Well.
First.
Of all.
Well thank you for having me it's
a pleasure to to visit again and.
On some level at this seminar will
be only as good as your questions.
I know that but I will need no
encouragement to ask or interrupt but
I hope the rest of you will as well not
merely the distinguished professors but
also the grad students and postdocs and so
forth so if there's anything that's
not clear please do stop me so
I want to introduce the subject
with some ancient history and
this history goes back to the one
nine hundred ninety S. And for there
by the time the ninety's rolled around a
series of linear power chains that a good
solvent had been generalized to treat
the statistical mechanics of something
rather different the statistical mechanics
of flexible flexible sheet polymers and
remarkably these objects these
tethered surfaces as they were called
because they have a shear modulus behave
very differently than their linear polymer
cousins they give it a low temperature
flat face a kind of wrinkled
sheet polymer face which reflected
a continuous broken symmetry and
at the time it was quite surprising
at least to many of us because.
If I imagine this piece of paper
as a kind of Rena in which to do
statistical mechanics I could imagine the
normals to the piece of paper is like I'm
on a ray of spin vectors and
these normal spin factors would be
flat in the flat face would be all
perpendicular would be very ordered but
you might imagine that entropy would
want to make this thing crumple up and
even if I were to uncrumpled a little bit
now that the normals to this
crumpled piece of paper.
Would be fluctuating be.
It would be like spin waves if I think of
my fingers being like a bunch of spins and
it was thought that in
the run in two dimensions.
We're doing of spins and
related systems will always be disrupted
by thermal fluctuations at any finite
temperature and it turns out for
reasons that are less I'll explain
as we go wrong that's not true for
these wrinkled sheet polymers and they
violate this expectation which had been
elevated to the name of the vermin
at the moment Wagner theorem and
they course don't violate any rigorous
their own they're just different
in their their own interesting way.
So this ancient history has had a bit of
a revival in the theoretical community
that studies grapheme in fact in this
lovely book by because that's Nelson.
There's a whole chapter on the role
that thermal wrinkles which I'll tell
you more about as we go along play in
graphene at finite temperature Here's some
papers on how these out of plain phone
ons that I'll be emphasizing in this talk
affect the electronic properties there are
quantum effects at low temperatures but
not many experiments of the kind
that really go after these
questions involving the mermen Wagner
theorem the reason is that graphene
beautiful though it may be is almost
always experimented upon either
pressed on a solid substrate or
stretched across the whole and
that irons out the very fluctuations
that I'm going to talk about.
That all changed and it changed with
the Nature paper that appeared about
a month and a half ago from
Paul McKeown for group there's Paul.
Cornell these are a bunch of his students
making three graphene hexagonal rings
as you can see here and what they did from
one perspective at least is the kind of
in a sense it's the first single molecule
polymer physics experiment done and
graphing because they liberated
this material from its substrate
as they grab maybe one end made
a graphene cantilever out of a ribbon and
they let it have such a weight around and
measured its mechanical properties or
see if I can get this movie to play so
this is a movie I have a barely visible
atomically thin graphene ribbon.
With a pendant magnetic pad and
there's just a magnet
that somewhere up here off the field
a few with which they're twisting around
this membrane this ribbon
that is one carbon atom thick
That's the first time that that I
believe has been done experimentally and
the way they were able to make this
happen as I said was to get the carbon
the graphene sheet well after substrate
and here's a little needle that is.
Pulling it off the substrate
on which this graphene.
Lattice work was produced and
it crumples up like a piece
of paper crumples However
a piece of paper that is
maybe two extremes thick so
that's never been done before as I said
this to my knowledge and I'd like to try
to figure out if we what sorts of
properties will be going on in the system
and maybe we can apply ideas about
the physics of sheet polymers
in a kind of space matter meets H.
ball hard condensed matter regime.
They are sort of merging
together these two disciplines so
I'm going to tell you about a theory of
critical phenomena without critical points
and it will be led to a Syria free
standing graphene ribbon Sometimes it'll
be graphene cantilevers I'll
tell you about a century old
problem of the non linear equations
of singed thin plate theory.
There were written down by fuck
all in one common in about
one thousand No five about ten
years before Einstein's theory of
general relativity to which
they have a close affinity.
The nonlinearities in these equations
are characterized by a combination of
of parameters a young smart you list
the size of the say graphing ribbon
squared divided by a bending rigidity
I'll tell you about those and
the problem is hard because this
coupling constant is very very
big in fact if you ask yourself what is
the coupling constant for the mechanical
properties of crumpling this piece of
paper it's about ten to the seventh so
I invite you to do perturbation
theory in terms of service and
try to figure out what's going to happen
it is an interesting challenging problem.
And then I want to tell you about what
happens if we take the century old
equations and add thermal fluctuations and
what will emerge is a kind
of self organize criticality a low
temperature phase which has its
own unique properties as it via
its the mermen Wagner theorem and
will discover strongly scale dependent
elastic parameters the Youngs modulus
the bending Witchetty now are no longer
constant they're not they're not
conventional lastic parameters they are
running coupling constants which can vary
by three to four orders of magnitude
as you change the link scale.
And finally I move on the graphene that's
not the unique atomically thin material.
There's Bon nitride live and die.
Die sulfide these other compounds could
also be studied in this context but
because graphene and
its relatives are so thin it's funny but
number isn't ten to the seventh like
a piece of paper it's ten to thirteen and
in a sense from the point of view of
the mechanical properties of cantilevers
we've reached the Moore's Law a limit
of thinness you can never get
thinner than this and we'll see
the thermal fluctuations dominated
run temperature whatever your
larger than a critical size and
this critical size measured in the plane
of the graphene is only an extra man to
have it all scales bigger You better pay
attention to these fluctuate phonons
their bending rigidity can be enhanced
by a factor of four thousand and
leads to anomalous properties of
ribbons crumpling transitions and
various other things this work began
many years ago in collaboration
I was fortunate enough to have with
look at Meran card Arioch of Cantore
these experiments renewed our
interest in this problem and
the recent series been done with
Mark Bowen Rasta skeptic and last but
not least because Mitch who is
a post-doc who did many of the new
calculations that I'll tell you about
who is now on the faculty at Princeton.
Any questions.
OK so.
Here's the ancient history.
This is fossil this is one
common both pioneers of
a number of subjects including
elasticity theory and it wrote down the.
Yes So I question.
It it could be and I am naked.
If I'm leaving out someone and if you
could send me the reference I see very
grateful good all right.
Other questions or comments.
OK.
So what are these.
What do these people do.
Well they thought about thin plates.
Now or just hearing of thin shells
engineers with might want to know only
what is the strength of
the Superdome stadium if there's
snow falls on top of it or
the strength of a submarine.
Hole as it goes deep into the ocean and in
the limit of fin thinness which is going
to be crucial to this very large shop of
uncommon number you can now do a kind
of solid state physics analysis of such
a thing by looking at in plain phone on so
you want a new two out
of plane displacement F.
as a function of internal Courtney's so X.
one an X. two if this were a piece
of graph paper would just be the Cartesian
coordinates of the graph paper but
they can get crumpled up into three
dimensions and a classic way of describing
this is in terms of bending and stretching
energy the stretching is described by
a strange sir this is the kind of strange
tensor that you might find the first two
terms in a solid state physics book
like the teller Ashcraft and mermen and
then this is a non-linear contribution
that has to do with the structuring
associated with this vertical displacement
F. coming out of the plane the reference
plane of this approximately flat sheet and
then the energy that
they wrote down was a bending energy
that's goes that goes actually has
a look plus the end of F. squared
plus lastic constants mewing lambda.
Sure modulus and then there's a bulk
modulus that you can make from them as
well and if you stare at this thing
there's here's the bending energy.
There's these elastic
stretching terms as well and
because this occurs as you I.J.
squared and there's this F.
term here there's a strong non-linearity.
Which has a huge effect on the physics
not only at zero temperature but
also at finite temperatures as we'll
see in the sense the take home message
of this problem from the perspective
of solid state physics for example.
Is that new effects arise
in transport properties but
also in the mechanical properties that
I'll talk about because of these Flexeril
phone ons they can escape in a very
soft way into the third dimension
so that's sort of where we're headed.
In fact you can take this
free energy here and.
Do some sort of functional
minimisation if you.
Take this strange free condition at zero
temperature right it has the symmetry eyes
derivative of an airy stress function
you end up with these famous fossils
on Karmen equations you can find them in
land Lifshitz for example Landau Lifshitz
write them down but they don't solve them
because they're so horribly non-linear So
here's a fourth the river of of cause
that's not a non-linear the Perceval
fourth of after on the right hand side
there are these non linear functions of F.
and in fact this nonlinearity here
is just the Gaussian curvature
of the surface that and there's a kind of
source term for the area stress function
that in turn couples back
to the stress tensor.
And if you play around with this.
These equations if you
re scale appropriately
there's a non-linear parameter.
Much like the Reynolds number for
an obvious Stokes equation.
That is this given by the Young's much a
list which is itself a combination of new
and lambda times the size of
the sheets credited guided by Kappa and
the thin sheet out he mentioned a piece of
paper has a font of uncommon number of.
Ten to the seventh this controls the
importance of these non-linearities And
as I said it's a kind of simplified
version of general relativity
because Gaussian curvature enters
this not exactly the same and
involves the embedding space so
in that sense it's it's different and
exact solutions are available
in very negligible set of cases
one exact solution is if you take out
a wedge and turn this into a cone.
That that's Aleutian is no but
not much beyond that except using
some of the techniques that I'll describe
and also some numerical simulations.
Serve these fought over uncommon ideas and
equations have been used to bottle
macroscopic things plant leaves egg
shells submarines sports
stadiums aluminum foil pipes and
so forth there's macroscopic
macroscopic applications
there's also more microscopic versions of
this I'll talk about those experiments and
graphene in a minute there's the spectrum
skeleton of red blood cells which is
a kind of tethered polymer
as well bacterial cell walls
viral cap suits are thin objects and
some of them are so
thin that Brownian motion is important and
of course.
Topple and one garment didn't worry
about Brownian motion back in those
days and
that's what I'd like to summarise here so
if I take that that
energy that I wrote down.
With bending and stretching and
I trace out those phone on degrees
of freedom you X. in you why.
There's an effect of free energy left
over in this out of plain phone and
displacement F. So of course there's
a bending energy out here this it's
soft because it involves a restoring
force because like the laplacian of F.
squared and then the Young's marvelous
comes in is a kind of coupling constant
that's forced the order in F. and this
allows the little pieces of the turtle
interface to interact with other
pieces of the tilted interface and
this nonlinearity is something that
we might hope to understand maybe
in perturbation theory so this is maybe
could I try to do perturbation theory and
why the five on common number we're
reroutes it's head shortly but
it's worth trying to do that perturbation
theory because it's rather instructive and
the idea is that at low temperatures
to grow Sumit the surface is flat and
will look at little phone on wiggles away
from being perfectly along the Z. axis.
And we can think of those a spin waves
if you like the low temperatures and
basically you set up a perturbation
theory Here's the Fineman
diagrams that correspond to that
perturbation theory here give me a giving
Let's say every normalized bending
rigidity you can do the same thing for
the we normalize young lose if you
like and the result is something that
looks small because at low temperatures
it goes like a BT over Kappa.
Kappa has units of energy as well K.B.
to over Kappa for many of these physical
systems subject to Brownian motion is like
one over fifty nine to five by
four pi cubed that's great this
looks really small evaluate this integral
but in fact the intercooler has a serious
infrared divergence and as a result
all the small stuff gets multiplied by
the fact of uncommon number which
we've seen can be ten to the seven.
Tend to the night and this is just
first order perturbation theory.
Interesting result though is that
this gives an enhancement of this
spending rigidity and
it could can be in principle
an enormous course we have to worry
about than all these extra terms.
And back in the ancient history
back in the one thousand nine
hundred six look at polluting
I actually study this
part of Asian Syrian summed all the graphs
the most there verging ones and
we found the bending rigidity that
was strongly scale dependent and
in fact that they first as the wave back
to goes to zero Those went to longer and
longer waits that wavelengths the bending
rigidity got stronger and stronger and
stronger and you can sort of see that.
In the following way.
I crumpled this thing just to show you
that I got thermal wrinkles and stuff but
you can see it's much stronger than
this under formed flat piece of paper so
that somehow this thing is getting
stronger because of these wrinkles and
if I think take these
wrinkles as a proxy for
thermal fluctuations you can see that
something interesting might be going on.
That didn't take too long before people
did more sophisticated renormalization
group calculations of this
formalized top of uncommon problem.
People like your run of it sometimes
the Bensky peer that is solemn he writes
a half ski and neighbors found resin deede
there are diverging Alas the constants
associated with this problem and
the fossils on Karmen fixpoint this thing
that controls the strength of submarines
and so forth is unstable to thermal
fluctuations there's another thermal
stop of uncommon fix point out
here no adjustment of parameters
is required this is a critical system
without that kind of adjustment.
And the results are indeed a running
coupling constant That depends
on length scale and it goes like.
The link scale you care about
divided by some scale given by this
parameter down here raised to a power the
power isn't one which is what police and
I found but it's more like point
eight two And similarly the Young's
much of this reference as
you go to larger scales and
then you can sort of see that with a piece
of paper it becomes soft to shear.
Because of the wrinkles because of
the stored area inside the wrinkles so
this is what was known and
the physics is kind of interesting.
If here here's the field theory that
describes what's going on and what's here
I didn't say earlier it turns out to be
the transverse projection OPERATOR And
you can think of these gradients of
half as being like little tilts of these
pencils held in this pair
of hands in the pictures So
here I'll attempt to
recreate this photograph.
So if I have my hands like this in their
little little neighboring regions of
the sheet polymer and I took them like
that that's a rancher to the phone.
And it's controlled by the bending
rigidity has a very low resistance
on the other hand if I make a try
to make a transfer so anon and
I took like this I have to tear
the surface or stretch and
so the lens modulus comes
into that defamation.
And as a result a simple
analogy with spins that
I started this talk with are quite right
and this is the mechanism basically
illustrated with these pairs of hands that
allow you to violate that old theorem
about the lack of broken symmetries and
long range water in two dimensions and
in fact if you work out the normal normal
correlation function summing up all these.
If that's the truck you find that this
normal normal correlation function is
reduced from one the value would have for
for a flat sheet by thermal fluctuations
by this prefecture here but
and it to case to a non-zero constant for
large large separations So
that's sort of the in
a nutshell the breakdown of
conventional elasticity theory and
here is again the example of a wrinkled
object merely regarded here as a proxy for
thermal arised sheet of
thermal ice sheet polymer.
With Andre we've actually made
controllable surfaces with frozen in
wrinkles with a three D.
printer and then we try to bend it and
indeed the wrinkles alone even if
they're frozen greatly enhance
the bending rigidity and as I said.
There are all sorts of interesting
things here one of them is the negative
coefficient of thermal expansion mean
you can see if this is regarded as or
something like a proxy for
throw my sheep polymer it has less area
than a reference eight half by eleven
sheet of paper pretty thick so
it shrinks much like
a rubber band shrinks due to
the thermal energies and
in fact you can work out.
In some detail what happens.
In the contraction of
the area this is a negative
call fission of thermal expansion shown
here due to the thermal fluctuations
this is the ordinary material
stretching if I put an outward force
on the edges if I subject this
thing to some sort of isotropic
strain tensor at the edges this is linear
response theory for the change in Delta
a restoring some of the the shrinkage
here it's just middle to.
Term that's quite intriguing and
this middle term is a breakdown of linear
response theory goes as a fractional power
of this parameter Sigma that's
pulling on the edges and
in fact that powers about point seven this
is pointed out in ancient times by by
these authors here so it's a very
strange object it behaves much like
a systematised critical point with these
power long response from sions much like
magnetization versus magnetic
field at a critical point.
And it turns out and this is only
became clear to me about a year and
a half ago the ideal system in
which to study this kind of stuff
is graphene.
Source Paul McKeown likes to say graphene
is the ultimate two D. Crystal and
membrane it's one atom sick
it's extremely stiff so
it has a that helps to have
a large prop of uncommon number
on the other hand it's also
extremely flexible and
since we can't get any sooner than
one atom thin we have to say for
a ten microns size square of graphene
the largest conceivable followon
common number that we could study in
the lab and it's about ten to thirteen.
Perhaps more important is that so
many links scale the link scale on which
if I go back here to this little
calculation down here one more.
This calculation right here.
This thermal link scale is the link
scale at which this correction becomes
comparable to one and
the bending rigidity begins to take off.
And so that formula for
the thermal length scale is lurking here
depends on temperature at the pens
on Kappa I put in the giant.
Young's modulus of graphene
the soft bending rigidity.
Here and put in room temperature and
was as was pointed out to me by by
Paul this thermal link scale for
graphene is about two extremes.
So on all scales bigger than two extremes
you better worry about
those fluctuate phones.
And.
Cornell group did this lovely cantilever
experiment they did a number of
experiments but you can't quite see the
graphene but there's a there's a gold pad
here gold pad here they stab one of
them they get it off the substrate so
it's floating around not being pressed it
can stand anything not being stretched
across the whole and they do this
classic can't believe or experiment.
I say classic because Galileo
was interested in this and
in his this work on discourses
relating to two new sciences he did he
discusses the theory of the can't
believe or at that time one of
his new sciences was physics
the other was mechanics.
And what these group with this group
did was to measure this deflection for
Ride variety of graphene ribbons.
They also turned it on its side and they
look at the the summer sucks you a sions
at right angles and
sort of used some side of.
Fluctuation just a patient's arm
to get an independent measure
of the bending stiffness the bending
stiffness according to the chemist at
zero temperature we certainly know
about the carbon carbon bonds so
we can calculate that was about
one point two electron folds but
if you do this experiment and
read off the effect of bending which it is
it's not one point two electron volts
it's about seven killall electron flow so
that's about a five thousand fold and
handsome and pretty nice.
This is happening at room temperature and
it's at least.
System with this formula if you
plug in the with of the graphene.
Can from leaders.
So.
We can ask a number of questions for
example.
If I take a graphene ribbon and I make
it longer and longer so a lot let's take
this this belt as a proxy for
a graphing ribbon.
If I made it long enough it
would start to flop around and
eventually probably would
behave like a polymer.
So how do we cross over from this
remarkable behavior of a short stubby
graphene cantilever to
go to the conventional
theory of linear polymer chains linear
in this case linear polymer ribbons
this is how the experiment was done
looking at say a deflection under the mass
of one of the pads at the end of
the graphene ribbon the deflection
cord into classical mechanics goes
like the cube of the length here and
then the mass is the force
downwards Here's the width and
then this is how you would read off where
we normalized bending which would it be in
the experiments it depends on the width of
the ribbon and the length of the ribbon.
If I now store it with ribbons of varying
link this is roughly what you would
expect on a log log scale.
If all the scales are much less than
this thermal lengths it'll just be
the the boring Alas the behavior cap
is one point two electron volts but
once I get into this regime larger
than else thermal as as the length
of the ribbon gets bigger and
bigger compared to the width.
It will go up according to this exponent
eight it's about point eight And
so it will grow up and this is this
four thousand fold and handsome and and
then when it gets out here for
a while it'll it'll be a constant
a very large constant.
The given by W. over L. thermo and
then eventually will be longer than
the persistence length which
is a famous length and
theory of linear polymer chains and it
will start flopping around isotropically.
Now this persistence length for
the original Cornell Cornell experiments
is pretty large Just like five metres.
But if they if they really
did the experiments for
say something one hundred angstroms across
it's only twenty five microns and so
eventually they should be able to figure
out what happens when you you torture
a flopping ribbon a floppy ribbon with
with perpendicular Force want you
to know force and so forth and
in a sense they're repeating beautiful
single molecule by our physics experiments
which were done by Steve Chu and
collaborators on D.N.A. But here they're
doing on the non biological material and
atomically thin graffitti ribbon
The question is what happens.
Well we can we can study let's just
tickle mechanics in a variety of ways.
There's a very elegant and
sensible description which you can
make and sort of inspired by this belt
if this passes meandering around three
dimensional space in various ways.
Relative to a boring flat belt.
There are a number of defamations there's
a very male energy defamation which is
bending so that's relatively
straightforward there's another fairly low
energy one which is twist so bend and
twist not too hard for a belt but
if I try to to spoil you
the belt it fights back
in fact a sort of straw is strong
because in that cylindrical hollow
geometry you can't bend a soda straw
without stretching it and that's
what happens when you try to spray these
other defamations only involve the low.
Energy bending motion and so
you can describe what goes on.
In terms not of a smilie a spin model but
for these ribbons it's better to think
about an orthonormal triad the statistical
mechanics of an orthonormal triad defined
at every point along the ribbon so
I go along the ribbon like this that
would be the direction let's say
three and then there is the perpendicular
direction to the ribbon each to and
then there's a long short
direction there's a run and
I can describe the motion of these
orthonormal triad by by a set of or
angles for example or a set of
infinitesimal rotations that are defined
in this way so you can you can
work out what happens in terms
of these these these Promise a little bit
like the frank description of the last to
City of a liquid crystal
has been twisted splay.
And in fact there are certain limits where
we can understand what's going on if I
take a rod at the shoelace for
example with a circular cross-section.
You can show that this case by
symmetry the stiffness is a one and
a to have to be equal to each other.
And then the formula that
the energy simplifies.
To energy that basically only
depends on the local tangent vector
to two the shoelace or the or or
the the soda straw or whatever and
this maps the physics of long wave
lengths of a ribbon onto a one D.
Heisenberg model Isenberg spin
chain with three components.
If on the other hand we
take a two to infinity and
we also take this this twist last that
constant to infinity then we're only
allowing a particular kind of bend and
this belt is sort of
snaking along in a plane this is this is
these are the only I can't twist I can't.
Spray it and so it's sort of as if it were
confined to the surface of the table and
now the tension factor is
basically like the order Fremen
are of a one dimensional X.Y.
model so there's a Heisenberg model
working in this theory of worth
a normal triad as well as a.
X. Y. model.
What we actually need though.
For these ribbons that we talked
about earlier is is a very
different regime it's
a regime where a one which is
the bending modulus is only
proportional to the W.
the single power W.
times Kappa So it's pretty soft so
is the twist much of the C C is the twist
modulus it's also very soft but
there's a two is gigantic because that's
the splayed modulus that I already talked
about for bending the spout like that and
in fact the ratio of a two to a one and
also to a two to see goes like
the fuck on karma number.
So you might just as well
send a two to infinity and
then try to do this to become a can accept
this constraint orthonormal try it.
Well some of you know it certainly people
with a background in polymer science know
that it can be very helpful people
like Jan and Sam Edwards pointed out
to map these kinds of problems onto
their quantum relatives and that's
done in statistical mechanics by something
called the transport tricks method and
if you move along the backbone of
the ss arc length along this ribbon and
try to think about what's going
on the physics is that of.
Quantum Richard Roeper.
With wildly an isotropic
moment of inertia tensor.
The quantum mechanical rigid rotor was
worked out all its I can values are known
for the relevant operator that describes
the quantum mechanics that goes
with the statistical mechanics in
the one nine hundred thirty S. and
the only slightly strange thing
is this last term F. is E.
which is a bending moment at
the end of the ribbon but F A Z
maps onto a quantum top spinning around
in the presence of a gravitational field.
And everything is known about that as well
and so to figure out how you cross over
from a semi flexible graphene ribbon
to a more isotropically wandering
polymer you need only to solve
the relevant transfer matrix here.
Find all the I can values you can't just
use the ground state dominance because
you want to understand short change
short ribbons and long ribbons.
And then you can try to figure out what's
going to happen so what under a did.
He did these calculations in great detail
calculating the deflection age for
example.
Of a cantilever as
a function of this length.
When L. is varying with respect to W.
is the study the physics as a function of
these two links is the thermal length so
I already talked about which for graphene
is maybe one or two angstroms There's also
the persistence like this is a classic
notion and let me polymer chains but
this persistence length is then
changed dramatically by this.
Non-trivial flop of uncommon thermal
thought of income number physics associate
with the exponent eight zero and when
this length is much greater than thermo.
Comes into play you end
up with three regimes.
And.
So here's the deflection scaled by.
This linear.
Distance L. and
once as a function of temperature so
when temperature gets low enough behaves
like a classical cantilever and is roughly
independent of temperature this is just
temperature measured in units of Kappa.
There's an intermediate regime
where where we're basically
the you end up being larger
than the stumble length and
you get this sort of
anomalous one over to.
Zero point four temperature dependence so
the experiments so
far have been done in this regime but
they haven't very temperature but
here's a prediction that would tell you
how this rain are remarkable four or
five thousand fold in almost ation would
scale with temperature on this log
log plot and finally if we take W.
to be if we take Al very very large larger
than this persistence length things start
to behave like a random walking ribbon but
even here in the temperature dependence
this thermal eyes top of
uncommon exponent sneaks in
to give you a power law like when
over to the one point six So
this is something that we we hope the
experimenters might consider exploring and
certainly very temperature
with these graphic sheets.
Together with Roscoe skep neck and
Mark Bowling We've also
started to study computer simulations
of these thermal eyes to ribbons.
Important aspect of this is to look at
tension tension correlation functions and
here there's an exponential expectation
of an exponential fall off but
now this persistence length is enhanced
over Kappa divided by K B T which is
the value it would have had four.
A simple very thin.
Polymer like ribbon by this W. over L.
thermal to the eight A power so
that's this four thousand fold and Hansen.
And you can ask what were what
would you expect for the for
the thermal fluctuations the the actual H.
deflection so that you could say
there's no force here there's
Interestingly she is a very kind and
hopefully I will get this thermal
simulation which is kind of finicky
to actually run once you get it year ago.
So there it is you can sort
of see the ribbon flexing
around sometimes even
leaving the screen at the end
there is a fairly low energy ripples
which are this the better the twisting
mode is certainly bending mode.
Bending modes as well and
very little structure.
And from this kind of simulation you can
measure a group mean square and
then distance squared deflection so
forth and what we.
Find there for
a whole menagerie of W.'s and
L's that's both thing for
the main square deflection of the end and
units of the persistence length is a
crossover from the behavior of a thermal I
can't a lever which is out here to
the beginnings of a bending of this curve
the beginnings of a crossover
to a some sort of one D. polymer
like behavior so we can get data collapse
we would need bigger simulations to.
Really test these ideas in detail but
this collapse was attained
with no what just the ball parameters.
Now where could we go with this
this is a kind of shotgun marriage
implemented by the Cornell group in
some sense between soft matter and and
hard condensed matter the to do graphene
is largely the province of each bar
related physics they're coming together
an interesting way I already talked about
looking at the normal normal correlation
function and that's kind of intriguing in
itself there's a long range order we break
the mermen Wagner theorem and so forth and
if you work out this normal normal
correlation function what I want to focus
on now is the reduction of this
asymptote this goes to zero for
large distances the reduction
in the asymptote.
When we have a finite temperature
it's reduced slightly at
very low temperatures compared
to Kappa and if that's the case
this is presumably roughly speaking
what the graph seen is doing but
you might imagine that the temperature
were large enough compared to Kappa eight
as order unity this is water unity
as well that I could make this zero.
In which case I didn't finally at high
enough temperatures lose long range order.
And maybe end up with something like that.
That is there would be a phase transition
between a high temperature flat phase so
I learned temperature flat phase and
a high temperature crumple phase.
This complex phase has
been conjectured about.
Theoretically.
This has a frightful dimension of two
This has a fractal dimension of about
two point five.
And.
It would have.
Many interesting mechanical properties you
might have to take the temperature to be
ridiculously large say twenty thousand
degrees Kelvin to get this to happen but
we have other tricks that we think we can
make it happen such that we can think we
can make it happen when temperature.
Here's a kind of interesting
direction that the group has
emphasized they sometimes call
this graphene Keurig Nami.
So this is a graphene spring.
Where laser cuts have
been used to cut a ribbon
down to size and
when the stress is off you see that
it's just a graphic ribbon with
a set of rectangular holes in it and
this is one of I think
twenty thousand holes and
releases associated with this atomically
thin spring sitting in water.
You can torture in other ways as well.
Through very strong material.
Doesn't break and as part of this exercise
they did something which
is quite natural in.
Related.
Condensed matter they measured
the current voltage characteristics
and in fact.
If you look at the the current
voltage characteristics
associated with this object of
the spring Here's your present strain.
Here's two hundred forty percent strain
this thing is still like a rubber band OK
you might think that putting on
a two hundred forty percent strain
which change the electrical properties.
Remarkably.
At least initially if you do this
experiment so the idea is you you want
to current across here and then you
apply gatefold to through the water so
it's there's a gate that has to do with.
Seeing things in the water and and
then if you just measure the conductance
as a function of the gate fault it's
quite radically different this relative to
to the zero here which I think is right
in the middle as you go from zero percent
strain to two hundred forty percent
strange almost nothing happens.
There are these two curves
in there they're not so
different now think about that.
First tight little disappointing you might
have expected a larger change to them.
But what's going on once you cut these
holes there's a defamation that's
totally dominated by bending bending
is the low energy mode here and
if you make if you cut out as they did.
From a piece of paper a little
a rare spring with these holes and
you pull on it it stretches not
at all it simply whips and forms.
By bending to produce this
this contorted object up here
so the electrons that are going from
one end of this bring to the other
both cases are going on
the same length path
despite the fact that the two
hundred forty percent stray so.
This is not a good experiment to torture
the electrons they're just doing their
thing what thing are they doing well
we know the graphene is described
by an electronic band structure which
has direct cones at the Fermi surface so
there's a fire state for
the fan a log of the.
Kind of relativistic.
Physics of electron.
This is embodied in graphene we have
crumpling that's kind of like a baby
version of general relativity and if all
we could find a way to couple these things
we be doing something possibly
quite interesting and.
One of the experiments if we
can make graphene crumple and
we have some ideas about putting in
cuts that allow it to to locally
tear and
restrains the moment Wagner serum so
to speak one of the things that we can
make example that would be fun to do
would be to subject a crumpled piece of
graphene suitably insulating on the top on
the bottom to avoid shorts
to a uniform magnetic field.
What would happen.
It's a crumpled object so now the
electrons are going through affectively
a magnetic field that's up down
left right various angles.
Then I think we can probably
begin to couple together.
These these two kinds of physics.
Speculation but I think it might be
worth pursuing and one last thing before
I close and ask your questions is there's
also the analog as I mentioned of these
frozen wrinkles that body in the piece
of paper that once it gets spread out
this was this is a surface made
with say a three D. printer.
And it looks like just something that
I crumpled up and spread out but
in fact it's a very special surface that
was constructed by design to have hike
high correlations that very
like one over Q The fourth.
One of a Q The fourth is what for
example the membrane in a vessel by layer
what the fluctuations in the membrane
looks like a bicycle by layer.
Would look like and then if you politicize
that thing if you fly ash plume arised
you get a stiff object.
Like this and I'm very well
though is a card carrying thirst
actually one of the lap to measure the
mechanical properties of this thing and
it's enormously enhanced.
So frozen ripples as well as thermal
ripples have this kind of effect.
Here is what I showed you earlier this
is actually a picture from you know so
I think this is a good time to close and
thank you for your
attention.
Yeah.
Well it would certainly stiffen
the graphene and I think if you
look at their nature paper there's a
mixture of combinations thermal stuff and
combination of perhaps
charge induced ripples.
As well so parents are certainly present
in the experiments what would be very
useful I hope they will do this
is to vary temperature and
get to some of fluctuations to go away
the experiments could be done in fact to
secure go down to very low temperatures
see what part is due to the frozen ripples
and what part is due to the thermal
fluctuations both of them will
create a large unhandsome
in the bending region.
Whoa.
Whoa
yeah.
Yeah so.
What I've told you about is
mere speculation if one were to
pursue this question one would certainly
worry about graphene graphene interact.
Since in the actual experiments there's
some kind of surfactant that goes in and
passive eights the upper and
lower layer of the graphene
to reduce the stickiness the self
stickiness of the graphene and
you certainly want to do that if you
are going to say to a hard measurement
on this crumpled piece of paper
you wouldn't want to short out so
you'd have to have some
insulation above and below.
I just like to know
the mechanical properties.
If I were hook I pick this up and I say
whoa is there a hoax law for this and.
Absolutely it bounces in very
interesting ways and it's so
similar it's a hierarchy of links
skills in this crumpled object
reflecting its own karma number
of ten to the seventh So
putting that together with
electrical conduction mechanics and
so forth seems like
an interesting direction to
me.
Yes.
The.
It's hasn't been fully explored but
you can take over beautiful ideas from
polymer linear polymer science and
add in tensors and
do rouse to see I'm like models and
this critical exponent ADA which
seems to control a lot of things
sort of plays a role little bit like
the Florey exponent in the flat face.
Enters in to the dynamics and
there are non-trivial power laws just
as the power laws you know and
many are polymer chains.
Yet.
Yeah.
Yes.
Yes Yes Thank you thanks for the question.
That's that's certainly true.
Room temperature there will
be transport consequences and
some of the Cirrus that are listed here in
particular Finkelstein
at all have studied.
Electron scattering basically off
these Flexeril phonons and strange.
Power laws in temperature for
the conduct pivoting emerge as a result
so it you know we know phonons are going
to be important in the electronics
like the scattering electrons as they're
transported across the material and
the only unusual thing about these
fluctuate phonons is there so soft
because they they can escape
into the third dimension and.
There's unfortunately no analogue of that
in three dimensional materials they cannot
escape into the fourth dimension so
we're stuck with what we have but
here we have these ripples.
That can lead to this unusual physics.
Yeah.
OK.
Yeah.
Yeah.
I think so.
In particular in no way a meter by
meter macroscopic sheet of mylar or.
Some kind of food.
Well specially if it's insulated.
You could.
Crumple it up put on a magnetic field and
see how the.
Transport a crumpled hall
a physics experiment behaves.
It would have an enormous problem on
common number the fossil her Naaman
one common number as you I think
anticipated scales as the linear dimension
divided by the sin a square so
you want that to be big and
graphene gets it big by having
atomically thin I think it could
be big as you said by making
a metre long mylar sheet.
And there are fascinating things to look
at I think when you crumple such a thing
but it will be hard to get similar
fluctuations to play a big role when
things start to get much bigger than
ten microns ten micron square let's say
it's just hard to get to some of
fluctuations to a color break properly so
I know that this this complicated paper
idly is not in thermal equilibrium
it's not enough thermal energy to make it
visit all the regions of face face and so
forth but there were these other.
Sort of non equilibrium crumpling
questions that you could go
after in the way that you
describe.
All K. thank you yeah so.
That's simply done for convenience.
We are we're even in the soft
condensed matter community
that graphene has a hexagonal
not a triangle last.
However.
If I imagine putting a normal to this
more realistic simulated graphene sheet
here in the center of every hexagon
then the hexagons themselves
make a triangular lattice.
And so Rick or screening a little bit and
just using the triangle lattice is
a convenient way to inject
a bending rigidity and
a Young's Margo's it's an excellent
question but the triangular and
the honeycomb lattice is or
dual to each other so
it's believed they would have
the same macroscopic physics.
OK so it could do the crucial thing.
Has to do with whether one of
the important things is whether
the elasticities theory is actually
going to be isotropic or not and so
this this is the underlying long
way Flink continuum theory and
when I read it this way in
terms of the strength tensors
this is a trace of a strange cancer
squared this is this is you right J.
squared I'm assuming isotropic
continuum elastic theory
that's an adequate assumption for
a triangle lattice or
a honeycomb loudest if I were to take
a square out US I'd have a third elastic
constant If I were to
take a highly anisotropic
rectangular lattice now I have the two
directions would be different and
maybe it would crumple one way but not
the other way not the other direction so
then the underlying lattice
would make a difference.
And that would also be interesting
to explore I'm hoping that.
My persuasive powers are limited but
they maybe could I could persuade some
experimental group to actually
take some graphene and
we already know that the Cornell
people can make cuts and
turn this into a spring by
cutting out rectangles rocks so
paste I made a set of cuts they're
all going in the same direction.
Bunch of random laser
cuts that they would.
He sort of weakening the tendons of
the graphene with respect to a certain set
of subset of bendings but
not new either direction so
maybe it could crumple up like
an accordion and we could end up with
something that looks like this.
So so that might be a way to get it to
come for a run temperature if you start
making cuts isotropically in
the plane you weakening it and
thereby cutting enough of these
carbon carbon bonds but not too many
years probably I'm hoping there's a really
rigidity percolation transition so
that what actually happens is
this even at room temperature and
that might be a way to get in this
couple days by judiciously chosen
maybe possibly random laser cut.
Thank you.