so let us the (( )) from where we left over
ah namely in solute segregation profiles and
we are in the second part of that and in the
second part what we are going to do is look
at the equation that we have derived for this
variation which is exponentially decaying
in nature and then see how that would allow
has to inspect the kind of micro structure
that would be formed in the fusion weldment
we would like to see how it could guide us
to say whether you would have a (( )) green
structure or in (( )) structure and in a weldment
you normally want to prefer (( )) and the
where we want to proceed as follows we have
already seen these small domain which is a
small part of the weldment ah aligning in
the direction of the maximum terms to gradient
and you are instructing this domain and seen
how the compression profile will be away as
view go from the solute to the liquid in the
direction of the maximum terms gradient
and this profile nature is already given to
you in the previous part we have discussed
it has the liquid composition given by these
expression 
expression decaying and v is the velocity
of the solute liquid inter phase in the fusion
zone it is conv ah it is related to the ah
the torch velocity by it a trigonometric function
depending on the distance from the surface
of the weld to the bottom of the weld so what
is what we know about v d is a diffusivity
of the solute in the liquid and x is the distance
from the inter phase into the liquid let means
this is profile is available only for liquid
region in case there are situations where
in the solid there is some amount of disquisition
that could be accounted for and then you would
called as back diffusion and that will be
ah regarded separately later on but for not
that is ignored and we are only looking at
limited amount of mixing in the liquid and
because of which the profile is their
i am showing you here a simple phase diagram
of a dilute alloy for which we would have
done this analysis and i have would drawn
the phase diagram showing with the solid region
solid plus liquid region liquid region and
the composition that we have chosen average
composition c naught is here so where we heat
the liquidous that is a temperature we are
calling as t l and where we heat the solid
we are calling as solid has we are calling
as solid as temperature and the initial the
first solute to form for a composition of
c naught is k c naught and i would write it
here this is k c naught and the last liquid
to certify from a unlike composition of c
naught will be c naught by k and we have seen
already that in this profile the maximum composition
at the interfaces c naught b k for away it
is c naught ok and so what we do is that in
this domain we want to draw the same plot
but not of compositions but of let us say
liquidous temperatures so how does that look
like what we are doing is liquidous temperature
is like a map ok we are just converting the
compositions in to the liquid us temperatures
use in the phase diagram so that we will do
like these same distance they way they have
drawn here ok the same distance so for comparison
let us just ensure that we are looking at
the same domain 
so how does this plot look like ok
so what we need to do is at the interface
at the at x is equal to zero the composition
of the liquid is given by c naught by k c
naught by k is such that the temperature of
the liquidous is very low which is t s and
high value is here far away you have got the
composition is c naught and c naught composition
will have a liquidous which is which is t
l so this is the t l and the variation is
an exponential decaying variation for the
composition and if you make an assumption
that the phase diagram is linear linearized
phase diagram it just means that the line
lines here the liquid ah send solute as lines
are straight lines so if you were assume that
these lines are straight then this compositional
profile which is exponential will be mapped
on to the liquidous profiles as also exponential
ok so which means that the liquidous profile
is gone a looked like that ok now this is
how the temperature at which the liquid will
start (( )) would look like
what would be the actual temperature actual
temperature should be then given by a plot
and that plot i am drawing in this manner
it would be drawn something like this this
is a actual temperature it would going through
the same point here because at the composition
we are talking about to the liquid us temperature
is matching as the condition at the equilibrium
for the solute and liquid to be their and
therefor this is how the profile now i am
not drawing the temperature profile with any
ah slope changes the reason being that the
thermal diffusivity is about four orders of
magnitude higher then the solute diffusivity
so therefor the temperature profile is going
to be even doubt much earlier then the solute
segregation profile so there is for the temperature
plot is always drawn in a straight line remember
how is a temperature plot and the weld zone
in the weld zone it look like these if you
what look at this distance over this distance
the temperature profile is going to look there
is a slope change however we are not looking
at such a distance we are actually looking
at only small distance and in that small distance
you are going to have the profile fairly linear
and thats what we are assuming and therefor
this plot we can just also put one gradient
g ok
now there are various possibilities that are
existing in a real weld if you have a heat
source that is very intense ah the heat source
distribution is very intense then what would
happen is that the temperature variation is
very steep so you may have g value so high
that you may have a situation like this 
ok g i would say high ok and you may have
a situation like this also g low ok now we
can inspect what will be the effect on this
solidification mode by looking at these two
profiles when you have a low gradient and
high gradient what you mean by high gradient
is situations like laser welding electromagnetic
welding low gradient we mean situations like
(( )) welding or (( )) welding ok so if you
change the heat source intensity temperature
gradients are changed from shallow to narrow
and therefor how you can ah look at the ah
interface is something that we can do now
now what i want you to imagine is as follows
imagine that this interface has got a perturbation
due to ah normal vibrations that are their
in the system thermal fluctuations etcetera
so let us say that this interface is not straight
but let us say it has got a fluctuation ok
let us say it has got a fluctuation i am exaggerating
it because i want to show clearly in the board
ok so the fluctuations are generally much
much smaller so if what is the fluctuation
then how would the temperatures look like
at the front of the fluctuation is something
known when you draw a vertical line ok and
you can see that there are two intersection
points ok i am putting this point has h and
this point has l ok and we can inspect what
is happening at these two points whats happening
at h that is the situation where ah high temperature
gradient under this situation at h what is
happening is that the actual temperature is
higher then the liquidous temperature you
can see that the blue line is below to green
which means that actual temperature is higher
then the liquidous temperature so 
this implies that this solid which happens
to get per tabbed and then get a bulge is
finding itself at a temperature that is suppose
to be in a fully liquid region so that solute
is not stable it suppose to melt back so it
means that the bump or the perturbation melts
back 
ok so what it means when you say the perturbation
melts back is that the front between the solute
and liquid is going to be flat it means that
the growth of the grains from the fusion zone
is going to be in a ah flat manner that means
that means that micro structure will not be
a (( )) and grandity it will be having flat
grains ok and it will be also columnar in
nature ok so thats what we mean by a high
gradient
we can inspect the situation it l and c what
would happen what would we meany by l we mean
by low temperature gradient 
if it was low temperature gradient you can
inspect what happens at l you can say at l
the bump accidentally the perturbation has
found itself at a temperature which is below
the liquidous ok so we can see that the actual
temperature is less then the liquidous temperature
which means bump is finding itself at a temperature
below liquidous means below liquidous you
are suppose to has solute growing which means
that the bump is actually now in a condition
suitable to grow which means that this break
breakage of the planer front of solidification
is going to take place and you are going to
have a perturbation that is grow which means
that in this case thew growth of the solid
is going to be either cellar or dendritic
in nature you can say cells or dendrites and
in this case you would get the planer
so this is how the impose temperature is going
to change the micro structure evolution in
the weldment as to solidify and you can see
that you are able to make this conclusion
only because we know the variation of the
liquidous and we are able to get the variation
of the liquidous in the liquid region for
the various composition in the liquid because
we know the composition profile also from
solving the equation ok so now we have this
then we can use it further to tell what would
happen what we can do is we can inspect this
height difference and ask what would happen
ok this height is basically being giving a
name this height is basically the difference
between the actual temperature and the liquidous
temperature and a temperature below the liquidous
where the solidification is enhanced or possible
then its called as under cooling and because
this is coming from the solute segregation
profile its called as solute under cooling
or solutal under cooling ok this delta t
ok what happens when this delta t is large
what happens when delta is large ahead of
the interface then what happens is that ahead
of the interface you have a situation where
the liquid is having a composition such that
actual temperature is much below the liquidous
temperature that means is all fully under
cooled it has an of driving force to solidify
so you may have situation where nucleation
can take place ahead of the interface which
means that its not accessory that the grains
of the fusion zone only can grow in you can
have nucleation of fresh grains ahead of the
interface and that means that you could have
equi axed micro structure 
ok and if delta t is very small or zero if
this is very small or zero in the sense if
it is also a situation like this like the
very high temperature gradient which is going
above the solute segregation profile causing
this liquid us temperature profile in the
blue line if the actual temperature is above
that then you have no solute under cooling
and in such situations what happens is that
it is impossible to nucleate a grain ahead
of the front and that means that the grains
belonging to the ah weldment only can grow
inside which means that you would have whats
called a columnar micro structure
so this is how we can relate the profile with
the under cooling given by the name solute
under cooling with the micro structure that
is going to be coming out there are situations
already we have seen in the reality people
have worked in welding ah would know that
electron beam welding usually gives you columnar
micro structure in the (( )) zone and if you
take g t a w we normally tend to see a equi
axed micro structure so this again conforming
that in elecron beam welding the grints are
very steep and therefor you get columnar micro
structure and in the g t a w or such processes
the temperature gradients is not so steep
and you can have under cooling and can have
a equi axed micro structure that can come
up now what governs the delta t and how can
be see that change over ok ah the implied
the applied temperature gradient is it good
an of to give you the equi axed micro structure
are not but can be known by seeing with respect
to the tangent where we are ok so i would
just draw the tangent in the dotted line if
this was a solute segregation profile we can
draw a tangent this is the tangent and we
can inspect this has a critical slope and
we can ask a question the imposed a temperature
gradient if it was less then the critical
critical gradient of a temperature profile
that is tangential to this liquidous profile
so then you have a situation where you have
equi axed green possible equi axed grow possible
so this criterion is also call as the the
ah criterion to know that solute under cooling
will be take place so its also call as a solute
under cooling criterion so what would be the
g star is important for a inspect
now g star is nothing but the tangent to this
slope and tangent to a slope can always be
inspected because you already know how to
inspect that here ok so we could then write
a simple expression and put that value here
and let me just clear the board at one and
then then show you their ok we can show it
here ok now what we need to do is basically
ah g star ok and g star is nothing but the
critical temperature gradient and that means
it is the liquidouos temperature gradient
like this t liquid itself is known for us
at the interface this itself is known t liquidous
it is known from looking up the liquid us
temperature from the compositions and if the
liquid us is says ah straight line we can
say that it is m that is slope of the line
and the solutous and liquidous lines are going
down so it is minus m into the compositional
profile ok and this already we have the expression
their so you could write it has dow by dow
x minus m and we have got that expression
c l is nothing but c naught into m plus one
minus k by k is e (( )) power of minus v x
by d and this has to be applied at i interface
which is having x is equal to zero so you
can then substitute that to see how it looks
like then that would be minus m c naught ok
and one is gone because you are differentiate
so you have one minus k by k coming a and
you would have e (( )) power minus v x by
d into minus v by d at x is equal to zero
and you know that x is equal to zero this
function in a goes to one so minus and minus
will get canceled and you would have a value
like this m c naught into one minus k into
v divided by k into d this is the value of
g star
what it implies is that once the g star is
known like that when you can inspect in an
actual weld what is the temperature gradient
i am getting at the fusion boundary if that
temperature gradient is less then this value
then i would be having the solutal under cooling
possible that means i will have equi axed
micro structure and in case the temperature
gradient interface is more then i will have
a columnar micro structure and that value
g star depends upon the phase diagram para
parameters such as the liquid us slope m the
actual composition c naught and the partion
coefficient k the velocity of the interface
v which is related to the torch velocity and
then d is a diffusion which means that related
to the process conditions as well as the phase
diagram now process conditions are basically
coming by a the velocity of the interface
because that is related to the torch velocity
and parameters like d c naught k and m are
coming from the phase diagram c naught is
something that we can choose as a process
parameter the alive composition so k m and
d are coming as a property of the material
so you can see that it is a condition that
can be tuned
we can know ask in the assignment various
questions what would happen when you change
each of those parameters to this condition
so given to alloys which have the alloys will
have a higher tendency to form for example
equi axed micro structure is something that
we can ask so you can inspect this relationship
with is to all of them and then attempt the
assignment problems and you would get in a
ah idea of how this can be interpreted and
i would also show you some videos of a real
time visualization of this solidification
under these conditions and you would get an
idea of how the micro structure is evolving
and what i actually mean by perturbation and
what does it mean to say a flat interface
growing or a interface that is growing as
a cellar and (( )) so i would show you some
videos which i have ah i have planning to
apply upload in the google website i will
show you that and then you can also see the
micro structures of typical ah weldments under
these two conditions and then you will appreciate
the meaning of this particular concept
so with that we come to the conclusion of
this solute aggregation profile and then if
you attend the assignment problems you will
get clarity on ah this concepts further on
so then ah we will continue in the next session
about the micro structural evolution further
so with that we close this lesson
thank you
