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2/23/06 Texas Colloquium
The Penn Team• Theory
– TCL– Randy Kamien– Andy Lau– Fangfu Ye– Ranjan
Mukhopadhyay– David Lacoste– Leo Radzihovsky– X.J. Xing
• Experiment– Arjun Yodh– Dennis Discher– Jerry Gollub– Mohammad Islam– Ahmed Alsayed– Zvonimir Dogic– Jian Zhang– Mauricio Nobili– Yilong Han– Paul Dalhaimer– Paul Janmey
2/23/06 Texas Colloquium
Outline• Some rods • Rotational and Translational
Diffusion of a rod - 100th Anniversary of Einstein’s 1906 paper
• Chiral granular gas • Semi-flexible Polymers in a nematic
solvent• Nematic Phase• Carbon Nanotube Nematic Gels
2/23/06 Texas Colloquium
Comments
• Experimental advances in microsocpy and imaging: real space visualization of fluctuating phenomena in colloidal systems
• Wonderful “playground” to for interaction between theory and experiment to test what we know and to discover new effects
• Simple theories - great experiments
2/23/06 Texas Colloquium
Rods
fd Virusfd Virus
100 nm – 10,000 nm100 nm – 10,000 nm
~1 nm~1 nm
900 nm900 nm
1 1 mm
PMMA Ellipsoid
Carbon Nanotube
2/23/06 Texas Colloquium
Einstein – Brownian Motion1. “On the Movement of Small Particles Suspended in Stationary Liquid Required by the Molecular Kinetic Theory of Heat”, Annalen der Physik 17, 549 (1905);
( )
1
2
; (6 )
2 ;B
x f a
x Dt D k T
ph -= G G=
D = = G
&
a = particle radius; = viscosity; f = force
2/23/06 Texas Colloquium
Langevin Oscillator Dynamics2
212 2p
H kxm
p mv mx
= +
= = &
1
22 2 2
;
( ) ( ') 2 ( ')
2| ( ) |
x kx
t t T t t
T kx
k k
x g
x x d
ww
-= - G + G=
= G -
G=
+G
&
Ignore inertial terms:
( ) ( ') 2 ( ')
6
p kx v
t t T t t
a
g V
V V g d
g ph
= - - +
= -
=
&
( )2 2| |2
Bd k Tx x
kw
wp
= =ò
Dynamics must retrieve equilibrium static fluctuations: sets scale of noise fluctuations to be 2T
P. Langevin, Comptes Rendues 146, 530 (1908) Uhlenbeck and Ornstein, Phys. Rev. 36, 823 (1930)
2/23/06 Texas Colloquium
Diffusion with no potential
( )
2
2
0:
[ ( ) (0)]
21
2 2
i tB
B
k x
x t x
d k Te
k T t Dt
w
x
wp w
-
® =
-
G= -
= G =
ò
&
AVG
16 6
Bk T RTD
a N aph ph= = Measurement of D
gives Avogrado’s number
2
( , )
1exp
4 4
P x t
xDt Dtp
=
æ ö÷ç- ÷ç ÷çè ø
Gaussian probability distribution
R = Gas constant
2/23/06 Texas Colloquium
Density Diffusion
( )
( )
( ) ( )
( )21 2 1 2
2
2
( , ) ( )
; ( ) ( ) ( ') '
( , ) ( )
( ) ( )
1( ) ( ) ( )
2
( , ) ( , )
( , ) ( , )
t t
t
x
t t t t
xt t
t x x
t
n x t x x t
x x t t x t t dt
n x t t x x t t
x x t x x x t
x x t t t dtdt
n x t nv D n x t
n t n D n t
aa
a a a a a
aa
a a aa a
a a aa
d
x x
d
d d
d x x
+D
+D +D
= -
= +D = +
+D = - +D
= - - ¶ -
+ ¶ -
¶ = - ¶ + ¶
¶ = - Ñ × + Ñ
å
ò
å
å å
å ò ò
r v r
&
&
2/23/06 Texas Colloquium
J. Perrin Expts. (1908)
NAVG=7.05 x 1023
( )
20
3/ 2
( , )
exp44
n r t
N rDtDtp
=
æ ö÷ç- ÷ç ÷çè ø
2/23/06 Texas Colloquium
Rotational Diffusion“On the Theory of Brownian Motion,” ibid. 19, 371
(1906): Idea of Brownian motion of arbitrary variable, application torotational diffusion of a spherical particle.
= torque
( ) 13
2
; 8
( ) 2 ;B
a
D t D k T
q q
q q q
q t ph
q
-= G G =
D = = G
&
P. Zeeman and Houdyk, Proc. Acad. Amsterdam, 28, 52 (1925)W. Gerlach, Naturwiss 15,15 (1927)G.E. Uhlenbeck and S. Goudsmit, Phys. Rev. 34,145 (1929)F. Perrin, Ann. de Physique 12, 169 (1929) W.A. Furry, “Isotropic Brownian Motion”, Phys. Rev. 107, 7 (1957)
2/23/06 Texas Colloquium
Diffusion of Anisotropic Particles
1. Brownian motion of an anisotropic particle: F. Perrin, J. de Phys. et Rad. V, 497 (1934); VII, 1 (1936).
Interaction of Rotational and Translational Diffusion: Stephen Prager, J. of Chem. Physics 23, 12 (1955)
Texas Colloquium2/23/06
Defining trajectories
x0
x1
x2
x3
x4
x5
x6
x7
2
1
34
5
6
7
Can extract lab- and body-frame displacements, trajectories at fixed initial angle and averaged over initial angles.
2/23/06 Texas Colloquium
Rotational Langevin (2d)
[ ]2 2
2
2
( ) ( ') 2 ( ')
( ) (0) [ ] 2
[ ]
2cos[ ] exp[ ]
B
HI p
H
t t k T t t
t D t
Dt
n n D t
q q q q
q q q
q q q
q
q
q
q q xq
q x q xq
x x d
q q q
q
q
¶= - - G +G =
¶¶
= - G + ® =¶
= G -
- º D =
D=
D = -
&& & &
& &
2/23/06 Texas Colloquium
Translation and Rotation
Anisotropic friction coefficients ;a bx f x f^ ^= - G = - GP P& &
0
( )
( ) ( ) ( ) [ ( ) ( )]
( ) ( ') 2 ( ( )) ( ')
i ij ij
aij i j ij i jb
i j ijB
Hx
x
n n n n
t t k T t t tq
q x
q q q d q q
x x q d
¶= - G +
¶
G = G +G -
= G -
&
Lab-frame equations
.In lab-frame, noise depends on angle: expect anisotropic crossover
F. Perrin, J. de Phys. et Rad. V, 497 (1934); VII, 1 (1936).
2/23/06 Texas Colloquium
Body-frame equations
[ ] [ ]2 2
cos sin
sin cos
( )
; ( ) ( ') 2
0
0
2 ; 2
i nin
t i i i i ijB
a
ijb
a b
x x
y y
x t x
x t t k T
x D t y D t
d q q d
d q q d
d
x x x
æ ö æ öæ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç=÷ ÷ ÷ç ç ç÷ ÷ ÷-÷ ÷ ÷ç ç çè ø è øè ø
=
¶ = = G
æ öG ÷ç ÷çG = ÷ç ÷G ÷çè ø
D = D =
å
%
%
% %
% % % %%
%
% %
x yBody frame : and are
are independent; simple
Langevin equations with
constant diffusion.
% %
x2
x1
1
2dx~
dy~
2/23/06 Texas Colloquium
Anisotropic Crossover[ ]
00
0 0
40 0
1( , ) ( ) ( )
2cos2 sin2
( )sin2 cos22
ij i j
ij
D t x t x tt
DD t
t
q qd t
q q
é ù= D Dë û
æ öD ÷ç ÷ç= + ÷ç ÷- ÷çè ø
( )/ 2
(1 )( )
a b
a b
nD t
n
D D D
D D D
et
nD
q
q
t-
= +
D = -
-=
1. Diffusion tensor averaged over angles is isotropic.
2. Lab-frame diffusion is anisotropic at short times and isotropic at long times
3. Body-frame diffusion tensor is constant and anistropic at all times
[ ] [ ]
0 0
2 2
1( , )
2
2 ; 2
ij ij ij
a b
D d D t D
x D t y D t
q q dp
= =
D = D =
ò% %
2/23/06 Texas Colloquium
Gaussian Body Frame Statistics
;
( ) ( ') 2
t i i
i i ijB
x
t t k T
x
x x
¶ =
= G
%%
% % %
21( ) exp
4 4i
ii i
xP x
Dt Dtp
æ ö÷ç ÷= -ç ÷ç ÷çè ø
%%
2/23/06 Texas Colloquium
Non-Gaussian lab-frame statistics[ ] [ ]
( )0
24 2(4)
2 212 4 4
204 16 4
( ) 3 ( )
[3( ( ) ( ))
( ( ) ( ) 3 ( ))cos4 ]
C x t x t
D t t t
t t t
q
q q
q q
t t t t
t t t t t q
= D - D
= D - -
+ - -
[ ]
( )( )
0
(4)
22
2
2
( )( )
3 ( )
2
a b
a b
C tp t
x t
D D
D D
q=
D
-®
+
Fixed q0: Gaussian at short times – same as body frame; Gaussian at long times – central limit theorem.Average q0: nonGaussian at short times, Gaussian at long times
2/23/06 Texas Colloquium
Non-Gaussian Distribution.
Probability distribution at fixed angle and small t is Gaussian. The average over initial angles is not
2/23/06 Texas Colloquium
Rattleback gasChiral Rattlebacks spin in a preferred direction; Achiral ones do not.
Chiral wires spin in a preferred direction on a vibrating substrate
Tsai, J.-C., Ye, Fangfu , Rodriguez, Juan , Gollub, J.P., and Lubensky, T.C., A Chiral Granular Gas, Phys. Rev. Lett. 94, 214301 (2005).
2/23/06 Texas Colloquium
Spin angular momentum dynamics
( )
Spin angular momentum
Spin angular frequency
( , ) ( )
l nI
l t p t
HI p
qa aa
q q q q
d
q q xq
= W=
W=
= -
¶= - - G +G =
¶
åx x x
&& & &
2
2
( ) ( ( )
( )
t i j ij
j j
l n D p t
n D l
lv D
q a qa aa
q
q d
W
¶ = - GW+¶ ¶ -
= - GW+ Ñ
= - ¶ - GW+ Ñ W
å x x
2/23/06 Texas Colloquium
Rattleback gas III
2( ) ( )t j jl lv Dw tW
W¶ = - ¶ - G W- GW- + Ñ W+
2 12( ) ( )v
t i j i j i i i ij jg gv p v vh e w¶ = - ¶ - ¶ + Ñ - G + ¶ GW-
( ) / 2
i i
z
l I
g vr
w
= W=
= =
W=
= Ñ ´ =v
Spin angular momentum
Center-of-mass mometum
Spin angular frequency
CM angular frequency
Substrate friction Spin-vorticity couplingVibrational torque
1( ) ( ) ( )/ ; rR R v R R v vw -W = = ¶ = - lB.C.:
2/23/06 Texas Colloquium
Nematic phase1 13 3( )
ij i j ij i j ijQ S nn d nn d= - = -
{ ( )
( )[ ]
( )[ ] }
2312 1
2
2
2
3
F d x K
K
K
n
n n
n n
= Ñ ×
+ ×Ñ ´
+ ´ Ñ ´
ò
Order Parameter
Splay Twist
Bend
)( nn n
)( nn
n =Frank director
Frank free energy
2/23/06 Texas Colloquium
Isotropic-to-Nematic Transition
4 1I N
D LL D
whenf-
= ?
D - rod diameterL – rod length
-
-I N
I N
rod concentration
at phase transitionorder parameter S :
orientational distribution functions
increasingconcentration
212 sin ( )(3cos 1)S dq qf q q= -ò
Onsager Ann. N. Y. Acad. Sci. 51, 627 (1949)
2/23/06 Texas Colloquium
Semi-flexible biopolymers
2 – 30 micron length 7-8 nm in diameter
~ 16 micron persistence length
ActinWormlike Micelle( polybutadiene-polyethyleneoxide )
10 – 50 micron length~ 15 nm in diameter~ 500 nm persistence
length
DNA
16 micron length 2 nm in diameter40 nm persistence
length
Neurofilament
5 - 20 micron length 12 nm in diameter
~ 220 nm persistence length
2/23/06 Texas Colloquium
Semi-flexible Polymer in Aligning Field
( )d
sdsR
t=2
| ( ) | 1;
( ) ( ( ), 1 | ( ) | )
s
s s s
t
t t t^ ^
=
= -
R(s)
t(s)
R
h=kBTG/
2/23/06 Texas Colloquium
Fluctuations2
2
22
| |2
| |2
BzP
BP
k T dH ds l
ds
k T dds l
ds
tt
tt^
^
é ùæ öê ú÷ç= - G÷ç ÷ê úè øë ûé ùæ öê ú÷ç» +G÷ç ÷çê úè øë û
ò
ò/
/
=
p B
B
p
l k T
h k T
l
Persistence length
Alignment parameter
Odijk deflection length
k
l
= =
G= =
=G
| |/( ) (0)2
zx x
p
t z t el
ll -=/ 2( ) (0) ps ls et t -á × ñ= = 0
> 0
2/23/06 Texas Colloquium
Polymers in fd-virus suspensionsActin16 m
WormlikeMicelle500 nm
DNA50 nm
Neurofilament200 nm
Isotropic Nematic
10 m 10 m
Hairpindefects
Actin in Nematic Fd
Dogic Z, Zhang J, Lau AWC, Aranda-Espinoza H, Dalhaimer P, Discher DE, Janmey PA, Kamien RD, Lubensky TC, Yodh AG, Phys. Rev. Lett. 92 (12): 2004
2/23/06 Texas Colloquium
Tangent-tangent correlations
actin in nematic phase
/ 2( ) (0) ps ls et t -á × ñ=Orientational correlationsdecay exponentiallylp – persistence length of actin in isotropic phase
Isotropic phase – quasi 2D
t(s’) t(s’+s)
h(s)
2/23/06 Texas Colloquium
Polymer in nematic solvent
( )2
2 23
0 0
1/ ( ) (0, )
2 2 2
L Lpl d
F T dz dz z z K d xdz
tt n nd^
^
æ ö G÷ é ùç= + - + Ñ÷ç ë û÷çè øò ò ò
pl
K
Persistence length
Coupling constant
Elastic constant
=
G=
=
Bending energy Coupling energy Elastic energy
Pl
Odijk lengthl = =G
2
/
2
2 2 2 2 2 20
1 cos( / )lo
( ) (
g(1 / )4 (1 )[1 log(1
)2
/ )]
xz
px
xz D xdx
K x
t z
x
z
x
z e
x
t
D
l
lp l a
l -
¥
¢+
++ + +
+
=
+ò
l