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Soft Motors:molecular to macroscopic
perspectives
Peter Palffy-Muhoray Liquid Crystal Institute Kent State University
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Outline: microscopic• What is a Motor?
• Molecular Motors 1. Anomalous Photoalignment– ideas of Landauer, Prost & Astumian
• Molecular Motors 2. Yokoyama - Tabe Experiment– Lehman effect
• Molecular Motors 3. ATP synthase– Boyer & Walker
• Molecular Motors 4. Myosin and Actin– muscles
• Molecular Motors 5. LC elastomers– elongation/contraction
• Broken symmetry– broken symmetry drives motors
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Outline: macroscopic
• molecular & macroscopic length scales
• shape change
• motors based on elongation
• motors based on bend
• summary
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what is a motor?
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• what is a motor?– anything that produces or imparts motion (dictionary.com)
– a machine that supplies motive power for a vehicle or other device (O.E.D.)
Motor
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Motor• what is a motor?
– a device which uses energy (but not momentum!) to cause motion
• how does motion come about?• car convinces the road to exert a force on it, and push it
forward
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Motor• what is a motor?
– a device which uses energy (but not momentum!) to cause motion
• motor causes one part of a system to exert a force/torque on another, causing motion
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• what is a motor?
– a device which uses energy (but not momentum!) to cause motion
• motor causes one part of a system to exert a force/torque on another, causing motion
Motor
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Molecular Motors 1.Anomalous Photoalignment
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Direct optical torque
laserray
table
_hp = λ τ = ×optvol D E
angular momentum transfer:
torque results fromchange in extrinsicangular momentumof light.
τ τ+ = 0el optvol vol
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Indirect optical torque
⇒ ×1
addition of 1% of anthroquinone dye reduction of threshold intensity by ~ 100!
laser
- dye
τ τ
τ τ
+ =
+ =+1100
without dye: 0with dye: ?? 0
el optvol vol
el optvol el
Torque on nematic causing elastic deformationCANNOT come from light! Source of torque??
1. I. Janossy, A.DD. Lloyd and B.S. Wherret, Mol. Cryst.Liq. Cryst. 179, 1, 1990.I. Janossy, Phys. Rev. E 49, 2957, 1994.
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• light causes the director to reorient, against a restoring elastic torque, essentially without the transfer of angular momentum.
• light causes rotation without exerting a torque!
Puzzle 1:
???
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Prost, Astumian
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The translational ratchet*
U
xU=0
xU
x
* R.D. Astumian et al: Phys.Rev.Lett.72, 1766, 1994. J. Prost et al: Phys. Rev. Lett. 72, 2652, 1994.
Particles in asymmetric periodic potential:
When potential is turned OFF, particles diffuse, no net current
When potential is turned ON,particles move towards minimum,giving rise to current.
J
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Simple model:• dye
• anisotropic molecule with orientation
• in ground state does not interact with nematic
• in excited state becomes 'nematic' molecule
• probability of excitation depends on
• lifetime of excited state is
• optical field• transfers energy, but no momentum
to the sample
dl
2ˆ( )dl Eot
k
E
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Torque on liquid crystal:
n
ld^
n
ld^
Average orientation of excited dye molecule is NOT along nematic director
interaction between liquid crystal and excited dye gives rise to torque!
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Rotating dye gives rise to shear flow
n
ld^
flow
n
ld^
flow
dye: - rotation source of vorticityliquid crystal: - aligned by nematic field of dye
⇒
viscous shear carries angular momentum from cell walls to dye
angular momentum current: dye nematic cell walls dyemolecular elastic shear
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Summary of Puzzle 1
– Janossy effect1 (source provides energy but not torque yet causes rotation) ⇒ orientational ratchet2
– dye is light-driven rotor with continuous rotation
– energy from optical field drives shear flow
– viscous stress carries angular momentum from dye to cell walls
1. T. Kosa and I. Janossy, Opt.Lett. 17, 1183, 1992; Opt. Lett. 20, 1231, 1995. 2. W. E and P. Palffy-Muhoray, Mol.Cryst.Liq.Cryst. 320, 193, 1998
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Puzzle 2. Photoinduced Twist
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Photoinduced Twist
laser OFF
nematic
photosensitive
alignment layer
strong anchoring
dye
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Photoinduced Twist
laser ON
nematic
photosensitivealignment layer
strong anchoring
E
polarized
light
dye
θ
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Origin of torque
• before deformation takes place, polarization is parallel to the director, hence optical torque D x E = 0.
• optical field stabilizes configuration!
• source of torque?
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Landauer
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Landauer’s Blowtorch
x
T(x)x
U(x)
R. Landauer, J. Stat. Phys. 53, 233 (1988)
particles in periodic potential
‘hot’ molecules are more excited & diffuse over barrier
steady current: system is molecular motor (pump) ∴
one part of system pushes on other;no momentum transfer from outside.
( )U x
( )T xx
x
M. Büttiker, Z. Phys. B 68, 161 (1987).
J
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Indirect optical torque: Brownian ratchet
x
T(x)x
U(x)dye molecules in nematic field
dye molecules parallel to pump polarization are excited & diffuse over barrier
steady rotation: dye is light driven molecular motor ∴
torque results from rotation of dye;no momentum transfer from light.
P. Palffy-Muhoray, T. Kosa and Weinan E, Appl. Phys. A 75, 294 (2002)
( )U θ
( )I θθ
θ
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Schematic of Brownian motor:
dye in trans-state starts to rotate towards director
it exerts torque on director & vice versa
as it becomes parallel to pump polarization
it is excited into cis-state, undergoes diffusion
relaxes into trans-state, rotates towards director...
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Dynamics• dye: Fokker-Planck
• transition rates
t
c
2t
2c
2U / 2
2U / 2
( )t
( )t
ˆ ˆ = e ( )ˆ ˆ = e ( )
t t t t t t t c c
c c c c c c c t t
kTt to t d
kTc co c d
D M U f f
D M U f f
f f e
f f e
ρ ρ ρ ρ ρ
ρ ρ ρ ρ ρ
ν
ν
∂= ∇ +∇⋅ ∇ − +
∂∂
= ∇ +∇⋅ ∇ − +∂
+ ⋅
+ ⋅
l E
l E
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Dynamics• liquid crystal
– bulk:
– surface:
2ˆ ˆ ˆ ˆ=K ( ) It
γ ∂ ∇ −∂n n nn
t c
ˆ ˆ ˆˆ ˆ={K( )
ˆ ˆ < > < >}( ) ˆ ˆ
s
t cd d
tU Ud d I
γ
ρ ρ
∂∇ ⋅ + ×∇×
∂∂ ∂
− − −∂ ∂
n N n N n
nnn n
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Simulations
• optical field drives orientational current
• dye exerts steady torque on nematic
0.0
0.5
1.0di
rect
or a
ngle
(rad
ians
)
0.0 5000.0 10000.0time
Field is turned off
0.0
0.5
1.0di
rect
or a
ngle
(rad
ians
)
0.0 5000.0 10000.0time
Field is turned off
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Summary of Anomalous Photoalignment
• light-driven molecular motors
• dye* molecules are rotors in nematic potential
• energy (but not momentum) from light
• steady unidirectional rotation
• orientational ratchet / Landauer’s blowtorch mechanism
• work: steady torque on director by viscous stress + dissipation
• achiral
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Molecular Motors 2. Yokoyama - Tabe Experiment
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• chiral Langmuir monolayer
• substrate: glycerol + H2O~3 nm
Yokoyama – Tabe Experiment
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Collective Molecular Rotor/Pump
Yuka Tabe and H. Yokoyama, Nature Materials, 2, 806(2003).
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Chiral LC molecules on glycerol
FELIX013 on pure glycerol (SmC* phase)
clockwise rotationCS4001 on pure
glycerol (SmCA* phase)
counter-clockwise rotation
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Chirality inversion
100µm
(S)-OPOB
(R)-OPOBon pure glycerol
on pure glycerol
CW
CCW
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Precession speed linearly depends on ∆PW/P0
∆Pw/P0=Pv-Ps
Pv: actual water vapor pressure
Ps: saturated water vapor pressure
(R)-OPOB(S)-OPOBCW
CCW
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Onsager Reciprocal Relations
υm: molecular volume of water in vaporγ: rotational viscosity of LC directorη: water mobility through the filmb: Lehmann coefficient (∝ chirality strength)
PJt
ST m∆⋅+∂∂
=•
υφτOnsager Relations:
Entropy Production:
P b dtd
m ∆+= υτγ
φ 1
PbJ m∆+= υη
τ 1
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zc
cc
∂∂
×+−=∂∂ PF
tν
δδγ
∫ ∑
+−∇=
= yxii
ucKdrF,
422 ||4
||2
||2
ccτLehman term
Shibata & Mikhailov, Europhys. Lett., 73 (2006)H. Brand et al., Phys. Rev. Lett. 96 (2006) Tsori & de Gennes, Euro.Phys. J. E 14 (2004)T. Okuzono, Kyoto Workshop (2005)
Simulated by T. Okuzono
Phenomenological Theory of Lehmann effect
Equation of motion :
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Summary of Yokoyama – Tabe experiment• substrate & monolayer system is motor
• energy stored in chemical potential of H2O
• vapor current drives rotation of chiral molecules
• work: produces circulating flow, dissipation
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Molecular Motors 3.ATP Synthase
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F1FO ATP Synthase
• enzyme that synthesizes adenosine-5'-triphosphate (ATP)
from
adenosine diphosphate (ADP) + phosphate
• reaction:
ADP + Pi → ATP
• ATP:"molecular unit of currency" of intracellular energy transfer
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F1FO ATP Synthase• energy from glucose sets up
H+ gradient• H+ current drives motor• stator opens & closes,
producing ATP
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F1FO ATP Synthase
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F1FO ATP Synthase• normal human produces
~100lbs of APT/day
• motor can run backwards
Boyer & Walker, Nobel Prize in Chemistry, 1997
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Why does rotor turn?• sequential protonation &
deprotonation of side groupscoupled with rotation
• chemical potential differenceof H+ drives current, whichdrags surface of C-rotor.
• pot. diff. →current →
rotation.A. Aksimentiev et al.Biophysical Journal 86 ,1332 (2004)
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Modeling
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A. Aksimentiev et al.Biophysical Journal 86 ,1332 (2004)
• system of Langevin equations
• Lennard-Jones + hydrophobic +screened Coulomb
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• proposed sequence
based on simulations
Modeling
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A. Aksimentiev et al. Biophysical Journal 86 ,1332 (2004)
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Summary of ATP Synthase• rotary molecular motor
• rotation driven by proton current
• energy from stored electrochemical potential
• work: creation of ATP + dissipation
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Molecular Motors 4.Actin & Myosin
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Biological transport• all organisms (eukaryotic cells of yeasts, plants, animals) contain
‘motor proteins’
• two well known examples:
– kinesins• two active heads• hydrolyses ATP• moves along microtubules
– myosins• two active heads• hydrolyses ATP• moves along actin fibers
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Muscle force and movement
F=5.3pN /cross-bridgestep-size=5.5nm1 step/ATPase reaction
myosin filament length=1.55µm
C.J. Pennycuick, Newton Rules Biology (Oxford, 1995)
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Summary of Actin &Myosin
• processive motor (unidirectional translation)
• hand-over-hand motion
• driven by chemical energy of ATP
• work: translation of cargo against force, dissipation
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Molecular Motors 5.LC elastomers
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director
If orientational order increases,
expansion contraction
If orientational order decreases,
expansioncontraction
LC elastomer network changes shape
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Summary of LC elastomers
• elongational motor:
– change in orientational order produces stress
– stress produces change in shape
• driven by heat, light, or impurities
• work: motion against force, dissipation
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Broken Symmetry
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Proper- and pseudo-tensors
tensorrank
0 1 2 3 4
proper
pseudo
π x
( )×A B C
E DP, , ,
, ,
,,
H M B
αβδ Qαβ
αβΓ
αβεαβµ
ijkε
dαβγ
scalar vector
etc.
changes sign on inversion
does not change sign on inversion handedness =pseudoscalar c*!
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Symmetry and motors
• symmetry of the system determines motion of motor!
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Translational motion
My rugs are alive and moving!?
I have wall to wall carpets in my apartment. I have placed large rugs on top of the carpet. Without damaging the carpet or the rug, how can I stop them from wrinkling and moving? Is it caused by static electricity?
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Translational motion
My rugs are alive and moving!?
I have wall to wall carpets in my apartment. I have placed large rugs on top of the carpet. Without damaging the carpet or the rug, how can I stop them from wrinkling and moving? Is it caused by static electricity?
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tilted strands
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Translation
My rugs are alive and moving!?
I have wall to wall carpets in my apartment. I have placed large rugs on top of the carpet. Without damaging the carpet or the rug, how can I stop them from wrinkling and moving? Is it caused by static electricity?
7/27/2011 62
tilted strands
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Translation
My rugs are alive and moving!?
I have wall to wall carpets in my apartment. I have placed large rugs on top of the carpet. Without damaging the carpet or the rug, how can I stop them from wrinkling and moving? Is it caused by static electricity?
7/27/2011 63
tilted strands
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Translation
My rugs are alive and moving!?
I have wall to wall carpets in my apartment. I have placed large rugs on top of the carpet. Without damaging the carpet or the rug, how can I stop them from wrinkling and moving? Is it caused by static electricity?
7/27/2011 64
tilted strands
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Translation
My rugs are alive and moving!?
I have wall to wall carpets in my apartment. I have placed large rugs on top of the carpet. Without damaging the carpet or the rug, how can I stop them from wrinkling and moving? Is it caused by static electricity?
7/27/2011 65
tilted strands
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Translation
My rugs are alive and moving!?
I have wall to wall carpets in my apartment. I have placed large rugs on top of the carpet. Without damaging the carpet or the rug, how can I stop them from wrinkling and moving? Is it caused by static electricity?
7/27/2011 66
tilted strands
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Translation
My rugs are alive and moving!?
I have wall to wall carpets in my apartment. I have placed large rugs on top of the carpet. Without damaging the carpet or the rug, how can I stop them from wrinkling and moving? Is it caused by static electricity?
7/27/2011 67
tilted strands
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Symmetry and motors
• symmetry of the system determines motion of motor!
• broken left-right symmetry: – proper vector exists in system – translation in direction of vector (carpet, kinesin, myosin)
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Symmetry and motors• symmetry of the system determines motion of motor!
• broken translational symmetry: – proper vector exists in system – translation in direction of vector (carpet, kinesin, myosin)[Q: is there a vector in Landauer's blow torch?]
• broken chiral symmetry – pseudoscalar c* exists in system
• broken chiral + broken translational symmetry– pseudoscalar + proper vector = pseudovector– rotation (Yokoyama – Tabe, ATP Synthase, etc.)
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Symmetry and motors• if system has proper dyad (director)
– strain (elongation)
• if system has proper dyad and gradient (vector), – proper vector (curvature, bend)
• if system has proper dyad, gradient & pseudoscalar– pseudovector (twist)
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Symmetry & deformation/motionSymmetry can construct deformation/motion material
vector translation in kinesin, actin
dyad uniaxial strain nematic networks
c* - -
c* pseudovector rotation Lehman, ATP, Tabe
c* pseudovector twist LCE twist
proper vector bend LCE photoactuation
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Connecting molecular & macrosopic scales• cumulative effect of many molecular motors →
macroscopic response– rotating dye system, Yokoyama-Tabe expt:
• distributed local vorticity → bulk torque & bulk flow
– ATP synthase:• produces bulk ATP
– myosin & actin• muscle fibers form long muscles, these contract
– effective shape change• contraction, elongation and bend and twist
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Smooth shape change: elongation
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• liquid crystal elastomers:– coupling of orientational order and strain– large mechanical response to excitations
• elongation due to temperature change
• motor based on elongation
H. Finkelmann, nematic LCE
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Another motor based on elongation • rubber band heat engine
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Another motor based on elongation
• collagen contracts in salt water
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Aharon Katzir-Katchalsky 1913-1972
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Another motor based on elongation
• a modified version
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M. V. Sussman and A. Katchalsky, Science, 167, 45 (1970)
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Dunking bird of the first kind• heat engine
– Carnot efficiency limit
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Dunking bird of the second kind• not a heat engine
• isothermal
• involves shape change
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N. Abraham, P. Palffy-Muhoray., Am.J. Phys. 72, 782 (2004)
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Bend Motors• samples of liquid crystal elastomers with azo dye
bend on exposure to light
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Yanlei Yu, Makoto Nakano, Tomiki Ikeda, Nature 425, 125 (2003)
timescale: 10 s
LC + diacrylate network+ functionalizedazo-chromophore
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Water vapor sensitive network
K. Harris, C. Bastiaansen, D. Broer, J. MEM Syst., 16, 480 (2007)
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Bend Motors
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sample: nematic elastomer EC4OCH3+ 0.1% dissolvedDisperse Orange 1 azo dye 5mm
300 mµ
Response time: 70ms
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• if sample bends both sides are illuminated, producing oscillations
Photoinduced oscillations
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nematic azo-elastomer
S. Serak, N. Tabiryan, R. Vergara, T. White, R. Vaia, T. Bunning, Soft Matter 6, 779–783 (2010)
90o>
max. frequency = 270Hz
sample size: 5 mm 0.8 mm 0.05 mm
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Bend Motors
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• floating nematic LCE sample illuminated from above
Laser beam
container
waterElastomer
M. Chamacho-Lopez, H. Finkelmann, P. Palffy-Muhoray, M. Shelley, Nature Mat. 3, 307, (2004)
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Bend Motors
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• robotic arm
CLCP Layers (Homogeneous)
CLCP Layers(Homeotropic) PE Film
T. Ikeda, Chuo University (priv. comm.)
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Bend Motor: focus of current interest
• azo-dye doped liquid crystal elastomer
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M. Yamada, M. Kondo, J. Mamiya, Y. Yu, M. Kinoshita, C. Barrett, T. Ikeda, Angew. Chem. 47, 4986 (2008)
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Bend Motor: focus of current interest• shape-memory Ni-Ti alloy,
– two phases:
• orthorhombic martensite (cold)
• cubic austenite (hot)
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Existing rotary bend motors
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LCE bends on exposure to light
wire becomes stiff when heated
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• bend vs. compression:– if ends are brought closer, will filament compress or bend?
Model
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d R
L
L L−∆
21 ( )24
L LL R∆
21/ ( )2 12b
d wdl ER
= 21/ ( )2c
Ll E wdL∆
=2 2~ ( ) ( )c
b
L Ld R
Bend is much cheaper energetically than stretch;→ length is constant.
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Model
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1s
3s
4s
2s
xsos
2θ
3θ1θ4θ
2R1R
arbitrary shape
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Model
– arc length along filament
– position vector of point on filament
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1s
3s
4s
2s
xsos
2θ
3θ1θ4θ
2R1R
s
( )sR z
x
y
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Model
– bend energy density:
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2 21 1/2b
cl EIR R
= =
1 2
1
3 4
2 3 4
22 2
21
22 2 2
22 1
1 ˆ( ) ( ( ))
1 1ˆ( ) ( ( ) ( ) ,
o
o
s s
s s
s s s
s s s
c ds c dsR s s
c ds c ds c dsR s s R
κ κ
κ κ κ
∂ ∂= − + − × +
∂ ∂
∂ ∂− + − × + −
∂ ∂
∫ ∫
∫ ∫ ∫
R R z
R R z
1s
3s
4s
2s
xsos
2θ
3θ1θ4θ
2R1R
stiffness natural curvaturefn. of positionfn. of position
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Model
– constraints:
• filament length is constant
• displacement in x-dir. is fixed
• displacement in y-dir. is fixed
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1s
3s
4s
2s
xsos
2θ
3θ1θ4θ
2R1R
2
11 1 2 2 1 2
ˆ ˆ(1 cos ) (1 cos ) ,s
sR ds R L R Rθ θ− − ⋅ + − = + +∫ t x
4
31 4 2 3 1 2
ˆ ˆ(1 cos ) (1 cos ) .s
sR ds R L R Rθ θ− + ⋅ + − = + +∫ t x
2
11 1 2 2
ˆ ˆsin sin 0,s
sR ds Rθ θ+ ⋅ − =∫ t y
4
31 4 2 3
ˆ ˆsin sin 0s
sR ds Rθ θ+ ⋅ − =∫ t y
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Model: Energy with constraints
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1 2 3
1 2
4
3 4
2
1
2
1
2 2 2
01 2
12 2
1
211 1 2 1 2
1 2
211 1 2
1 2
1 1ˆ ˆ ˆ( ) ( ( )) ( )
1ˆ ˆ ˆ( ( )) ( )
ˆ ˆ{ (1 cos ) (1 cos( )) ( )}
ˆ ˆ{ sin sin(
s s s
s s
s
s s
s xx s
s xy s
c ds c ds c dsR R
c ds c dsR
s ssf R ds R L R RR R
s ssf R ds RR R
κ κ κ
κ κ
′
′
= − + − + −
+ − + −
−+ − − ⋅ + − − + +
−+ + ⋅ −
∫ ∫ ∫
∫ ∫
∫
∫
t t z
t t z
t x
t y
×
×
4
3
4
3
342 1 2 1 2
1 2
342 1 2
1 2
)}
1 ˆ ˆ{ (1 cos( )) (1 cos( )) ( )}
1 ˆ ˆ{ sin( ) sin( )}
s xx s
s xy s
s ssf R ds R L R RR R
s ssf R ds RR R
−−+ − + ⋅ + − − + +
−−+ + ⋅ −
∫
∫
t x
t y
1s
3s
4s
2s
xsos
2θ
3θ1θ4θ
2R1R
Lagrange multipliers are components of forces in top and bottom filaments.
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Model: Energy minimization
• ODEs are obtained by minimizing w.r.t.
• these govern the shape of the filaments on top and bottom
– BC for is obtained by minimizing w.r.t.
– this gives continuous tangent and curvature at contact pts.
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ˆ (cos ,sin )θ θ=t
1 1 1 22( ( ) ) sin cos 0, if ( , )x yc c c f f s s sθ θ κ θ θ′′ ′ ′ ′− + + + + = ∈
1s
3s
4s
2s
xsos
2θ
3θ1θ4θ
2R1R
2 2 3 42( ( ) ) sin cos 0, if ( , ).x yc c c f f s s sθ θ κ θ θ′′ ′ ′ ′− + + − + = ∈
θ 1 2 3 4, , ,s s s s
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Numerical Solution1. Assign initial values for and and solve ODE
with BCs.
2. Assign initial values for and and solve ODE
with BCs.
3. Vary the forces to satisfy constraints (root finding)
4. Vary and go to 1. until
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1 2, , xs s s 1 1,x yf f
1 12( ( ) ) sin cos 0x yc c c f fθ θ κ θ θ′′ ′ ′ ′− + + + + =
3 4,s s 2 2,x yf f
2 22( ( ) ) sin cos 0x yc c c f fθ θ κ θ θ′′ ′ ′ ′− + + − + =
1 1 2 2, , ,x y x yf f f f
1 2 3 4, , , , xs s s s s 0, 1,...4, .i
i xs∂
= =∂
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Numerical Solution: varying the curvature
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local spontaneous curvature
κ
s
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Numerical Solution: varying the stiffness
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local stiffening
E
s
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Calculating torques
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– to calculate the net torque, we consider part of system
– look at forces
– look at point torques
1xf1yf
2 yf
2xf
1τ
2τ
ˆ ˆ ˆ2 ( ( ))cτ κ′= −t t z×
ODE is precisely the condition that the torque on wheelremain constant, regardless of where the filament is cut!
1 1 10 02 ( ) sin ( ) cos ( ) , (upper part)
s s
x yc f s ds f s ds Kθ κ θ θ′ + − − =∫ ∫
Note: integrating the ODE w.r.t. to s gives
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What happens on the pulleys?
• Problem: elastic beam on rigid cylinder
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Interesting observation– straight beam touches cylinder along one line
– curved beam touches cylinder over extended area
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F F Transition?
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Experiment
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Determination of point of contact
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Data
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0 10 20 30 40 50
-80
-60
-40
-20
0
20
40
60
80Co
ntac
t Ang
le (d
eg.)
Force (N)
Contact angle vs. total force
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Modelling: as before
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NOT a pitchfork bifurcation!
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Force distribution on pulleys
• when force is applied to filament,
– point contact if – extended contact if
• torque balance:
• Volterra integral equation of first kind.
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force density on pulley
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Force distribution on pulleys
• When force is applied to filament,
– point contact if – extended contact if
• Solution for force density:
where
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F
cF F≤
cF F> cEIFLR
=
( ) ( ( ) cos sin )c c cFfR
θ δ θ θ θ θ= − +
sin ( )cc
F F LF R
θ −=
R
F Fθ
L
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Force distribution
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How does rotation come about?
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How does rotation come about?
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• away from the small wheel, – longer lever arm on one side
• near small wheel– greater point torque on one side
• curvature is the same, – but stiffer on one side
– have pontaneous curvature on one side
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How does rotation come about?
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• away from the small wheel, – longer lever arm on one side
• near small wheel– greater point torque on one side
• curvature is the same, – but stiffer on one side
– have pontaneous curvature on one side
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Motion due to point torques• how can a point torque in filament cause translation?
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Motion due to point torques• how can a point torque in filament cause bulk translation?
– net force is zero,
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1F
2F
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Motion due to point torques• how can a point torque in filament cause translation?
– net force is zero, but torque causes constraint force to appear
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1F
2F
3F
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Motion due to point torques• how can a point torque in filament cause translation?
– net force is zero, but torque causes constraint force to appear
– net force along
– translation
• angular momentum transport!
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1F
2F
3F
1F
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Motion due to point torques• how can a point torque in filament cause translation?
– net force is zero, but torque causes constraint force to appear
– net force along
– translation
• no rotation without friction between filament & wheel!
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1F
2F
3F
1F
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• point torque in filament causes constraint force F to appear*
• reaction force -F appears;exerts torque on hand
• angular momentum currentflows into bend motor from hand
• * via stress transport:
Angular momentum current
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v
-F2
2( )E tρ ∂
∇ ∇ =∂σσ
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Summary• detailed model for bend motors driven by
– local spontaneous curvature caused by light (LCE-Ikeda)– local stiffening caused by heat (Nitinol - Wang)
• corrects/completes existing qualitative explanations
• prediction: – LCE motor with perp. alignment runs the other way:
confirmed!
• understand angular momentum transport
• efficiency calculation: remains to be done
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Angular momentum
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NN
stimulus is applied
filament bends
torque appears
angular momentum flows into motor
rotation starts
angular momentum is conserved &earth speeds up
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Conclusions• variety of molecular -scale mechanisms can be exploited
– rotary (dyes, Yokoyama-Tabe)– need better molec. motors with processive translation – elongation (LC elastomers/networks)
• symmetry plays important role in determining motion
• macroscopic phenomena are cumulative effects of molecular motors
• understand bend motors well
• many exciting possibilties to explore!
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