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than is someone
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and late things like relations
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fundamentals and applications in
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for perry also
so I I first who's this idea
pay workplace way are home
and do this in a very simple contact very thing
cock and
why do they want to I can't cover everything in 2012 after that is kinda
with that as a basis show you
where the bodies very in the
in the Facebook and Miami
one for you he information you need to understand when you look at a few
what questions to ask as to whether or not
calculation correct now first done
done a good job insuring
yet and that
through that me Facebook
well okay
so their their number reviews on
in literature these to famous was this one
is focused primarily on solidification I'll applies
this one is focused primarily on the physical metallurgy
these settlement grew up almost independently
communities in and the the way these communities look at these models are
there for a little bit different
a text book now by providers in Alberta that
we have used at northwestern your cup
class and out month also has more
0 visits oriented papon and so they said I want to give you an introduction to
this
bargain-basement both model think up
and how that how you build up some and their and
then talk a little bit more about the challenges
and limitation basically how you know
that you getting the right answer and so that the idea here
on Isabel efficient computational method
to allow you to track the evolution so
which is like to be able to do to protect the magneto microstructure
ariel's with shape the faces are with the compositions are
the 92 former uniformity in the composition
this can be from Ed be in film ferroelectrics
can be magnetic materials the nice thing about baseball bat there
Roy narrowly easy had
new is to place therefore model
other other 28 and gold
course use the models as one stage in
material design and to try to fix ideas let me show you the challenges
that we faced before face feel
met
and this is a picture of a dead right it's growing into a liquid
and if you want to describe the evolution this dendrite
in other words predict with the local or velocity is at that point
what you need to do is determined to solve the diffusion equation
classical now them City both the solid-liquid which is not so bad
I mean if you look up Numerical Recipes sample
the problem with a
with these interface I
interfaces or these patients racial problems is your boundary conditions the
and in this case the temperature at that point on the interface is penned by
melting point
minus the capillary life
energy constant mean curvature and why this is such a challenging problem
well setting up this problem people have Numerical Recipes
show you how to solve this equation way about it
0 are about it 0
we found mission L and i cant sell
you think you are how you doing this problem well
here is the interface with this interface is evolved with time
right so the boundary conditions where you could
chain time and the question is will have to determine locations that interface
you determine locations face by saying that the velocity that point
times the lake you know if he will leave the face boo
is going to be related to the temperature gradient simple
side and so I am
this just tell velocity
tell okay and the prop so therefore what you have prop this couple problem where
you need a boundary condition you pleased with the back
but you don't know where to put the bandage because you have been you don't
know
you have to solve this equation but that term depending on the gradient
temperature which comes from this equation you can solve this will be good
with this okay and the call rebounded problem
all and in this case I their their highly nonlinear because the curvature
here's a
nominee from shape as of people trying to do
for for face still met they would take a
then do it like this and they're trying to mess this up tried track
sir right and then who mesh points around
according to this kind of a long have a mess point
wizard their time and I solve this equation
and that basically basically
limited these problems the one dimension
so that the the problem of course is a dendrite is that the curvature
areas with position hymie curves along the curvature
it was even more complicated in the sense that now you have to
my boundary conditions it very with space along the interface
so not only you don't know the location
but the actual value the up the ban
depend strongly on the shape would you also don't know word
let me give you an example for
something that were rounded polymers or other
other materials where that where you have by continues like structures
and so on this looks like a sponge what it really is
solution to phase transformation problem where you have
diffusion Bowl the inside and outside face the outside faces
I'll you see so you get issues in this case is skeptical
used in this is the concentration the gas station at the interface goes like
the Meeker richard is
just so for with the temperature this is the composition
black to deface given by the phase diagram against the interfacial energy
and the mass balance at the interfaces
jump in the concentration camp is going to be balanced by the
lot blowing away that's the great composition
and again a match his problem this all
you have to measure complicated
virtually this is what was not want this these interfaces will climb
in higher magnification for me
and so on
is small section that action
something like this up it's pretty much impossible
to do the interface between phase as to what they still messes
allowed us to do is to compute the evolution structures like this
and that's why it's become such a powerful
numerical me and the place to let that goes back
all the way to handle back the hundreds
con know you know the name famous names associated with this line down ginsburg
for
activity you the idea here is very simple yet to face
you have order parameter composition
depends upon with how you want to view it I'll call them order parameters
%um compositions on acts and
on you have a diffuse interface or
move change in composition
or order parameter go through the internet the nice thing about this
about that now if you can right places
evolution this order parameter you don't have to track the location
you know if you have an equation that hold all
positions in space you don't have to worry about we
okay and to avoid the explicit racking up the
you can follow the top complex structures show you
and typological transition follow naturally what I mean by couple
transitions is simple up to be
feel population to particle sitting next to each other
and as time goes on
see what happens at this point the
two particles coalescence pathological transition
Spears 21 to single
if you doing this he's met with you have to do it
something with the match here that described in 20
changed contact face mild followed
and that's where you can do things that it's that complicated
in the morphology so here's
if I taken on a very simple they still models and arrive to them for you
because they form the basis have every faith field model that
today and the idea that you have a lol free energy
so this is a free energy density two phases alpha and beta
I there's a hot barrier height between these two
and you have to face in the system
alpha and beta and you want to compute the total free energy
up this system your at and you could do that by
integrating this free energy density over this box and every time you're now
up you go over here you pick out the free energy now
I'll which is in this case where together
you go over here in the Bay Area and every time you have eighty that free
energy
you at you integrate the whole thing up and I think it's pretty clear up
your nevertheless for you basically go
particular spot in the system the term with the value of the order parameter is
that spot
put that into the free energy Catherine
that that
doing on 9 new phone system free energy as it was a
want describe the interface between the alpha and beta phase
see we describe the interface between the outer the base
so how'd you do that and you want to minimize the energy
bf this is want to minimize at and you want to describe an interface between
the two phases
as the way you do that is pretty simple right
so in the beta phase the composition we were the order parameter be uniform
season and now and then when you go to the alpha phase
these people over the interface over the interface in the office now
you have composition uniform up to the interface there
the energy this system is still there
I because you're every place in the beta phase everyplace the alpha phase
and you have this discontinue Jul this
is what I was calling sharpen your face this is why I
all the boundary conditions actually before
temperatures composition sector assume that you have is
composition here composition the outpaces
here if you use this is good for you have to follow the old
what are the problems here one of the things it becomes clear is that the free
energy not only the functional decomposition
right here but it's also fun gradients in the car
what I mean by that is that lets in a goal a particular point in the crystal
I put my finger at that point to the crystal I determine the order parameter
is and from that i can get both would
well say here is the best now
in order to get the energy system in order to get the energy
system I had also have to know the gradient in composition
and on a strange idea
Wednesday is that I put my finger at this point in the crystal
I don't know the free energy tell me
with the gradient in the composition order parameter
is point and physically what his
wide so many had
at this point and a certain range iraq on that
drafted this Adam year
when the system is is uniform
average the composition this is around is
drafted the same at this point
you have a gradient you'll you do that same at
Ridge United value you had
uniforms year so because the final iraq
range here this is knows the grade
the energy here will back the great
composition will that again urged
therefore that the energy said brain
that argument very important already
it said that these great you know back scale
with the interacted had so they're very small
sale that and you had only years
neighboring Iraq you wouldn't have graded
and so you at great youth
them 300 raid
this idea year which goes back to battle
home as important what goes on in
and he had for
that hell this said
minimize had would
her and i'd
so you have a tip
here parameter here and what happened
head grade
ahead yep
and so what this is okay
well I have to minimize her would
well-versed limited ok my sister
that no great right you can't do that
hit on one side on the
you are allowed to have zero grade as you reach a compromise
and that compromises his
and back here with a great
0 balkan
year and close to the interface
interacted
graded important what that does
the free energy ball free energy other material
go from here here up over this barrier
back down
and so it traverses
is unstable read
abated
your who won another
you go up over that W become the poor as
W related properties years
the Interfaith related Absalon
which is lead W
which is barrier between actually
that barrier as the
the actual million in this energy here
is a little different which normally would
I'll so what we're looking at
who problem gigli your mom
day him
and classic cap for rid
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on location we were interested in the
both hell his his composition
both or on
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so when you have to do lead
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period and on
he say the valid
would
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mister with this brackets
this this method here
will always work all red
vector okay
so this here is very riveted
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as I say well I have to set for 2012
extra and ed
back hero
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want them here fine just the right way
said well did
period composition
if you wanted generally said
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variation
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we have this here hero
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if you have a couple of people
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cool and so
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if you have
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process and result
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comes from one wanted
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quoted you can tell whether third
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there's up
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this this equation
fine difference saluted
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work you grid
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and this his more and
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exactly the same Hilliard
worked there that Don sir
and this is motivated by hell were
on Wed I'll
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as or disorder where you have
handle classes the
their domain he said
her exacted
you go through exactly the same period
with not considered said
great him
basically with said the order parameter
is moved I'm Radian or
no long rated
so yep
or you at the park if I I thought this is
more here the law
here the great by is
posited mine
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5 Nate and so York parameter
used and this this
on smile essentially what happened
put a marble 3m here
and we'll roll down
well and can roll down to the bottom of the well
because 9 third
locally well said
the compositions and Jane
online both
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next year his that meant
sitting right here doesn't know whether role in that
to read direction America very good
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go back to the problem interested
yet right growing liquid
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curvature late released your face
away by you want
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and non did would
so this looks like the classical
right and so what you do is you
source good that related
time for a change the order PSL
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and when you're here he
moving in this direction here
with five function like
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cool when he calls to the
if here you
is pretty clear that by me you
Jamie this turned on he
this here gives you
fusion when you far away ended
when you up cap
we this term
here here
or right whenever epsilon hindered
against the polynomial in
the parameter and on the temperature
them is the back to
this smile that more exactly Langer
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and when I was supposed to when the
Institute theoretical physics and 1
nope on is mile for
occasion now and
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whole I'd
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so the idea
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he and then go into this
field model at Salon we kinda boring
let you know how would how
polynomial such that lucid
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is reentered and well I want to calculate
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and what he would tell you that
know if like like and the problem is
that this free energy this this apart the free energy function
easily unstable
other yet in other words
while material composition here
in between the two at all it will who'd
showed said and
you'll with me said
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and so the really is
that'd hit hard
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at all because as you get closer and closer hood
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i'm aware. while
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in the in the ball ball and
coarse-grained very clever way really very close here
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do a calculation resolved
with me have placed med
in this use in put at least six did
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and this is a hundred microns
or red
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if you doing now scale structures it'll
good my head
and that's really the problem
with basil that pal that
people said that here
we start with a simple model lied
okay so unique way
or we're Burnett we have this problem
hun parameter to reproduce the real problem
and we know it's hopeless to be a real
so want do
he
the parameters said
curvature is mine
word rather stay on your bed
on the really good I miss the old
radius of curvature here which would be a big deal
right because now I'm the Rays her
here's 10 microns my scale measures on the order
micron not only
and that most people
did here
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you 1 I'll now a parameter
did
that you have to worry about and if you do the math
correctly didn't
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happen
okay
and so if you do that on you can
you can basically cabinet with goes on your and
this is you people them better than this what
but not too much better this recalculation very active
would using this idea where you write the equations with
the interface times me curvature much
okay
I'm and so they're 28 doing this met asymptotic
extensions water than this to say well I wanna let this interface that is very
small
and you do that by letting Absalon go 0
your website logo 0 you want this to be fine let you go to
infinity same time and so when you look at the literature you'll find to kind of
classifying
expansion one associated with lane with the race going
0 turns out to be not back computation helpful
but if you a still enables you to connect
these equations sharpener basically the other one is more useful computationally
United 10
me okay
with the back to this equation so here we have I'm monologue a
Houston so you have now
a dendrite or anybody with it yes the world
composition process we have a liquid and solid the diffusion coefficient liquid
is tended hired call solid
you liquid to solid you have a big jump
the peak of well use of diffusion equations and
in the liquid and solid I'm sorry the mask found at the interface
this is the velocity times the amount material rejected
the interfaces use the way I concentration gradients
you have the country goes like which you have a flat on your face times
if me richer an
and so again this is a dendrite but it's growing by song
if you doing a persona que growing it was super saturated solution
this would be the modeled
film said and
who want to point out is that this is you have this enormous change
the
and so what do you do so you have a diffusion coefficient goes from
d ltd.'s across the interface right thing that you tend to do therefore
is to say well well just let the diffusion coefficient
very with the order parameter your face like this
and you'll see this all the time well
the star your 08 and one
and what you gonna do it to take the free energy the system now
is a linear combination are different
look good and the free energy the solid with these parameters
5 in 1 -5 so if you're in the liquid
York parameter is 0 yet
created if you have a solid
and browner's one this goes away
in said
when you look good you could
that very do
Northwood day saying that
in the interface here free energy
the ball cheerleader combination
dollar word and he
away from what he's going on his
met now really
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company free energy well phase
up
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hickel in the interface some intermediate value between with
solid what is a very specific kind
variation
so this
the some very
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first said
this is
that solid his
on position hi how alone composition
composition bloom his
rejected ability
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if you look at it any microstructure up
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or is the oldest said
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this is the challenge with with a single the big problem is
marathon did
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their across here terribly
you'll get busy your day
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and this particular problem he goes
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this is the page wanna velocity
material projected goes like he very
and what you get said is this equation
and this equation said
job creation you have parameter
interface age what is that thank you very simple
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with the interface as it grows the material
and this material how
what was supposed to go into the law said
encourage over here on the diffusion side
then bolted this his
full you get serviced and
their exactly correct this is the interface
hell go to zero he said
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the don't appear but there's no way did
third and two so to get rid of these terms
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is that this thickness which allowed
you all more in other words blood
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comes with it is penalty term
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fourth
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right up
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the problem is that the place is a highly nonlinear
even in the a.m the oil a drive equations the nominees to
so the key is to find solutions to the oil agrandir equations
mmm min
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this complicated on and both nominee
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what would be a good one there very strong connection
you could is a very strong connection between
this and any and the evolution
and with that me that all these equations
decrease in and general
energy framework that allows right so if you doing
men headed let magnetic dude
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very general but that through them
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all Alan yeah
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but look at this thing where's I'll the prelude
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the problem with a lot term is
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data management hard that for example you have law
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its own principle if you model your call lessons with these equations
you could actually pick out the right don't use an abseil
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yeah yeah to the closest thing that people have died actually
getting homes long-chain
very your life structures but it
is a major problem right
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