ancient history: from linear polymers to tethered...

Ancient History: from linear polymers to tethered surfacesBy the 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 (violates Mermin-Wagner-Hohenberg theorem)

F. A

brah

am a

nd d

rn, S

cien

ce 2

49, 3

93 (1

990)

sheet polymer = soft interface permeable to solvent molecules on either side…

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 McEuengroup Cornell: “Single molecule polymer physics” for graphene

Experiments of the McEuengroup Cornell: “Single molecule polymer physics” for graphene

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

Graphene (and BN, MoS2, WS2 , …?)

Remarkable effect of thermal fluctuations-- entire low temperature flat phase characterized by critical fluctuations; “self-organized criticality”--strongly scale-dependent bending elastic parameters

Luca PelitiMehran KardarYantor Kantor

Recent experiments:Paul McEuen group (Cornell)

Recent theory:Mark BowickRastko Sknepnek &

Critical phenomena without critical points: Theory of free-standing graphene ribbons

-- vK ~ 1013! “Moore’s Law limit of thinness…--Thermal fluctuations dominate for l > lth = 0.15nm…-- Bending rigidity at room temperature enhanced 6000-fold-- Anomalous properties of ribbons, crumpling transition, etc.

Andrej Kosmrlj

August Föppl(1854-1924)Pioneer of elasticity theory

Truly ancient history: in 1904, Föppl & von Kármánstudied 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 modulus 2

bending rigidity

Y

=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

2dimensionless "Foeppl-von Karman number" / 1 ( linear dimension) (compare Reynold's number Re / in fluid mechanics)

resembles a simplified form of general relativity (developed 10 yea

vK YL LuL

rs later....)exact solutions available only in very special cases

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 modulus 2

bending rigidity

Y

macroscopic (1cm - 10m)

Applications: thin solid shells and structures

macroscopic (1cm - 10m)

& microscopic (0.1nm - 1μm, but what about Brownian motion??)

Applications: thin solid shells and structures

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

/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

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

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

/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

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!

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

Negative thermal expansion coefficient

Negative coefficient of thermal expansion and nonlinear stress strain curves

Out of plane fluctuations suppressed by an external tension σ

E Guitter, F David, S Leibler, L Peliti, PRL 61, 2949 (1988)

Graphene is the ultimate 2D crystalline membrane:

• One atom thick

• Very stiff in-plane (Young’s modulus Y = 500 GPa)

Freely supported graphene is an ideal test bed…

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

L = 10μ, h = 3 A, vK = 1013 !!

Graphene is the ultimate 2D crystalline membrane:

• One atom thick

• Very stiff in-plane (Young’s modulus Y = 500 GPa)

Freely supported graphene is an ideal test bed…

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

L = 10μ, h = 3 A, vK = 1013 !!2

fluctuations dominate for

/ ( ) 0.2nm!th

th B

L l

l k TY

Graphene cantilever experiment

Melina Blees, Arthur Barnard, Samantha Roberts, Josh Kevek, Alex Ruyack, Jenna Wardini, Peijie Ong, Aliaksandr Zaretski, 

Si Ping Wang, and Paul L. McEuen

Cornell University Ithaca NY

Graphene cantilever experiment

Melina Blees, Arthur Barnard, Samantha Roberts, Josh Kevek, Alex Ruyack, Jenna Wardini, Peijie Ong, Aliaksandr Zaretski, 

Si Ping Wang, and Paul L. McEuen

Cantilever experimentGalileo Galilei (1638)“Discourses and Mathematical Demonstrations Relating to Two New Sciences”

Cornell University Ithaca NY

P. McEuen group, preprint

33 /Rk W L

P. McEuen group, preprint

Bending rigidity enhanced 6000 fold.Agrees with

(lth ~ 0.2nm for graphene)

0.8( ) ( / )R thl W l 33 /Rk W L

L = ribbon lengthW = ribbon widthlp= persistance length

5 ; W = 10 , but

25 ;W = 10nm!p

p

l m

l

McEuengroup

Theory of thermalized cantilever ribbons (A. Kosmrlj and drn)

21

32 1

(1 ) /12; C 2 (1 )

/12 ,

A W W

A YW A C

E = 2d Young’s modulus μ = 2d shear modulusν = 2d Poisson ratio

2 2 21 1 2 2 32

dsE A A C Fz

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 (A. Kosmrlj and drn)

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

Temperature dependence of cantilever deflection3 2( ) 4 / ( )

( ) ( )( )

th B

p thB th

l T k TY

Wl T W l Tk T l T

Above T0 , L exceeds the persistence lengthBelow TW , the thermal length exceeds W

log( / )T

Random walking ribbon

Thermalized cantilever

Classical cantilever

~1/T2-η/2~1/T1.6

~1/Tη/2~1/T0.4~const.

T =

T 0T =

T W

current experiments

1

( ) ( ) exp[ | | / ]

persistence lengthp

p

pB th

t s t s x x l

l

Wlk Tl

Computer simulations of graphene ribbons

Rastko Skepnek, University of Dundee

Mark Bowick, Syracuse University

Future directions: flat vs. crumpled phases

OR

Future directions: flat vs. crumpled phases

ORProjectedArea ~ L2

low T

Future directions: flat vs. crumpled phases

ORProjectedArea ~ L2

low T

Projectedarea ~ L8/5

high T

10 µm

Future directions: new, atomically thin springs

Electrical conduction of graphene spring as a function of strain & gate voltage

Experiments by McEuen group

Crumpled paper experiment

Figure: Melina Blees, Cornell

Thank you!!

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

Graphene (and BN, MoS2, WS2 , …?)

Remarkable effect of thermal fluctuations-- entire low temperature flat phase characterized by critical fluctuations; “self-organized criticality”--strongly scale-dependent bending elastic parameters

Luca PelitiMehran KardarYantor Kantor

Recent experiments:Paul McEuen group (Cornell)

Recent theory:Mark BowickRastko Sknepnek &

Critical phenomena without critical points: Theory of free-standing graphene ribbons

-- vK ~ 1013! “Moore’s Law limit of thinness…--Thermal fluctuations dominate for l > lth = 0.15nm…-- Bending rigidity at room temperature enhanced 6000-fold-- Anomalous properties of ribbons, crumpling transition, etc.

Andrej Kosmrlj