ancient history: from linear polymers to tethered...
TRANSCRIPT
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