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Soft Active Materials
Zhigang SuoHarvard University
James F. Bell1914-1995
The James F. Bell Memorial Lecture in Continuum MechanicsDepartment of Mechanical Engineering, Johns Hopkins University5 November 2009 1
2
Professors Sharp and Bell (1983)
Professor Bell (about 1990)
3
elastomer = networkgel = network + solvent
reversible
solvent
networkgel
4
Super absorbent diaper
Sodium polyacrylate: polyelectrolyte
5
Gels regulate flow in plants
Missy Holbrook
Zwieniecki, Melcher, Holbrook,Hydrogel control of xylem hydraulic resistance in plants Science 291, 1095 (2001)
6
Self-regulating fluidics
David Beebe
•pH•Salt•Temperature•light
Responsive toPhysiological variables:
•Many stimuli cause deformation.•Deformation regulates flow.
Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)
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octopus
Mäthger, Hanlon, Kuzirian – Marine Biological Laboratory, Woods Hole MA
8
Squid changes color
Expand pigmented sacs by contracting muscles
Mathger, Denton, Marshall, Hanlon, J. R. Soc. Interface 6, S149 (2009)
9
Adaptive OpticsAh, optics!
•Many stimuli cause deformation.•Deformation affects optics.
Wilbur, Jackman, Whitesides, Cheung, Lee, PrentissElastomeric opticsChem. Mater. 1996, 8, 1380-1385
Aschwanden, StemmerOptics letters 31, 2610 (2006)
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Soft Active Materials (SAM)
Soft: large deformation in response to small forces(rubbers, gels,…)
Active: large deformation in response to diverse stimuli(electric field, temperature, pH, salt,…)
A stimulus causes deformation. Deformation provides a function.
StimulipH, E, T, C…
Functionsoptics, flow…
SAMdeforms
11
Very well, but how does stimulus X cause deformation?
12
Dielectric elastomer
Reference State Current State
Φ
l
a Q+
Q−Compliant Electrode
Dielectric Elastomer
L
A
Pelrine, Kornbluh, Pei, Joseph High-speed electrically actuated elastomers with strain greater than 100%. Science 287, 836 (2000).
13
Parallel-plate capacitor P
Φ
Q+
Q−batterya
electrode
lvacuum
electrode
P force
ED 0ε=
ε0, permittivity of vacuum
202
1Eεσ = Maxwell stress
lE
Φ=
a
QD =
a
P=σ
Electric field
Electric displacement field
stress field
14
Field equations in vacuum, Maxwell (1873)
ii x
E∂Φ∂
−=0εq
x
E
i
i =∂∂
Electrostatic field
ii qEF =A field of forces maintain equilibrium of a field of charges
⎟⎠⎞
⎜⎝⎛ −
∂∂
= ijkkijj
i EEEEx
F δεε20
0
ijkkijij EEEE δεεσ20
0 −=
Q−
20
2E
εσ =
P
E
Q+ Maxwell stressP
15
James Clerk Maxwell (1831-1879)
“I have not been able to make the next step, namely, to account by mechanical considerations for these stresses in the dielectric. I therefore leave the theory at this point…”
A Treatise on Electricity & Magnetism (1873), Article 111
16
Trouble with Maxwell stress in dielectrics
+ + + + + + + + +
- - - - - - - - - - - - - - - - - - - - - - - -
+ + + + + + + + +
+-
Maxwell stressElectrostriction2
33 2Eεσ −=
±
Our complaints:
•In general, ε varies with deformation.•In general, E2 dependence has no special significance.•Wrong sign?
17
Trouble with electric force in dielectrics
In a vacuum, external force is needed to maintain equilibrium of charges
+Q +Q
P Pii qEF =
+Q +Q
In a solid dielectric,force between charges is NOT an operational concept
ii qEF =
The Feynman Lectures on PhysicsVolume II, p.10-8 (1964)
“It is a difficult matter, generally speaking, to make a unique distinction between the electrical forces and mechanical forces due to solid material itself. Fortunately, no one ever really needs to know the answer to the question proposed. He may sometimes want to know how much strain there is going to be in a solid, and that can be worked out. But it is much more complicated than the simple result we got for liquids.”
18
19
All troubles are gone if we use measurable quantities
Current StateReference State
L
A
P
l
a Q+
Q−
Φ
QlPF δδδ Φ+=equilibrate elastomer and loads
Nominal
Ll /=λ
APs /=
LE /~
Φ=
AQD /~=
aP /=σ
lE /Φ=
aQD /=
( )ALFW /=
DEsW~~δδλδ +=
LA
Q
AL
lP
AL
F δδδ Φ+=divide by volume
name quantitiesTrue
( )λλ∂
∂=
DWs
~, ( )
D
DWE ~
~,~
∂∂
=λ
equations of state
20
Fine, Zhigang, but what is ?( )DW~
,λ
21
Dielectric constant is insensitive to stretch
VHB 4910
Kofod, Sommer-Larsen, Kornbluh, PelrineJournal of Intelligent Material Systems and Structures 14, 787 (2003).
22
Ideal dielectric elastomer
( ) ( )ε
λλλλ2
,~
,,2
2121
DWDW stretch +=
Elasticity Polarization
a
QD =
A
QD =~
21
~
λλD
D =
1321 =λλλ
Dielectric behavior is liquid-like, unaffected by deformation.
For an ideal dielectric elastomer, electromechanical coupling is purely a geometric effect:
incompressibility
1
11
1λ2λ
3λ
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
23
Ideal dielectric elastomer
2P 1P
22Lλ
33Lλ
11LλΦ
Q
2P 1P
22Lλ
33Lλ
11LλΦ
Q( ) ( ) 22
21
22
22
122
2121 2
~3
2
~,, −−−− +−++= λλ
ελλλλµλλ D
DW
( )1
211
~,,
λλλ
∂∂
=DW
s( )
2
212
~,,
λλλ
∂∂
=DW
s
In terms of nominal quantities
In terms of true quantities
( ) 222
21
211 Eελλλµσ −−= −− ( ) 22
22
1222 Eελλλµσ −−= −−
24
Young man, don’t be too fast. The theory sounds fine, but can you relate the theory to experiments?
25
Two modes of failureElectromechanical instabilitySoft material
Φ
Q
l
Q
Φ
polarizing thinning
Q+
Q−l
Q
Φ
polarizing
Electrical breakdownHard material
Stark & Garton, Nature 176, 1225 (1955)
26
A dilemma
•To deform appreciably without electrical breakdown, the elastomer must be soft.
•But a soft elastomer is susceptible to electromechanical instability.
How large can deformation of actuation be?
27
Electromechanical instability limits deformation of actuation
ΦQ+
Q−
1λ
λ1
26.12 3/1 ≈=cλ
Experiment: Pelrine, Kornbluh, Joseph, Sensors & Actuators A 64, 77 (1998)
Theory: Zhao & Suo Appl. Phys Lett. 91, 061921 (2007)
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Pre-stretch increases deformation of actuation
2P 1P
22Lλ
33Lλ
11LλΦ
Q
2P 1P
22Lλ
33Lλ
11LλΦ
Q
Theory: Zhao, SuoAPL 91, 061921 (2007)
Experiment: Pelrine, Kornbluh, Pei, JosephScience 287, 836 (2000).
29
Coexistent states stiffening
Φ
l
Q+
Q−
Interpretation: Zhao, Hong, SuoPhysical Review B 76, 134113 (2007)
Coexistent states: flat and wrinkledObservation: Plante, Dubowsky, Int. J. Solids and Structures 43, 7727 (2006)
Qthick thin
Φ
thinningpolarizing
Cross sectionTop view
30
When stretched, a polymer stiffens
n links
nb
n
λ
Langevin
Gauss
releasedb
P
P
stretched
P
fully stretched
bnPP
31
Stiffening stabilizes elastomer
( ) ( )( ) ⎥⎦⎤
⎢⎣⎡ +−+++−++ ...9
20
13
2
1 223
22
21
23
22
21 λλλλλλµ
n( )3
223
22
21 −++ λλλµ
LangevinGauss
Arruda, Boyce, J. Mech. Phys. Solids 41, 389 (1993)
Q
Φ
Φ
l
Q+
Q−
Zhao, Hong, SuoPhysical Review B 76, 134113 (2007)
32
Giant deformation of actuation?Φ
λ
λ
E
BE
1
1
ΦQ+
Q−
λ
λ
33
Interpenetrating networks
•Short chains provide a safety net.•Long chains fill the space.
Experiment: Ha, Yuan, Pei, PelrineAdv. Mater. 18, 887 (2006).
Theory: Suo, Zhu. Unpublished
34
3D inhomogeneous field
dAdVqdAxTdVxBWdV iiii ∫∫∫∫∫ Φ+Φ++= δωδδδδ
Φ
P
Qδ
lδ
Need to specify a material model
( )DF~
,W( )DW~
,λ
( ) KKiKiK DEFsW~~~
, δδδ +=DF
( )iK
iK FW
s∂
∂=
DF~
, ( )K
KD
WE ~
~,~
∂∂
=DF
Condition of equilibrium
PDEs
( )K
K Xt
E∂Φ∂
−=,~ X( )
K
iiK X
txF
∂∂
=,X
( ) ( ) 0,,
=+∂
∂tB
X
tsi
K
iK XX ( ) ( )tq
X
tD
K
K ,,
~X
X=
∂∂
( )K
KD
WE ~
~,~
∂∂
=DF( )
iKiK F
Ws
∂∂
=DF~
,
Toupin (1956), Eringen (1963), Tiersten (1971), Goulbourne, Mockensturm and Frecker (2005), Dorfmann & Ogden (2005), McMeeking & Landis (2005)…
35
Suo, Zhao, Greene,J. Mech. Phys. Solids 56, 467 (2008)
The nominal vs the true
Reference State Current State
P
Φ
l
a Q+
Q−L
A
36
( ) iKjK
ij sF
Fdet=σ
( ) KiK
i DF
D~
det F=
KiiK EEF~
=
APs /=
LE /~ Φ=
AQD /~ =
lE /Φ=
aQD /=
aP /=σ
37
Ideal dielectric elastomersLiquid-like dielectric behavior, unaffected by deformation
( ) ( )ε2
~,
2DWW s += FDF
Stretch Polarization
( ) KiK
i DF
D~
det F=
Ideal electromechanical coupling is purely a geometric effect:
( ) ( )iK
iK F
Ws
∂∂
=DF
DF~
,~,( ) ( )
K
KD
WE ~
~,~
,~
∂∂
=DF
DF
( )( )
⎟⎠⎞
⎜⎝⎛ −+
∂∂
= ijkkji
jK
siKij EEEE
F
WF δεσ2
1
det
F
Fii ED ε=
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
38
Programmable deformation
Design:Kofod, Wirges, Paajanen, BauerAPL 90, 081916 (2007)
Simulation:Zhao, SuoAPL 93, 251902 (2008)
39
Electrostriction
+ + + + + + + + +
- - - - - - - - - - - - - - - - - - - - - - - -
+ + + + + + + + +
+-
Maxwell stressElectrostriction2
33 2Eεσ −=
±
40
Non-ideal dielectric elastomer
( )[ ]31 321 −+++= λλλεε c
Die
lect
ric c
onst
ant
Area ratio
c=-0.062
deformation affects dielectric constant
Wissler, Mazza, Sens. Actuators, A 138, 384 (2007).
41
Quasi-linear dielectric elastomer
( )( )31 321 −+++= λλλεε c
( ) ( )ε
λλλµλλ2
32
~,,
223
22
2121
DDW +−++=
( )1
211
~,,
λλλ
∂∂
=DW
s
Zhao, Suo, JAP 104, 123530 (2008).
42
Energy harvesting
Generate electricity from walking Generate electricity from waves
Pelrine, Kornbluh…
43
Maximal energy that can be converted by a dielectric elastomer
R
LT
EB
E=0
EMI
6 J/g, Elastomer (theory)0.4 J/g, Elastomer (experiment)0.01 J/g ceramics
Really that large? Someone should do more experiments.
Koh, Zhao, Suo, Appl. Phys. Lett. 94, 262902 (2009)
44
Summary• Convergence of the two worlds (soft biology, hard engineering).
• Soft active materials (SAMs) have many functions (soft robots, adaptive optics, self-regulating fluidics, programmable haptics, oil recovery, energy harvesting, drug delivery, tissue regeneration, low-cost diagnosis…).
• Mechanics of SAMs is interesting and challenging (mechanics, electricity, chemistry; large deformation, mass transport, instability).
• The field is wide open (fabrication, computation, application).
3 lectures on SAMs: http://imechanica.org/node/3215•Dielectric elastomers•Gels•Polyelectrolytes
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