ancient history: from linear polymers to tethered...
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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
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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