so first so good morning everybody
first I can't ask you maybe to come
closer so you will not be so dispersed
around the room it is a great pleasure
for me to start the tutorial session we
have 55 registrants much more than the
attendees in practice so I guess that
someone's long doing you're talking
later so it is a great pleasure for me
to welcome and introduce Professor Peter
who is from the Department of material
science and engineering at the
Northwestern University Peter is one of
the leaders worldwide in the area of
physical metallurgy and related things
like response formations and so on and
this tour will be faithful methods and
fundamentals and applications
okay so with apologies to the experts in
the audience this is intended for a
graduate student level material so so I
first I first want to introduce this
idea of a face people what face field
equations are and doing this in the very
simplest context of two very famous ones
the Kahn Hilliard and alençon equations
and then and then what I want to do is I
want to I can't cover everything and so
what I want to do after that is kind of
with that as a basis show you where the
bodies are buried in the in the face
built community and by that I mean I
want to show you give you the
information you need to understand when
you look at a phase field paper what
questions to ask as to whether or not
the calculations are correct and whether
or not the person's done done a good job
in ensuring that the answers that they
get are real and that turns out to be a
major issue in the Facebook community
that isn't well appreciated this so okay
so there there are a number of reviews
in the literature these are two famous
ones this one is focused primarily on
solidification as the title implies this
one is focused primarily on the physical
metallurgy these face field methods grew
up almost independently into these two
communities and and they the way these
communities look at these models are
therefore a little bit different so
that's interesting there's a textbook
now by a provide us an elder that we
have used at Northwestern in our
computational materials class and Alex
Aman Seb also has a more theoretical
physics oriented take on on Facebook and
so as I said I want to give you an
introduction to this to the
bargain-basement simplest phase field
models you can think of and how that how
do you build up something and there
and then talk a little bit more about
the challenges and limitations and
basically how you know that you're
getting the right answers and so the the
idea here is to develop efficient
computational methods to allow you to
track the evolution of interfaces and so
what you'd like to be able to do is to
predict the Nano and micro structure of
materials what the shapes of the
interfaces are what the compositions are
the non uniformity or uniformity of the
compositions and this can be from
anything this can be in thin films it
can be in ferroelectrics it can be in
magnetic materials the nice thing about
phase field methods is that they're
extraordinarily easy to add new physics
to the to the equations that they're for
model other other situations and the
goal here is of course to use these
models as one stage of a materials
design material design effort and so to
try to fix ideas let me show you the
challenges that we faced before phase
field methods were around and this is a
picture of a dendrite that's growing
into a liquid and if you wanted to
describe the evolution of this dendrite
in other words predict what the local
normal velocity is at that point what
you need to do is to determine to solve
a diffusion equation classical diffusion
alphas a thermal diffusivity in both the
solid and the liquid which is not so bad
I mean if you look up numerical recipes
for example they'll have a method to
solve the fusion equation that's the
problem with the fate with with these
interface interfaces or these these
phase transformation problems is that
you know boundary conditions on the
interfaces and in this case the
temperature at a point on the interface
depends upon the melting point - a
capillary length which was related to
the interfacial energy times the mean
curvature and why is this such a
challenging problem well if you think
about setting up this problem if you
pull out numerical recipes they'll show
you how to solve this equation they'll
say where are you going to put the
boundary conditions okay if I have a bar
I'll put a boundary condition zero I'll
put a boundary condition at L and I can
solve that problem okay you think about
how you do it in this problem well here
is the
your face but this interface is evolving
with time right so with the boundary
conditions where you put the battery
conditions for the solution of the
diffusion equation change with time and
the question is well how do you
determine the location of that interface
and you determine the location of that
interface by saying that the velocity at
that point times the latent heat in
other words the heat released as the
interface moves is going to be related
to the to the temperature gradients in
both the solid and the liquid and so
this just tells you the velocity it
doesn't tell you the location and the
problem so therefore what you have is
this prompt this couple problem where
you need a boundary condition you need a
place to put the boundary condition but
you don't know where to put the boundary
condition because you haven't you don't
know you have to solve this equation but
that term effects depending on the
gradients and the temperature which
comes from this equation but you can't
solve this one because you don't know
don't know where to put this one okay
and these are called free bounded
problems as a result and in this case
they're hot they're they're highly
nonlinear because the curvature here is
a nonlinear function of the shape and so
what people try to do before before face
fields methods is that they would take a
dendrite like this and they were try to
mesh this up try to try to actually put
a mesh on the surface of the dendrite
and then move the mesh points around
according to this kind of a law which
say I have a mess point that I move at a
certain distance in a certain time and
then I solve this equation again and
that basically is pretty hopeless it
basically limited these problems to one
dimension so the the the problem of
course in a dendrite is that the
curvature varies with position and it
says hi mean curvature low mean
curvature and so it's even more
complicated in the sense that now you
have to apply boundary conditions that
vary with space along the interface so
not only do you don't know the location
but the actual value of the boundary
condition depends strongly on the shape
which you also don't know that's a very
difficult problem let me give you an
example for something that's more
relevant let's say two polymers or other
other materials where you have by
continuous like structures and so this
looks like a sponge and water
really is is a solution to a phase
transformation problem where you have
diffusion in both the inside and outside
phase the outside phase is hollow gear
so you can see into it so you have again
diffusion in this case it's chemical
diffusion and this is a concentration
the concentration at the interface goes
like the mean curvature this is just as
we saw before with the temperature this
is the composition at a flat interface
given by the phase diagram again that's
the interfacial energy and the mass
balance at the interface it says the
jump in the concentration times the
velocity is going to be balanced by the
flux flowing away and that's the
gradients and composition and again
imagine you have this problem to solve
and you have to mesh up these
complicated interfaces virtually
impossible to do this is what one of
this these interfaces look like in
higher magnification for my cursor and
so this is just a small section of that
big picture right so how do you actually
measure something like this up it's
pretty much impossible to do so this is
the interface between the two phases and
so what phase field methods is allowed
us to do is to compute the evolution of
structures that look like this and
that's why it's become such a powerful
numerical technique and the phase field
method goes back all the way to Van der
Waals back in the 1800s kannan Hilliard
is another name famous names associated
with this Landau Ginzburg
and the super conductivity field the
idea here is very simple you have two
phases and you have an order parameter
or composition depends upon which how
you want to view it I'll call them order
parameters you could also use them as
compositions as a function of X and you
have a diffuse interface or a smooth
change in composition or order parameter
as you go through the interface and the
nice thing about this if you think about
this a bit is that now if you can write
equations for the evolution of this
order parameter you don't have to track
the location of this interface in other
words you have an equation at all
at all positions in space you don't have
to worry about where the interface is
actually located and so it avoids the
explicit tracking of the interface you
can follow these topologically complex
structures which I'll show you and
topological transitions that follow
naturally and what do I mean by comp
illogical transitions this is a simple
2d phase field calculation of two
particles sitting next to each other and
as time goes on you can see what happens
at this point the two particles coalesce
and this is the topological transition
because you go from spheres to bodies to
a single body if you're doing this using
a sharp interface method what you'd have
to do is you'd have to do something with
the mesh here that describes each one of
these shapes when these contact phase
field models it just does it naturally
so we can follow these topological
transitions and that's why you can do
things that are such complicated in
their morphology and topology and so
here's here's my take on the very simple
phase field models and I'm going to
derive two of them for you because they
form the basis of every faith-filled
model that is in existence today and the
idea is that you have a double well free
energy so this is a free energy density
two phases alpha and beta and there's a
barrier height between these two and you
have two phases in the system alpha and
beta and you wanted to compute the total
free energy of this system your F and
you do can do that by integrating this
free energy density over this box and
every time you are an alpha you go over
to here and you pick out the free energy
of alpha of alpha which is in this case
zero the way I put it together you go
over here to beta and every time your
beta you pick out that free energy use
that you integrate the whole thing up
and I think it's pretty clear that the
integral of that whole system is going
to be zero so the free energy of the
system is zero but nevertheless that's
the way you do it you basically go to a
particular spot in the system to turn
with the value of the order parameter is
at that spot put that into the free
energy calculate the free energy and
then
so that's that's the way you would do a
non non uniform system with the free
energy density and so let's say you
wanted to have them describe the
interface between the Alpha and the beta
phases okay so you want to describe the
interface between the Alpha and the beta
phases and so how would you do that and
you want it to minimize the energy being
of this system you wanted to minimize F
and you wanted to describe an interface
between the two phases and so the way
you do that is pretty simple right so in
the beta phase the composition or the
order parameter be uniform so you sit in
that well and then when you go to the
alpha phase you skip over the interface
skip over the interface in the alpha
phase now you'd have composition that's
uniform up to the interface and you sit
there the energy of this system is still
zero right because you're every place in
the beta phase every place in the alpha
phase and you have this discontinuous
jump at the interface this is what I was
calling the sharp interface method this
this is what all of the boundary
conditions that I showed you before with
temperatures compositions etc assume
that you have this composition in the
beta phase that sits here composition
the alpha phase assists here at an alpha
beta interface if you use this
description again you have to follow the
evolution of the interface which is
difficult one of the problems here are
one of the things that becomes clear is
that the free energy not only is a
function of the composition like here
but it's also a function of gradients in
the composition and what I mean by that
is that let's say I go to a particular
point in the crystal I put my finger at
that point in the crystal I determined
what the order parameter is and from
that I can get the bulk free energy of
the free end of density what I'm saying
here is that that's not enough in order
to get the energy of the system in order
to get the energy of the system I have
also have to know the gradient in
composition and this is kind of a
strange idea because what I'm saying is
that if I put my finger at this point in
the crystal I don't know the free energy
until you tell me what the gradient in
the composition or order parameter
at that point and physically what is
Italy tell you physically why this might
be the case so imagine you had atoms at
this point and there's a certain range
of interaction on those atoms certain
interaction distance and so this atom
here when the system is is uniform if
you average the composition in the
assistance around this interaction
distance it's the same as at this point
but if you have a gradient this yellow
one you do that same average you get a
different value than if you had in the
uniform system and actually sits about
up here and so because of the finite
interaction range of the atoms the
system knows that the gradient of the
energy here will affect the gradient the
composition will affect the energy of
the system and since therefore the the
energy of the system depends upon
gradients and that argument tells you
something very important already it says
that these gradient effects scale with
the interaction distance of atoms so
they're very small scale effects right
and if you had only nearest neighbor
interactions you wouldn't have a great
energy coefficient footage we have
second-years neighbor third year in
other words you do and so if you add
gradients to the system the free energy
now depends upon the gradients and this
idea here which goes back to van der
waals actually has an important
influence on what goes on at the
interface and so if you go back down to
what we had before you can have beta
alpha this is the free energy of the
system now and now that's the question
let's minimize F with this new term and
now you have a problem right so you have
beta that goes up to here copy the order
parameter beta goes up to here and what
a parameter alpha goes up to here and
what happened is at the interface here
to this term so the gradient of a
Heaviside step function is a delta
function so you have problems and so
what the system says it's all okay well
I have to minimize this term what am I
going to do well the first thing that
you could do okay I know how to minimize
this term this have no Brady's right but
you can't do that because you have to
have beta on one side and alpha on the
other so you aren't allowed to have zero
gradients in the system and so you reach
a compromise and that compromise is that
you have a diffuse interface and back
here where the gradients are zero the
bulk energy wins here and if you've been
close to the interface in this
interaction distance this a Gotama scale
interaction distance the gradients begin
to begin to come important and what that
does is that the free energy the bulk
free energy of the material as you go
from here to here actually goes up over
this barrier and back down to here okay
and so it traverses this unstable region
of the phase diagram that in between the
two spindles which are here in here and
so as you go from one to another you go
up and over and that W becomes important
and that's why I put it here because
this W basically is related to the
properties of the interface as it goes
to here so the properties of the
interface and related to the epsilon
which is this related to the interaction
distances and it's going to be related W
which is the height of the barrier
between these two because you actually
sample that barrier as you go from one
place to another and so the the the
actual minimum in this this energy here
is a little different than what you
normally would see in a standard
calculus problem so what we're looking
at in the normal K and the normal
calculus problem give me I give you a
function and I say find the minimum or
maximum and that's a classic calculus
problem where you just take the first
derivative and set it to zero here what
we're interested in is not the value of
the function of the location of the mid
maxximum we're interested in the
function itself in other words this tree
Anna this composition is a function of
distance or the water parameters of
function distance that minimizes F and
so we're interested in the function and
not the value of the function and so
what you have to do you have to think in
order to figure out what the what the
minimum is is that you have to take a
functional derivative of this equation
and not a regular derivative it's a
variational derivative functional
derivative depends upon the community
that you come from and let me show you
how this works and I apologize so if you
have a free energy that looks like this
is one dimension this is defi DX so
we're assuming a one-dimensional profile
through the interface and you take a
variational derivative and my apologies
for the mathematicians in the audience
this is not rigorous but it gets you the
right answer quickly so if you take the
variational derivative of the note this
by Delta and so you take the directional
derivative of this integral you pull
Delta through and it operates on this
term here it operates on the derivative
and so the functional derivative the
variational derivative of F it's just
the partial of F with respect to v times
of variation of Phi it operates kind of
like a differential operator in this in
this kind of approach which is why the
mathematicians don't like it and so and
then you have to do something with this
derivative term over here and again it
operates like a differential operator so
you get 2 D Phi DX times the variation
of the stuff on the inside variation 2
Phi DX like that and so that's what we
have before so you have you have to
figure out what to do with this so how
do you take the variation of the
derivative well what you do is you just
interchange Delta and D so you write it
as d by DX and you move the Delta on the
inside like this and now what you can do
is you could just integrate by parts and
so you take the derivative if you throw
it over to here which is what you
basically when you integrate by parts so
the derivative over here and you get
some terms associated with the
boundaries if you integrate by parts and
now since you throw the derivative over
here you get a second derivative times
the variation in fine if you say that
the value of the function of the
derivative is zero with the boundaries
in other words we don't have any free
surfaces of plus or minus infinity you
get the variation in via you can pull
this term together with this and you get
this brackets times delta v equals like
this this method that i showed you here
will always work it will always give you
the right answer it doesn't matter
whether it's a tensor or a vector you
always get the right answer to work
through this okay
so this thing here is the variational
derivative of that function or it
involves the first two the derivative of
F with respect to Phi the gradients and
the free energy and the composition
gradient the composition okay and so now
you say well I have to set this first
first variation to zero we want to extra
Mize it an extra Mises we might in a
minimum or a maximum and so you have to
set this to zero and now you're faced
with how do you set this integral to
zero there are two ways of sending is
integral to zero one of them is to
choose the variation and find just the
right way so that says set that integral
is zero and you can certainly do that
but that depends upon a very specific
choice of the composition is a function
of position if you want an answer that's
general you just set this up in the
brackets to zero and you get this and
this is what's called the variational
derivative or functional derivative
these are also called Euler's garage
equations in the mathematics community
and so the the first variational
derivative of F with respect to Phi is
just this and so now we have this thing
here equal to zero which tells you
something about the composition profile
through the interface if you have a
double wall free energy function that
looks like this
it turns out very
that you can get closed form solutions
to this equation and so if you set it
equal to zero you plug in the value of
five f is a function of Phi and you have
this ordinary differential equation for
Phi as a function of X you can integrate
it and lo and behold you get a
hyperbolic tangent solution and this
underlies all phase field methods and so
now we have a diffuse interface that
goes between the Alpha and the beta
phase it's a hyperbolic tangent and it's
the length scale of the for the width of
the interfaces that will become a crude
incredibly important is given by the
square root of the height of the barrier
times the interfacial inner times the
gradient Energy Coalition kind of as I
the beginning what would happen if you
have if you have a system with a
conservative parameter you have to put
in conservation and you do that with the
Lagrange multiplier we repeat the
process and now you get the result of
the functional derivative the DFT Phi is
equal to lambda which is basically now
it's sometimes called the generalized
chemical potential is equal to right
hand side so if you wanted to describe
diffusion you say that the flux is going
to be given by n times the gradient and
the chemical potential or n that should
be there should be gradient there times
the derivative of the variational
derivative you'll see that written in
the literature a lot and so that's the
expression for the flux and so this
first term here looks like a classic
diffusion equation in the second term
here is the correction to the
interfacial energy concentrations are
conserved you put this into here and so
this is a conserved quantity on I'll be
using conserved and not conserved order
parameters a fair bit as we go so this
is a conserved case so it says that the
flux is conserved you put this into here
and this is what's called the continuity
equation
this is one of the two basic equations
that make up all face field models is
the continuity equation it's basically a
little laplacian of the variational
derivative of F with respect to fine and
this is the key thing here is a physical
laplacian and the laplacian comes from
one of them one of the gradients comes
from conservation the other one comes
from the gradient of the chemical
potential with the velocity in front and
so whenever you see a phase field
equation written you can tell whether
it's conservative not just by looking at
whether or not there's a laplacian that
sits in front of the brackets here and
so this this kind of equation was
motivated by there's convoluted study of
spinozzi composition and the idea here
is you quench a material from some from
high temperature to low temperature
phase separates and the phase separation
process is described by this equation
this this equation if you wrote a finite
difference solution to this equation in
fact you basically find a difference of
times forward Euler spatial finite
differences this equation requires about
25 lines of MATLAB code everyone in my
phase transformations class my kinetics
class for graduate students they all
write an equation program to solve this
equation it's extremely easy these days
to do and I'm going to show you some
results of that we've done with
large-scale systems with 1024 cubed
currencies a very very large calculation
but I wanted to just show you what
happens if you start off with a volume
fraction of about 50% the system
undergoes corseting and this is only a
small section of this data and this
would be completely impossible if you
did it any other way than if they still
method there's another equation that's
important that one of the two that may
and one of the other one is called an
alençon equation and it's exactly the
same as the continuity equation except
for one important difference and that
what important difference is that it's
non conserved and this is motivated by
Allen and Khan's work on ordered sweater
transformations it also comes from
Ginsburg Landau theory in the
superconductivity area and so this is a
picture of an order disorder
transformation where you have atoms that
live on sub lattices and the anti phase
domain boundaries so if these could
easily be spins magnetic spins you see
this all the time in the magnetic
materials literature these kind of
Alencon equations exactly the same free
energy function you go through exactly
the same variational derivative to
determine the equilibrium state of the
system but since it's not conserved
instead of a great instead of a
laplacian sitting after em there's
nothing and basically what this says is
that the order parameter just moves
depending upon gradients and the bulk
free energy so there's no long range
diffusion so it's just gradients at the
bulk for energy so so if so let me give
you an idea of how this works
so here's F is a function Phi I start
the system off here
let's see if this works okay I start the
system up here the gradient F with
respect to Phi is positive there's a
minus sign here and that says that the
time rate of change of Phi with respect
to time is negative and so the order
parameter moves down in this direction
and so this this this model is
essentially what happens if you put a
marble on top of this free energy
surface put a marble here and it will
roll down to the bottom
the wells and it can roll down to the
bottom of the wells because it's
non-conserved it just kind of doesn't
have to worry about what all the rest of
the stuff is doing in the structure
because locally I can just flip the
spins or I can just flip the
compositions and it doesn't matter that
doesn't change the free energy of
anyplace else in the system unlike the
case when it's conservative and so those
are the two basic two basic face build
equations alençon and con earlier and
if you do the simulations you can get
you get slightly different structures
this is Alan Khan this is con hell you
can see that they look a little bit
different this what we did here is we
chose a 50/50 mixture of the two phases
and so what that meant is that we were
sitting right here and the system
doesn't know whether to roll in that
direction or that direction and if
you're numerix are very good it will sit
there and you can you get very different
structures as is the Gaussian curvature
as a function of position and this is
really quite a different structure and
so now you go back to the problem that
we're interested in so you have this
dendrite that's growing into the liquid
you have diffusion interfacial energy
that the temperature goes like
interfacial hundred tons of mean
curvature latent heat released at the
interface that's diffused away by by
temperature gradients and you want a
description of this in a phase field
model and it looks like this and you
just take the conserved equation for the
heat and a non conserved equation for
the interfacial energy and let me show
you how these two things work so this
looks like the classical diffusion
equation right and so what you do is you
add a source term to the diffusion
equation that's related to the time rate
of change of the order parameter P is
some polynomial and so what how does
this equation to work so this is five
function of X and when you're over here
and let's say the interface is moving in
this direction you're sitting over here
what's pi is a function of t2 Phi DT
over here it's near zero right you're
way far away from the interface if it's
zero over here this term isn't present
and you get
when you get close to the interface
let's say you're sitting right here your
observer sitting the interface is moving
past it's pretty clear that five begins
to change as five vidiians to change
this term turn is on and that gives you
latent heat and so this equation here
gives you diffusion when you're far away
and it gives you this condition when
you're up close to the interface so it
captures the release of latent heat it
captures this and so where do you get
this term here that comes from here
here's epsilon right
whenever you see epsilon you think of
inter facial energy so here's the
interfacial energy and here's the order
parameter terms and again this is a
polynomial in the order parameter and it
depends upon the temperature and so
these two things have number this goes
back to this model goes back to some
work and fixed actually Langer in 86
wrote these equations down and when I
was a postdoc I went to the Institute
for Theoretical Physics and Jim lying
here came up to me said I have these
notes on this this model for
solidification would you be interested
in looking at them I said what a mistake
and and actually the cool thing in his
notes actually at the very bottom of the
last page of his notes he said this
model is too nonlinear and complicated
to ever be of any use and of course you
told me this when I was a postdoc so
anyway and and what's happened is that
computers have become more powerful and
as a result these things actually useful
and so the idea here so let's go back to
here okay so the idea is I give you the
interfacial energy I give you the
diffusion coefficient I give you the
light and heat and thermal conductivity
and then I go into this phase field
model and I pick epsilon we kind of
already know how to do that
know how it connects the interfacial
energy Tao and the polynomial such that
a solution to these equations is related
to a solution of those equations okay
let me show you the implications here is
the implication so this is a Renee
McGrath he is a Belgian surrealist
painter this actually sits at the Art
Institute of Chicago so for all those
who haven't visited Chicago you should
visit the Art Institute of Chicago this
is actually recently it was actually
ranked the number one museum in the
world and for those of you don't read
French it says this is not a pipe and so
the question here is what does this have
to do with phase field models and the
problem with phase field models is that
you can get things that look just like a
pipe but in reality they're nothing like
a pipe and I want to show you what I
mean by that and so they're two two
schools of thought and this basically is
is is reflected in the way these phase
field models grew up one in the
solidification literature one in us
women in the face view in the physical
metallurgy community one of them is it's
a computational engine and I kind of
showed you that that that's the case in
solidification the other is that it's
somehow related to a first principles
method and this comes from the physics
and physical metallurgy community I want
to talk about these two schools of
thought to give you some perspective
years to what the problems are and this
is my entree into where the bodies are
buried in the faces of method so I
showed you that if you have a free
energy function looks like this you have
a diffuse interface you have a
characteristic length scale that I
showed you before with that hyperbolic
tangent function that looks like epsilon
over the related epsilon W interfacial
energy that's surprised with epsilon and
the square
this is classic con halyard
vanderwaal's theory of fluids the
problem here if you consider this to be
a first principles method is that if you
walked down the hall to your friendly
statistical physicist and you showed him
or heard this free energy function and
you said well I want to calculate W from
force principles I want to calculate W
and what she would tell you is that no
the free energy function doesn't look
like this it looks like that and the
problem is that this free energy this
this part of the free energy function is
in the unstable region of the Prieta of
the system in other words if I put up a
little bit of material of composition
here in between these two spinodal it
will spin normally decompose and that's
what I showed those pictures I just
showed you I quenched to 50% and I ended
up with this beautiful spinel structure
and what that means is that it's
impossible to measure the free energy
because as soon as you put a system
let's say you're doing a molecular
dynamics history the system as soon as
you put that system together and you
will allow the atoms to begin to move
and you want to measure the
thermodynamic fluctuations in the
entropy what you need to do together
free energy it would face separate it
would change into it was normally
decompose and so the answer really is
that is what the free energy looks like
and in fact it's even hard from a
statistical mechanics sense to go from
here even into this metastable regime
and get closer and closer to these
panels because as you get closer and
closer to these photos what are you
doing you approaching the system that's
becoming more and more unstable so the
problem here is that you need to know
after the unstable region the
interfacial energy depends upon W so
it's difficult or impossible to get
these free energy functions first
principles the closest thing that people
have done that I'm aware of is this very
nice work by
Donnellan young Lavoie and what they did
is they started in the in the in the
bulk the bulk and then and
coarse-grained in a very clever way to
get really very close here and tried to
patch in this unstable a free energy
function based upon the based upon this
idea of basically scaling the
fluctuations in a certain way but it's a
very difficult problem and it's the only
problems that have done before and so
you look at this and you say okay we
have a problem here right I can't get W
well the answer that people said okay
wait a minute I know the diffusion
coefficient the interfacial energy and
the thickness of the interface and if I
give you these three parameters I can go
back into the phase field method let's
just think to know so interface energy
and thickness if I give you the
thickness and the interfacial energy
experimentally measured quantities so
you can go to David Simon's lab and you
measure the thickness of the interface
or you do wait a mo the dynamics
calculation of an interface and you
measure the breadth of the interface
which you can do that's a well posed
problem so I get this and I get this I
have two equations and two unknowns and
I get epsilon and the W that I can't
calculate the first principle music okay
and then you go and you find out well
what's the real problem here okay okay
so I'm going to therefore have our
interface thickness and to do a
calculation I have to resolve this
diffuse interface and what that means is
that I have to place mesh points in my
calculation in this diffuse interface I
have to put at least six to ten mesh
points in that confuse interface in
order to resolve okay so let's see the
implications of that
so how thick is an interface pick a
number
anatomy okay thanks a nanometer this is
a hundred micron scale bar so you take a
range of 100 microns to one nanometer
that's ten to the fifth I have to put
ten match points within that
with so that means 10 to the 6th so to
do a box 3d box of this by this by this
that's 10 to the 6th cubed but I want to
do this let's say that's 10 by 10 I need
a grid on the order of 10 to the 7th
cute
so Langer may have been right you can't
use this method to do anything if you're
doing nanoscale structures it gets a
little bit easier right you cut this
down by a factor of a hundred let's say
you might be able to get away with it
using a real interface thickness but in
general you can't and that's really the
problem with Facebook lungs and that's
really the problem with basic models in
fact some people have said that that if
we had discovered if we started with
base field models we would have been
delighted to figure out how to do sharp
interference calculations ok so here the
equations I gave you before and we're in
this conundrum that we have this problem
of how do you pick these parameters to
reproduce the real problem and we know
that it's hopeless to pick a real
interface thickness and so what you
might want to try to do is to say that
well what I'm going to do is I'm going
to pick the parameters such that the
thickness of the interface times the
curvature is much much less than 1 what
do I mean by that so in other words
rather than scaling your mesh on the
real interface thickness I'm going to
scale the mesh spacing based upon the
radius of curvature of the object here
which would be a big deal right because
now I'm if the radius of curvature here
is 10 microns my scale my mesh is on the
order of microns now and not on the
order of nanometers and that's what most
people do to make it computationally
tractable and so the idea here is to
look at this limit of the equations when
the thickness of the interface times the
mean curvature is much much less than 1
hopefully about 0.1 or so and you want
to look so this is now a parameter and
amassed asymptotic expansion of the face
field equations that that you have to
worry about it
and if you do the maths asymptotics
correctly that's interesting I didn't
turn Skype off that's okay and so if you
do that you can you can basically
calculate what goes on up here and this
is people done better than this now but
but not too much better and so this is a
3d calculation of a very typical of a
dendrite using this idea where you write
the equations outweigh the thickness of
the interface times the mean curvature
is much much less than 1
okay
and so there are two ways of doing this
this Matteson taught expansions one of
them is to say well I want to let this
interface thickness be very small and
you can do that by letting epsilon go to
0 and if you let epsilon go to 0 if you
want this to be finite you let W go to
infinity at the same time and so when
you look at the literature you'll find
two kind of asymptotic expansions one
associated with letting the width of the
interface going to 0 which turns out to
be not that computationally helpful but
it gives you the it gives you it still
enables you to connect these equations
to the sharp interface equations the
other one is more useful computationally
it's where you just say that the unit
ice thickness is small on the scale of
the mean curvature what over the mean
curvature okay so let's go back to this
equation so here we have a model of a
inter diffusion so you have now a
dendrite or anybody let's say that the
literal decomposition process where you
have a liquid and a solid
the diffusion coefficient in the liquid
is 10 to the fifth higher than the
diffusion coefficient in the solid so
you go from liquid to solid you have a
big jump from the diffusion coefficient
that's kind of well-known you solve
diffusion equations and then in the
liquid and the solid I'm sorry you have
a mass balance at the interface that
says the velocity times the amount of
material rejected if the interface is
diffused away by concentration gradients
you have the concentration goes like
what you have at a flat interface times
the interfacial energy times the me
curvature and you have diffusion
equations and so again this is a
dendrite but it's growing by solute if
you're doing a precipitate growing into
a supersaturated solution this would be
the model you would use evolution films
it's the same sort of a same sort of
deal and the key thing I want to point
out is that this is this you have this
enormous change in the diffusion
coefficient and so what do you do so you
have a diffusion coefficient that goes
from DL to D s across the interface
right and so the thing that you're
tempted to do therefore is to say well
we'll just let the diffusion coefficient
vary with the order parameter through
the interface like this and you'll see
this all the time in phase
as well so you start off you're at zero
it and 1 and what you're going to do is
you're gonna take the free energy of the
system now is a linear combination of
the free energy of the liquid and the
free energy of the solid with these
parameters pi + 1 - fine so if you're in
the liquid the other parameter is zero
you get the free energy of the liquid if
you're in the solid and if an order
parameter is 1 this goes away and you
get the free energy of the solvent same
thing for the diffusion coefficient when
you're in the liquid you get the
defeated cooperations liquid in the in
solving that particular coefficient
itself this seems like a very natural
thing to do and notice what it's saying
is saying that in the interface here the
free energy of the the bulk material in
the interface is some sort of linear
combination between the solid and the
liquid and you can see that you're
already beginning to get away from what
really is going on here right you're
kind of saying that well this is a
computational method now right it
doesn't really have much to do with the
physics of what goes on at real
interfaces because I'm just taking a
linear combination of the free energies
the bulk phases describe the interface
right so it's kind of a computational
method now and not what goes on a
reality the diffusion coefficients the
same way right so it's saying that the
diffusion coefficient the interface is
some intermediate value between the
liquid and the solid which is probably
reasonable but it's saying a very
specific kind of variation through the
interface and so this these kind of
assumptions turn out to have very big
effects on what's going on at the
interface and I'll show you that in a
second so using this method Jim Warren
and Bill Bettinger did one of the first
faith-filled calculations of alloys
fortification in the dendrites like this
and so this is liquid that's solid
obviously what you're looking at here is
the composition is a function of
position hiked up low composition
composition blue is high you can see
that the solute is rejected at the
interface because the solubility inside
the dendrites less than outside
you get non-uniform compositions in the
dendrite as a function of position kind
of is exactly if you look at any micro
structure of a dendritic solidification
process you can see precisely this you
see these stripes that go down the
middle it's called chlorine you see all
of this stuff that is segregated between
the second between these little
secondary alarms exactly as you see so
it looks really good and this business
this paper if you look looked at ACTA
met it has 500 citations yeah okay okay
so so now think about the pipe right
this calculation is not quantitatively
correct it looks like a dendrite that
smells like a dendrite but it is not a
dinner and in fact I built that Jordan
John Warren would admit this it is not
coordinated with correct so so the
amount of solid segregation that you get
here is not correct the amount of solute
segregation you get in here is not
correct and so this is the challenge
with with phase field methods and the
big problem and this is a very subtle
thing the big problem that they were
unaware of when they use this this kind
of a model is that this reasonable
assumption on how you interpolate from
solid to liquid as you go through the
interface is gives you spurious terms
and actually how you interpolate
interfaces when you go from one to
another
in a face build model is still an open
question but the takeaway message that I
want you to be aware of is that whenever
you see a face build equation come on
this could be ferroelectric thin films
this could be magnetic materials if you
have structures that bearing across the
interface and you're interpolating
across the interface you will get
physics at the interfaces that depend
upon these interpolation functions that
sometimes can be a problem and in this
particular problem what happens is if
you take those equations that I just
showed you and you did these asymptotic
expansions in this Delta parameter Delta
where L divided by R is much much less
than one this is the equation that you
want that the velocity times the
material rejected goes like the
concentration gradients and what you get
instead is this equation and this
equation says that you have instead of
the jump of the concentration you have a
parameter times the thickness of the
interface times H and so what is that
it's actually a very simple physical
explanation so if you take a spear think
about the individual area of a sphere
here and a sphere here right
so as the sphere grows the interfacial
area increases right if you have a very
thick interface and lots of material
associated with the interface as it
grows you'd have to feed the material in
the interface and this sucks material
out of what was supposed to go into the
bulk and so you get a term this related
to the thickness times the mean
curvature over here on the diffusion
side instead of getting bulk diffusion
this interface is so thick you get
surface diffusion and and they are
exactly correct that if you let the
thickness of the interface L go to zero
these terms disappear
so if you use physically realistic
interface sizes these terms don't appear
but there's no way that you
you get these spurious terms and - so to
get rid of these terms you have to do
something very very clever and very very
non-intuitive and reasonably complicated
and it goes back to some work that these
folks here to get to get to get rid of
these equations to get rid of these
problems David brought up the fourfold
anisotropy and that's that it turns out
not to be a problem with these equations
but it turns out to be a problem with
what happened in the early stages of
faced with laws and you notice the
orientation of this dendrite is not
along the XY directions of the grid okay
so the very first phase field models
that people do and there's a company
that's mom I'm not is that there's a
company that markets a base field code
that you can buy and when they first
came out with his toga has beautiful
dendrites and you got beautiful
dendrites when they grow along the grid
the ex-wife accordance to the grid
because what was happening here is that
the anisotropy of the grid would
introduce enough anisotropy in the
interfacial energy that you get
beautiful dendrites growing along these
directions and as soon as they refined
the grid the secondary thing do I want
to disappear and so so Jim and Bill
we're very much aware of that which is
why it's canted at a strange angle with
respect to the computational box so they
don't have that problem but they had
this far more subtle problem that that
basically because they didn't do the
math asymptotic expansions that no
their calculations and so the idea here
and I think I'll conclude I'll think I
what's up with this line is that this
thickness which allows you to do all
this marvelous stuff in other words
topological transitions very complicated
morphologies you can follow them very
very easily comes with it this penalty
of extra terms in the diffusion equation
another situation that you have to watch
out for and this is a generic effect
they're always going to be there thanks
very much for be happy to answer some
more questions so why is the so hard
yeah so the major problem is I didn't
the problem is that the equations are
highly nonlinear even in the the Euler
garage equations of nonlinear so so the
key is to find solutions of the oiler
garage equations so if you just look at
this equation if you finding T goes like
M fourth a fourth the fourth derivative
of Phi this is just a standard by
harmonic linear constant-coefficient you
do it after that's that's no problems
the problem is this term it's a
derivative of this complicated
polynomial function and so it's
nonlinear and it's and the non-linearity
sits inside the it's operated on by l
squared Y and so is that non-linearity
that makes them hard to solve you can do
it in one dimension as I showed you to
get the interface this one here so in
one dimension this is what this is what
the polynomial looks like the derivative
in the simplest case and you can get a
solution but in general you can't solve
the equation so you can write that what
what you're pointing out is a good one
there's a very strong connection between
the means to come back to the free
energy yeah there's a very strong
connection between this and this and the
and the evolution equations and what
that means is that there's that all
these equations decrease the energy in
the system and so you have this general
energy framework that allows you to
write them so if you are doing magnetics
you could add a lot magnetic energy if
you were doing electrostatics you can
add electrostatic energy if you're doing
all the elasticity just add more
so it's very easy to generalize this but
that's the reason why it's so hard it's
just nonlinear and in the end in the
solidification case they're two coupled
nonlinear first this this this is this
is a very good question yes because okay
yeah so if I look at this thing where is
it well the friendiy function is I was
just using 5 squared 1 minus 5 squared
right right and it and in general you
would have these logarithmic terms and
in fact if you're doing a real face of
the model you do put in a logarithmic
terms the problem with the logarithmic
terms is it makes the numerix much much
harder and the reason why it makes
numerix much much harder is that for
example if you have log X so if you have
a freebie function looks like
like that right if you have this is your
this is plus some other terms in fine
think about what happens when Y goes to
zero that divergence right
this doesn't divert but the chemical
potential the variational derivative of
this the first derivative at 1 over fine
and and so you have terms of call like 1
over 5 and if you look at the free
energy function and this is why when you
draw free energy function it actually is
slope you can fit a median it comes down
and there's something like this this
slope of infinity 1 over infinity
infinity is Phi goes to 0 makes implicit
numerical methods of these equations
extraordinarily difficult and it's an
unsolved problem and so though so people
will say all the time all you just write
the dominant PD's especially if the
voter with continuing mechanics who
they're very familiar with implicit
methods which basically make it easy to
do time stepping and said well well just
take these face field equations and take
my favorite finite element code write an
implicit solver and as soon as you put
in this thing it doesn't work and and
it's an unsolved it's an unsolved
problem so a realistic free energy
functions absolutely in fact if you want
to connect with thermodynamic databases
you have to deal with these problems and
you just hope that any of the
concentrations in the system
what we do which is at the atomistic
scale of coalescence of parameters
changing with time and that we think
that the physical mechanism is that the
vacancies spending most of its time in
the precipitates because the
interactions the atomic interactions are
going out the nearest neighbors control
yeah yeah no I think it's quite
different no that's right one of the
really one of the really interesting
ideas is that during coalescence what
you do is you have two particles for
those who don't know time you take two
particles you bring them close together
and you think about what's happening you
know there's diffuse interface over here
you have this diffuse interface over
here one particle particle you start
bringing them close together
and it's begin to come together and this
this point here eventually disappears
the two particles coalesce so this is
the order parameter as a function of X
and so this is the coalescence region as
you bring these two particles close
together and so what you're doing in
here when the particles coalesce you're
sampling the free energy in this region
between 1 and 0 and so you're in this
region here and so as these particles
coalesce what you're doing is you're
sampling the free energy in this
unstable region and so in principle if
you model your coalescence with these
equations you could actually pick out
the right W's and epsilon to describe
the coalescence process overlapping
these concentration profile so the free
energy locally in the system is high
compared to the case when they would
connect the two particles yeah yeah I
think so and so in principle you could
you could actually model it carefully
yeah
yeah yeah that's right
I mean and I think the the connection
Airport is with these macroscopic
variables that people measure and so if
I don't have these values I'm kind of
sunk this parrot is very difficult for
me to say what with that really what
that really looks like doesn't yeah yeah
the the closest thing that people have
done to actually getting some ideas to
what that looks like they're polymer
systems we're these big long chains like
structures but it's a major mechanics
and that's right that's right and and
they're the key is the key is in that
case is to put together a model that
accurately reflects the interface with
in a real system and you can do that if
you choose the parameters the right way