Rods, Rotations, Gels
Soft-Matter Experiment and
Theory from Penn.
2/23/06
Texas Colloquium
The Penn Team
• Experiment
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• Theory
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TCL
Randy Kamien
Andy Lau
Fangfu Ye
Ranjan
Mukhopadhyay
– David Lacoste
– Leo Radzihovsky
– X.J. Xing
2/23/06
Texas Colloquium
Arjun Yodh
Dennis Discher
Jerry Gollub
Mohammad Islam
Ahmed Alsayed
Zvonimir Dogic
Jian Zhang
Mauricio Nobili
Yilong Han
Paul Dalhaimer
Paul Janmey
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
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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
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Rods
PMMA Ellipsoid
fd Virus
1 mm
900 nm
~1 nm
Carbon Nanotube
100 nm – 10,000 nm
2/23/06
Texas Colloquium
Einstein – Brownian Motion
1. “On the Movement of Small Particles Suspended in
Stationary Liquid Required by the Molecular Kinetic
Theory of Heat”, Annalen der Physik 17, 549 (1905);
x&= Gf ;
G = (6pha )- 1
2
(D x ) = 2Dt ; D = kBT G
a = particle radius; h = viscosity; f = force
2/23/06
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Langevin Oscillator Dynamics
p&= - kx - gv + V
p2
1 2
H =
+ kx
2m 2
p = mv = mx&
V(t )V(t ') = 2gT d(t - t ')
g = 6pha
Ignore inertial terms:
x&= - Gkx + x;
G = g- 1
x
2
x(t )x(t ') = 2T Gd(t - t ')
| x ( w) |
2
2T
Gk
=
k w2 + G2k 2
P. Langevin, Comptes Rendues 146, 530 (1908)
Uhlenbeck and Ornstein, Phys. Rev. 36, 823 (1930)
2/23/06
dw
kBT
2
(
)
= ò
|x w | =
2p
k
Dynamics must retrieve
equilibrium static
fluctuations: sets scale of
noise fluctuations to be
2TG
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Diffusion with no potential
k ® 0:
x&= x
Gaussian probability
distribution
[x (t ) - x (0)]2
P (x , t ) =
d w 2kBT G
- i wt
= ò
(1 - e )
2
æ x2 ö
p w
1
÷
exp çç÷
çè 4Dt ÷
= 2kBT Gt = 2Dt
4p Dt
ø
kBT
1 RT
D=
=
6pha
N AVG 6pha
Measurement of D gives
Avogrado’s number
R = Gas constant
2/23/06
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Density Diffusion
n (x , t ) =
å
d (x - x a (t ))
a
x&a = xa ;
x a (t + D t ) = x a (t ) +
å
n (x , t + D t ) =
ò
t + Dt
xa (t ')dt '
t
d (x - x a (t + D t ))
a
=
å
d (x - x a (t )) - ¶ x
å
a
1
+ ¶ 2x
2
x&a d (x - x a (t ))
a
å
a
d (x - x a (t )) ò
t + Dt
t
¶ t n (x , t ) = - ¶ x nv + D ¶ x2 n (x , t )
¶ t n ( r, t ) = - Ñ ×n v + D Ñ 2n ( r, t )
2/23/06
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ò
t
t + Dt
xa (t 1 )xa (t 2 ) dt 1dt 2
J. Perrin Expts. (1908)
n (r , t ) =
æ r2 ö
N0
÷
ççexp
÷
3/ 2
ç
è 4Dt ÷
ø
(4p Dt )
NAVG=7.05 x 1023
2/23/06
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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.
3 - 1
&
q = G t ; G = (8pha )
q
q
(D q)2 = 2Dqt ;
Dq = kBT Gq
t = torque
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
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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)
2/23/06
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Diffusion of a rod
Han, Alsayed,Nobili,Zhang,TCL,Yodh
2/23/06
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Defining trajectories
Can extract lab- and
body-frame
displacements,
trajectories at fixed initial
angle and averaged over
initial angles.
2/23/06
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Rotational Langevin (2d)
¶H
&
&
Iq = - Gqq&+ Gq xq = p&q
¶q
¶H
&
q = - Gq
+ xq ® q&= xq
¶q
xq (t )xq (t ') = 2k BT Gqd(t - t ')
2
[q(t ) - q(0) ] º [D q] = 2Dqt
Dq =
2
[D q ]2
2t
2
cos[n D q ] = exp[- n Dqt ]
2/23/06
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Translation and Rotation
Anisotropic friction coefficients x&
P = - Ga fP;
x&^ = - Gb f^
Lab-frame equations
¶H
x&i = - Gij (q)
+ xi
¶xj
Gij (q) = Gan i (q)n j (q) + Gb[dij - n i ( q)n j ( q)]
xi (t )xj (t ')
q0
= 2k BT Gij ( q(t ))d(t - t ')
In lab-frame, noise depends on
angle: expect
. anisotropic crossover
2/23/06
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F. Perrin, J. de Phys.
et Rad. V, 497 (1934);
VII, 1 (1936).
Body-frame equations
ædx%ö
çç ÷
÷=
ççdy%÷
è ø÷
æ cos q sin q öædx ö
÷
çç
çç ÷
÷
÷
÷
çç- sin q cos q÷
ç
÷
d
y
è
øèç ø÷
x%i (t ) =
å
dx%ni
n
¶ t x%i = x%
;
i
Body frame : x%and y%are
are independent; simple
Langevin equations with
æGa
% = çç
G
ij
çç 0
è
2
ö
0÷
÷
÷
Gb ÷
ø
[D x%] = 2Dat ;
constant diffusion.
2/23/06
%(t ') = 2k T G
%
x%
(
t
)
x
i
i
ij
B
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2
[D y%] = 2Dbt
Anisotropic Crossover
D = (Da + Db ) / 2
1
é
ù
Dij (t , q0 ) =
[D x i (t )]ëD x j (t )û q
0
2t
D D = Da - Db
æcos 2q0 sin 2q0 ö
DD
- nDqt
÷
ç
÷
(1
e
)
= D dij +
t 4 (t ) çç
÷
÷ t n (t ) =
2t
çèsin 2q0 - cos 2q0 ø
nD
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
1
Dij =
d q0Dij (t , q0 ) = D dij
ò
2p
2
[D x%] = 2Dat ;
2/23/06
2
[D y%] = 2Dbt
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q
Lab- and body-frame diffusion
2/23/06
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Gaussian Body Frame Statistics
¶ t x%i = x%
;
i
%(t ') = 2k T G
%
x%
(
t
)
x
i
i
ij
B
2/23/06
P (x%i ) =
Texas Colloquium
æ x%2 ö
1
i ÷
÷
exp ççç÷
çè 4Dit ÷
4 p Di t
ø
Non-Gaussian lab-frame statistics
C
( 4)
q0
4
2 2
= [D x (t ) ] - 3 [D x (t ) ]
=
1
2
p(t ) =
+ ( t q t 4 (t ) - t q t 16 (t ) - 3t (t )) cos 4q0 ]
0
2 2
3 [D x (t ) ]
2
(D D ) [3( t qt - t q t 4 (t ) - t 42 (t ))
2
4
C q( 4) (t )
2
®
(Da - Db )
2
2 (Da + Db )
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
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Non-Gaussian Distribution.
Probability distribution
at fixed angle and
small t is Gaussian.
The average over
initial angles is not
2/23/06
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Rattleback gas
Tsai, J.-C., Ye, Fangfu , Rodriguez, Juan , Gollub,
J.P., and Lubensky, T.C., A Chiral Granular Gas,
Phys. Rev. Lett. 94, 214301 (2005).
Chiral Rattlebacks
spin in a preferred
direction; Achiral
ones do not.
Chiral wires spin in a preferred direction on
a vibrating substrate
2/23/06
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Rattleback gas II
2/23/06
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Spin angular momentum dynamics
l = nI W= Spin angular momentum
W= Spin angular frequency
l( x, t ) =
å
pqa d (x - x a (t ))
a
&= I q&
¶H
- Gqq&+ Gq xq = p&q
¶q
å
¶ t l = - n GqW+ ¶ i ¶ j
Dij (qa )pqa d( x - x a (t )
a
= - n GqW+ D Ñ 2l
= - ¶ j (lv j ) - GW+ DWÑ 2W
2/23/06
Texas Colloquium
Rattleback gas III
l = I W= Spin angular momentum
gi = r vi = Center-of-mass mometum
W= Spin angular frequency
w = (Ñ ´ v)z / 2 = CM angular frequency
¶ tl = - ¶ j (lv j ) - GWW- G(W- w) + DWÑ 2W+ t
Substrate friction
Spin-vorticity coupling
¶ t gi = - ¶ j (giv j ) - ¶ i p + hÑ vi - G vi +
2
B.C.: W(R ) = w(R ) = v(R ) / R ;
2/23/06
v
¶ r v = - l - 1v
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Vibrational torque
1
2
eij ¶ j G(W- w)
Nematic phase
Order
Qij = S (n in j Parameter
F =
1
2
ò d x {K
1
(Ñ ×n )
Twist
+ K 2 [n ×(Ñ ´ n )]
2
+ K 3 [n ´ (Ñ ´ n )]
}
Frank free energy
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1
3
dij
n  (  n)
Bend
n  (  n)
2
2/23/06
dij ) = ni n j -
Splay   n
2
3
1
3
n =Frank director
Isotropic-to-Nematic Transition
increasing
concentration
(q)orientational distribution
functions
D - rod diameter
L – rod length
I N - rod concentration
f I- N
2/23/06
order parameter S :
at I - N phase transition
D
L
= 4
when
? 1
L
D
S =
1
2
2
dq
sin
q
f
(
q
)(3
cos
q - 1)
ò
Onsager Ann. N. Y. Acad. Sci. 51, 627 (1949)
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Semi-flexible biopolymers
DNA
Neurofilament
16 micron length
2 nm in diameter
40 nm persistence length
5 - 20 micron length
12 nm in diameter
~ 220 nm persistence length
Wormlike Micelle
Actin
( polybutadiene-polyethyleneoxide )
10 – 50 micron length
~ 15 nm in diameter
~ 500 nm persistence length
2/23/06
2 – 30 micron length
7-8 nm in diameter
~ 16 micron persistence length
Texas Colloquium
Semi-flexible Polymer in Aligning
Field
h= kB T G/ 
t(s)
R(s)
R
dR
= t(s )
ds
2/23/06
| t(s ) |= 1;
t(s ) = ( t ^ (s ), 1- | t ^ (s ) |2 )
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Fluctuations
é ædt ö2
ù
2
ê
ú
÷
ç
H =
ds
l
G
|
t
|
z
÷
ú
2 ò êëP çèds ø
û
2
é
ù
kBT
ædt ^ ö
2
êl ç ÷
ú
»
ds
+
G
|
t
|
÷
^
ú
ø
2 ò êëP ççè ds ÷
û
kBT
lp = k / kBT = Persistence length
G = h / k BT = Alignment parameter
l =
lp
G
= Odijk deflection length
G=0
át(s ) ×t(0)ñ= e
2/23/06
- s / 2lp
Texas Colloquium
t x (z )t x (0) =
G>0
l - |z |/ l
e
2lp
Polymers in fd-virus suspensions
Isotropic
Nematic
Actin in Nematic Fd
Actin
16 mm
Wormlike
Micelle
500 nm
Neurofilament
200 nm
DNA
50 nm
Hairpin
defects
10 mm
2/23/06
10 mm
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
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Tangent-tangent correlations
t(s’)
t(s’+s)
h(s)
actin in nematic phase
Isotropic phase – quasi 2D
át(s ) ×t(0)ñ= e
- s / 2lp
Orientational correlations
decay exponentially
lp – persistence length of
actin in isotropic phase
2/23/06
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Polymer in nematic solvent
Bending energy
F/T =
lp
2
ò
L
0
ædt ^
dz çç
çè dz
Coupling energy
Elastic energy
2
ö G L
2
1
2
3
÷
é
ù
+ ò dz ët ^ (z ) - dn(0, z )û + K ò d x (Ñ n )
÷
÷
ø
2 0
2
lp = Persistence length
G = Coupling constant
l =
lP
= Odijk length
G
K = Elastic constant
l - z/ l
t x (z )t x (z + z ¢) =
e
2l p
1
+
4p 2K l
2/23/06
¥
cos(xz / l ) log(1 + D 2 / x 2 )
ò dx (1 + x 2 )[1 + x 2 + a x 2 log(1 + D 2 / x 2 )]
0
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