that of pure metal. So, this kind of deviations
from an ideal isomorphous system, they are

result of deviations from ideal deviations
from ideal solid solutions. And we will later

on, we will see how we can express this deviations
in a more quantitative fashion.

Now, binary eutectic will looked at eutectics,
an in this particular cases, where where we

have two fields, alpha, beta these are terminal
solid solution, this is liquid; here you have

alpha plus eutectic, here you have eutectic
plus beta. And the two cases, you know, there

are certain binary alloy system will exhibit
this kind of behavior like silver-copper,

aluminum-silicon, led-tin, they have this
type of phase diagram; and this is a common

solider alloy, this is used as led-tin is
is a solder alloy, it is a low melting material.

And the type of structure that you can get
in eutectic is shown here; and we can look

at this, say one type of structure in eutectics,
say this alloy, here it is totally liquid;

and once it solidifies, solidification is
complete, you will have a structure, which

will give you really an intimate mixture of
two phases alpha and beta; here, suppose if

see alpha solidifies, surrounding liquid becomes
rich in beta, so automatically, the beta solidifies;

so in fact, one after the other, it will start
farming as so, you have a layer of alpha and

that a layer of beta will form; say something
like this, you you have a layer of alpha forming,

then within that, you will have beta, then
again alpha. So, this is be a lamella structure

can form.
Similarly, you can also have this type of

structure; if alpha is precipitates out, surrounding
if alpha precipitates out, surrounding is

beta, again alpha precipitates out. So, you
can have this kind of a very intimate mixture

of this kind of a rod like this precipitate,
which is chromatically shown in this diagram

mixed one here. You can also have certain
irregular structures are also possible, where

maybe one phase is like this; regular structures
are also possible; but what it means that

eutectic will be an intimate mixture of two
phases. So, this is the say beta, this is

alpha, and this pattern can be regular or
irregular, and this depends on certain physical

characteristics of the two metric A and B.

Now, you can it is also possible to have one
extreme case, where the solids are totally

invisible like here; this is a level diagram
of an binary eutectic, where there is no solubility

in the solid state. There are certain clear
examples like cadmium-bismuth, antimony-led,

which exhibit this kind of phase diagram.
And here, the eutectic will be an intimate

mixture; the hypoeutectic, you will have primary
alpha, primary that means, A crystal of A

and eutectics; and here in the hypereutectic
alloy, you will have primary B crystals of

B and then eutectic.
Now, use in thermodynamics, it will also possible

to calculate if suppose, we take the case
of cadmium-bismuth; if you know the melting

point of cadmium and bismuth unknown, the
in a latent heat of fusion of cadmium and

bismuth is known; in that case, if we assume
that liquid is an ideal solid solution, it

is possible to calculate from this thermodynamic
data that means, melting point and heat of

fusion. The composition as well as the temperature,
that eutectic temperature and eutectic composition,

it is also possible to plot this liquid as
line, this liquid as line as well as this.

And we will see look at one such example of
calculating this type of phase diagram; in

fact, we have seen one such example in the
case of isomorphous system, where both liquid

and solid are assume to be Raoult’s law;
that means, both liquid and solid are assume

to be ideal solution.

So, let us look at the determination of eutectic
diagram from thermodynamic properties. Now

this is a problem bismuth-cadmium, they are
soluble in liquid state, but insoluble in

solid state; estimates it eutectic composition
and temperature; the melting points and latent

heats of fusion of bismuth and cadmium, they
are given below. Assume the liquid to be an

ideal solution; and since pure bismuth and
pure cadmium are precipitating out, we can

assume that their activity, you will be equal
to 1. And these are given over here; this

is the melting point of bismuth is 271 degree
centigrade, its heats of fusion that is latent

heat of fusion is 2.6 kilo calorie per mole;
cadmium melting point 321 degree centigrade;

and its latent heat of fusion is 1.53 kilo
calorie per mole. And let us see, how we proceed

with the calculation. So, here the main principle
is that you have to calculate the free energy

of this transformation.

And look at a case in the phase diagram, consider
a temperature over here. If you are in this

region, you have pure A a equilibrium, pure
A is in equilibrium with solution; that liquid

consisting of liquid or as a liquid solution
of B in A. Now, how do we calculate the free

energy of transformation,

which is illustrated
here; say suppose in this particular case,

you take that pure A as standard state; in
free energy calculation, definition of standard

state is quite important; and we take the
standard state that is pure A at T, temperature

T.
Now, in this case in that case, what is the

free energy of pure A? Say since it is a mixture,
it is RT ln activity of A; so we write it

mole fraction of A and activity is equal to
mole fraction. So, in that case, this is 1,

because it is pure A, so this is 1, so therefore,
for pure A, this is 0. Now how do you calculate

the free energy of that liquid or let us say
partial we want to find out partial free energy

of A in liquid; how will you find out? In
that case, you take first pure A - solid,

it transforms into liquid; and then we assume
that this A liquid goes into solution. So,

in that case, you can easily write it down,
you can check up your earlier notes.

And this particular case, here that free energy
will be delta H of that melting of A times

1 minus T over melting point of A; whereas,
this is RT ln we assume ideal solution; solution

to be ideal, then this is the atom fraction
A in liquid. So, you add the two, this is

the free energy, so this is the partial model
free energy of A in liquid, and this should

be equal to this. So, that is the chemical
that partial molar free energy is also known

as the chemical potential; and this two potential
should be 0. So that means, over this, if

you sum these together, so you assume that
this is some kind of a chemical reaction,

and you add the two, and they are in equilibrium.
So, therefore, these two should balance; and

with this, what you get is listed here.

In this particular case, you can see that
atom fraction, the large atom fraction A in

liquid will be a function of which is shown
over here. So, this you can just algebraic

simplification, this is equal to this; and
you can apply the same concept to this region

of the diagram, and then you get similar expression
for this. Now, if here, so all these are known,

this is known, this is known, this is known,
temperature is fixed. So, it is possible to

calculate N A that is composition of the liquid
as a function at any given temperature T.

And if you do that, you will be able to generate
this plot.

In this same way, if you try and solve this,
then you will be able to generate this plot;

and wherever these two meet, so this is where
composition of the liquid, which is in equilibrium

with A is exactly same as the composition
of the liquid, which is in equilibrium with

B. So, in this particular case, the sum total
of this will be 1. So, in this case that that

that you have to possible to find out the
eutectic temperature as well as eutectic composition,

and for this you can easily set it up in a
spreadsheet, and then and these results which

are shown here, on the next one.

So, these are the different temperature; this
is the melting point of 1 or the 2, which

is the higher; and this is in kelvin, degree
kelvin; and you calculate that N B - atom

fraction B; and this is the other one.

And if you plot the two, then you get this
is the diagram you get, and this eutectic

composition comes out to be close to around
let us say 0.55 atom fraction, and this temperature

comes out to be 408 degree kelvin.

So in fact, if you check up this diagram,
you will find you will find that although

this, there is a good match between this atom
fraction, this composition is exactly seen

as as given in the phase diagram, which is
experimentally found out. But this temperature

is possibly, is a slightly lower; what is
there we have found out by this. So, in that

case, why is this deviation, why it is not
matching? So, one possible explanation could

be we have made on assumption that the liquid
is ideal, second is we are assumed that there

is no solubility.
So now, it is a question of is there a slight,

which is extremely small amount of A is soluble
in B or B in A, these are some of the possibilities,

we can think about; so that means, you have
to question this assumptions only, but in

the procedure, this is a procedure which is
applied; and in one way, you can see that

thermodynamic properties are also evaluated
from the phase diagram. So, actual experimentally

you determine, and then you try to conclude
whether that liquid, a solution is an ideal

or not; if it is not ideal, from this it is
possible to calculate, how much is its deviation

from ideality?

Now, next we also looked at the binary peritectic
system. Now, binary peritectic system you

know, this is one way, you can when you look
at the phase diagram, you said this is a very

useful tool, you know by which you can know
about the phase percentages, you make some

quantitative estimates, like say in this particular
case, say this alpha beta region, you know

how well its property depend? Suppose we assume
that alpha is softer, beta is harder. So,

there is a possibility, what we can see? You
can apply the rule of mixture to find out

is properties.
So, if you know percentage alpha, it is possible

to… And to find out the property of the
mixture; say suppose we are we say that a

particular composition say say and this, we
try and find out that on this axis, we super

impose tensile strength or hardness. Say suppose
beta has a certain hardness, say of beta of

a particular composition, this composition
has certain hardness, say let us say, this

is the value of this hardness; and then alpha
has certain hardness, in that case in this

region, as amount of beta increases, if you
go along this, it will follow is some something

like this, a linear dependent. So, if you
know percentage beta, you will be able to

find out such properties.
And also, if you look at the micro structure;

say suppose of this diagram is not known,
it is also possible to find out or locate

this composition these compositions from micro
structure, analysis of the micro structure.

So, what you have to do? Look at two micro
structure or two compositions, look at this

micro structure, find out percentage alpha
here; find out percentage alpha here. And

in both cases, both the two phases alpha and
beta, say here as it is shown here, they will

have a composition, alpha will have this composition,
beta will have this composition. So, using

few a lever rules, you can set up one equation
with these compositions.

And secondly, follow the second alloy, you
can write a similar expression; in that case,

and once you know have two equation, two unknown,
you will be able to find out these compositions.

So, it is both way possible; if the phase
diagram is known, you can find out its properties

from phase percentages if or else if phase
diagram is not known that you find that, this

is the region you have two phase structure;
then you find out at least make two alloy

in the similar in the region, and then find
out percentage alpha and beta. And from that,

you will be able to find out the compositions
that is of this solvers, that is solubility

limit of alpha and beta.

And now, let us look at we have looked at
two types of invariant reaction; see one is

a eutectic, where liquid dissociate into a
mixture of two different phases, two distinct

phases having different physical and properties
physical mechanical properties; a crystal

structure like alpha and beta. And we also
have looked at a peritectic system, where

two phases say liquid reacts with alpha giving
giving a totally different crystal say beta,

a different phase called beta. So, these two
cases, we had looked at.

But real phase diagrams are not see basically,
not not simple solid solution or eutectic

or peritectic, they often consist of several
two phase regions, and invariant reactions

involving three phases in a binary system.
Now, these commonly invariant reactions, you

have some invariant reactions other than eutectic
and peritectic, and we will look at them.

And we can broadly classified this invariants
into two groups in one case, one phase on

cooling separates into two different phases
like that in eutectic; and another case, two

phases on cooling reacts to form a different
phase like in peritectic.

Now, let us look at the first that is one
phase dissociating into two phases; now this

example we have looked at in detail, say that
is eutectic; and here this shows a small part

of that eutectic phase diagram we eutectic
isotherm. So, like here this is eutectic,

you have two solid solution alpha and beta,
and this is the liquid, and this dissociates

here. So, here it is totally liquid, and here
it is a mixture of alpha and beta; and they

are intimate mixture. And now we we can esteem
this, in cases where the liquids also can

have solubility limit, there are several examples
in liquid, which you can think about try and

makes all and water, they do not makes.
So, study the possibility that there can be

in these two phases, if you are trying to
makes in the liquid state, they may not be

soluble in each other; they may, and there
or they may be partly soluble in each other.

So, there can be several such examples. And
let us generalize that we have a liquid, and

it dissociates into two different liquids
having different composition; one we represent

as L 1 and another we represent as a L 2.
And if we have a reaction say one liquid L

1 dissociates into a solid another different
liquid, and which is shown over here; here

this is alpha, this is L 1, and this side,
it is liquid is L 2, and this is the miscibility

gap; so that means, liquid here, in this particular
composition liquid, if it has this composition

at this temperature, they will not mix and
they are not totally miscible, they are not…

They will dissociate into two liquid L 1 and
L 2.

So, if you extend this, there may be a possibility
that higher temperature chance of miscibility

is higher. So, here they are miscible, if
you go down this temperature, there is a miscibility

gap, it is dissociates into two liquids L
1 and L 2, this is a miscibility gap. So,

in this particular case, this liquid is here,
it is reacting, it is it is dissociating as

you cool this temperature is dissociating
into alpha, another liquid; and this type

of reaction is called monotectic reaction
or this is also an invariant, because you

have three phases coexisting in a binary system.
And you can have amount of variant, it is

not bad, it is only liquid can dissociate
on cooling into a mixture of two phases, you

can also have a solid, say gamma; on cooling,
it dissociates into two phases, two other

solid different solid phases, alpha and beta;
and this diagram is shown is here. So, gamma

is dissociating into alpha and beta, so here.
So,

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