Ancient History: from linear polymers to tethered surfacesBy the early 1990’s, theories of linear polymer chains in a good solvent were generalized to treat the statistical mechanics of flexible sheet polymers

Remarkably, “tethered surfaces” with a shear modulus are able to resist thermal crumpling and exhibit a low temperature “wrinkled”, flat phase…

A continuous broken symmetry -- long range order in the surface normals --arises in two dimensions (the Mermin-Wagner-Hohenberg theorem is evaded!)

Experiments on the spectrin by Christoph Schmidt & Cyrus Safinya

F. A

brah

am a

nd d

rn, S

cien

ce 2

49, 3

93 (1

990)

sheet polymerlinear polymer

Lots of recent interest in the flat phase among graphene theorists, but (until recently) not many experiments….

Influence of out-of-plane phonons on electronic properties:

1. E. Mariani and F. von Oppen, Phys. Rev. Lett. 100, 076801 (2008).

2. K. S. Tkhonov, W. L. Z. Zhao and A. M. Finkel’stein, Phys. Rev. Lett. 113, 076601 (2014).

Quantum effects at low temperatures:

1. E. I. Kats and V. V. Lebedev, Phys. Rev. B89, 125433 (2014).

2. B. Amorim, R. Roldan, E. Cappelluti, F. Guinea, A. Fasolino and M. I. Katsnelson, Phys. Rev. B 89, 224307 (2014)

Experiments of the McEuen group at Cornell: “Single molecule polymer physics” for graphene

M. Blees et al., Nature 524, 204 (2015)

10μ

Equations of thin plate theory--nonlinear bending and stretching energies--vK = Föppl-von Karman number = YR2/κ >> 1

Physics of thermalized sheets -- “self-organized criticality” of the flat phase-- strongly scale-dependent elastic parameters-- hard condensed matter applications : graphene , --MoS2, BN, WS2 ,….

Thermalized sheets and shells: topology matters

Andrej Kosmrlj

Physics of thermalized shells -- spherical shells accessible via soft matter

experiments on diblock copolymers.-- shells with zero pressure difference are

crushed by thermal fluctuations for R > Rc(T)!

Recent experiments:Paul McEuen group (Cornell)

G. GompperG. Vliegenthart

Jayson Paulose

August Föppl(1854-1924)Pioneer of elasticity theory

Elastic membranes: in 1904, Föppl & von Kármán studied large deflections of elastic plates

Theodore von Kármán(1881-1963) Hungarian-American physicist & aeronautical engineer

To study deformed surfaces, expand about a flat reference state…

2 20 2 ij i jdr dr u dx dx

2 2 2 2 21 [ ( ( )) 2 ( ) ( )]2 ij kkE d x f x u x u x

bending rigidityshear modulus

+ = bulk modulus

1 2( , )f x x

1x

2x

1 2

1 2 0 1 1 2

2 1 2

( , )( , ) ( , )

( , )

f x xr x x r u x x

u x x

( )( )1 ( ) ( )( )2

jiij

j i i j

u xu x f x f xu xx x x x

bending energy

stretching energy

Flexural phonons can escape softly into the 3rd dimension…

=0 ( ) 2 ( ) ( ) ( )

( ) Airy stress functioni ij ij ij ij kk im jn m nx u x u x x

x

Nonlinear Föppl -von Karman Equations (1905)

Bending modes ( ) coupled to stretching modes ( ); Minimize energy over ( ) and ( )...

f x u xf x x

2 2 2 2 2 24

2 2 2 2

22 2 24

2 2

2

1 Gaussian curvature

f f ffy x x y x y x y

f f fY x y x y

4 ( ) Young's modulus2

= bending rigidity

Y

2

resembles a simplified form of general relativity (developed 10 years later....)exact solutions available only in very special cases dimensionless "Foeppl-von Karman number" / 1 (linear sizevK YL

= L) [compare Reynolds number in fluid mechanics, Re / ]uL

Hope big is the Foeppl von-Karman number?22 2 2 2 2 2 2 2 2

4 4 22 2 2 2 2 2

12 ; let , ...f f f f f ff f Lf YLy x x y x y x y Y x y x y

2 2 2 2 2 24

2 2 2 2

22 2 24

2 22

2

1

/

f f ffy x x y x y x y

f f fx

vK

vK YLy x y

1/vK << 1 multiplies highest derivative in first equation

2

2

12

10 square of graphene30 / ; 1

102

/ !!.

LY eV A eVvK YL

2 2

33 3

3

2

2

7/ 12( / ) (1 ) 4 1

crumpled foil or paper of thickness / [12(1 )];

3d isotropic Young's modulus

0

hE h v Y E h

vK YL L h

E

8.5 in, 0.1mmPoisson ratio 0.25

L h

macroscopic (1cm - 100m)

& microscopic (0.1nm - 1μm) -- is Brownian motion important ??

Applications: thin solid shells and structures

What About Thermally Excited Membranes? (L. Peliti & drn)

Assume kBT /κ << 1, and do low temperature perturbation theory

2

2 4

3

ˆ ˆ( )( ) ....

(2 ) | |[1 / (4 ) ... ) ( ]

Ti ij j

R B

B

q P k qd kq k T

vK

Yq k

k T

2 2/ ( / ) 1membrane sizemembrane thickness

vK YL L hLh

Self-consistent bending rigidity, κR(q)~1/q & diverges as q0!

/eff ln ( , ) ( , ) BE k T

B x yF k T D u x y D u x y e

Tracing out in-plane phonon degrees of freedom yields a massless nonlinear field theory

22 2 2 20 1 2

1 1( ) ( ) ; 2 4

i jT Teff ij i j ij ijF d x f Y d x P f f F F P

4 (Y= Young's modulus2

1 2 1 2( , ) ( , ) exp( / )BZ u x x f x x E k T D D

2 2 2 2 21 [ ( ( )) 2 ( ) ( )]2 ij kkE d x f x u x u x

( )( )1 ( ) ( )( )2

jiij

j i i j

u xu x f x f xu xx x x x

Renormalization Group for Thermally Excited Sheets

L. Peliti & drn (~1987)J. Aronovitz and T. LubenksyP. Le Doussal and L. Radzihovsky

0

2 2( ) / ;Bl k T a 0

2 2( ) /Bl k T a

( )l

( )l

F.-von K. fixed point

Thermal FvKfixed point

define running coupling constants....

scale dependent Young's modulus 4 ( )[ ( ) ( )( )

2 ( ) ( )l l lY l

l l

( ) ( / )

( ) ( / ) u

R th

R th

l l l

Y l Y l l

0.82, 0.36u

2

Thermal fluctuations dominate whenever

/ ( )th

th B

L l

l k TY

Why long range order?Longitudinal vs. transverse spin waves

10ˆ ˆ( ) ( ) 1 ln( / )

2thB B

a b tha b

lk T k Tn r n r l a Cr r

22 2 2 20 1 2

1 1( ) ( ) ; 2 4

i jT Teff ij i j ij ijF d x f Y d x P f f F F P

ˆ( )an r ˆ( )an rˆ( )bn r

ˆ( )bn r

Graphene is the ultimate 2D crystalline membrane:

• One atom thick• Very stiff in-plane (Young’s

modulus Y = 20eV/A2)• L = 10μ• κ = 1.2eV

With graphene, we have reached the “Moore’s Law” limit of large Foppl-von Karman numbers

vK =YL2/κ ~1012

2

fluctuations dominate for

/ ( ) 0.2nm!th

th B

L l

l k TY

L

Freely supported (σij = 0) graphene tests the theory

Galileo Galilei (1638)Cantilever experiment

Bending rigidity enhanced ~5000 foldAgrees with

(lth ~ 0.2nm for graphene)

0.8( ) ( / )R thl W l

M. Blees et al., Nature 524, 204 (2015)

15

Thermal fluctuations ofwarped membranes

Adapt self-consistent screening approx. of Le Doussaland Radzihovsky,Phys.Rev. B48, 3548 (1993).

geometrydisorder

dominates

thermal fluctuations

dominate

thermal fluctuations

dominateA. Košmrlj and drn, PRE 89, 022126 (2014)

McEuengroup

Map 1d path integral statistical mechanics onto the quantum mechanics of a rigid rotor in an external gravitational field

Theory of thermalized cantilever ribbons (see Andrej’s talk, late february)

21

32 1

(1 ) /12; C 2 (1 )

/12 ,

A W W

A YW A C

Y = 2d Young’s modulus κ = 2d bending rigidityν = 2d Poisson ratio

2 2 21 1 2 2 32

dsE A A C Fz

Shum et al., JACS 2008, 130, 9543

Microfluidic fabrication of polymersomes

Polymersome Radius, R = 30 µm Thickness, h = 10 nm

2 8

Foppl-von Karman number ( / ) 10 R h

Start with “double emulsion” of ampiphillic diblock copolymers (PEG-b-PLA).Tune wetting properties to eject thin *crystalline* bilayer shells.Result is a delivery vehicle for dugs, flavors, colorings and fragrances that can be crushed by osmotic pressure

Thermal fluctuations again matter…

To study thermal deformations of spherical shells, we use shallow

shell theory….

( , ) ( , ) ( , )( , ) ( , ) ( , )

( , ) ( , ) ( , )

x x

y y

x u x y f x y Z x yxy y u x y f x y Z x y

Z x y Z x y f x y

jiij dxdxudsds 2' 22

1xx yyZ Z

R

2 2 2

( , )z Z x y

z R x y

Initial shape:

x

y

2 2 2 2 21 [ ( ( )) 2 ( ) ( )]2 ij kkE d x f x u x u x

( )( )1 ( ) ( )( )2

jiij

j i i j

u xu x f x f xu xx x x x

( )( )1 ( ) ( )( )2 ij

jiij

j i i j

u xu x f x f xu x fRx x x x

TkFyxB

BedfyxuDyxuDTkF /0eff ),(),(ln

Nonlinear Field Theory for Thermally Excited Shells…

New for curved membranes!

0 1effF F F

20

2 22

2 2 ( )1 (2

)2

pF f Rd x Yf fR

2 21

1 ( )4

Tij i j

Tij i j

f P f fF d x P f fR

Trace out in-plane phonons and radial shrinkage f0 …

*

1/4

curvature length scale

= /R vK Rh R

)(2 xxqq fexdf i

f(qx, qy)

q 2

Gaussian Fluctuation Spectrum in Fourier Space

2 2 2 2 20 2

2 24 22 2

2 2

0 22 4

2

1 ( ) ( )2 212 (2 ) 21 ( ) (,2 2 2)

)(

pR YF d x f f fR

d q pR Yq q

Y

fR

d qF pRq q qqR

f

q

q

There is a soft phonon mode above a critical pressure!

( )q

q

2/Y R

1/ R *q

2

4c

Yp pR

1/4* / 1/ *q vK R l 2( / )vK YR

Macroscopic Buckling Instability Arrested by a Wax Mandrel…

R. L. Carlson et al., Exp. Mech. 7, 281 (1962)

* 1/4/R vK Rh R

thickness = h

R

Koiter, 1963 (1945)Hutchinson, 1967

Classical shell theory

*

R = 4.25 in., R/h ~ 2000

Evaluate effect of nonlinearities with perturbation theory…J. Paulose, G. Vliegenthart, Gompper, drn, PNAS 109, 19551 (2012)

2 1/4

Assume bare pressure = 0 (e.g., hemisphere!) and use renormalized elastic parameters ( *) and ( *), * ( /Y) , to evaluate the pressure generated by thermal

1fluctations: ( *)24 ( *

R R

RR

Y RkTp

( *) ( *); Thermal fluctuations will crush)

spheres or hemispheres whenever ( *) ( *)

c

R c

p vK

p p

2 2

61 31 ; 14096 256

1 ; / ; 4 /24

B BR R

BR c c

k T k TvK Y Y vK

k Tp p vK vK YR p Y R

1/23

2Criterion for self-crushing is 54 ; .( )

60 for graphene....

c cB B

c

R R R const hY k T k T

R nm

(see talk of Andrej Kosmrlj in late February)

Thermalized sheets and shells: topology matters

Andrej Kosmrlj

Jayson Paulose

F. A

brah

am

G. G

ompp

er&

G. V

liege

ntha

rt

vs.

Scale-dependent bending rigidity ~5000 fold enhancement for grapheneat room temperture

3 2 1/2

Enhancement of bending rigidity only ~70 fold atroom temperature; but spheres (and hemispheres) crush themselves for 54[ / ( ) ]c BR R Y k T

10 µm

Future directions: new, atomically thin springs

10 µm

Future directions: flat vs. crumpled phases

ORProjectedArea ~ L2

low T

Projectedarea ~ L8/5

high T

Crumpled paper experiment

Figure: Melina Blees, Cornell

Thank you!!

Equations of thin plate theory--nonlinear bending and stretching energies--vK = Föppl-von Karman number = YR2/κ >> 1

Physics of thermalized sheets -- “self-organized criticality” of the flat phase-- strongly scale-dependent elastic parameters-- hard condensed matter physics : graphene , -------MoS2, BN, WS2 ,….

Thermalized sheets and shells: topology matters

Andrej Kosmrlj

Physics of thermalized shells -- spherical shells accessible via soft matter

experiments on diblock copolymers.-- shells with zero pressure difference are

crushed by thermal fluctuations for R > Rc(T)!

Recent experiments:Paul McEuen group (Cornell)

G. GompperG. Vliegenthart

Jayson Paulose