Soft Active Materials
Zhigang Suo
Harvard University
The James F. Bell Memorial Lecture in Continuum Mechanics
Department of Mechanical Engineering, Johns Hopkins University
5 November 2009
James F. Bell
1914-1995
1
Professors Sharp and Bell (1983)
Professor Bell (about 1990)
2
elastomer = network
gel = network + solvent
solvent
reversible
gel
network
3
Super absorbent diaper
Sodium polyacrylate: polyelectrolyte
4
Gels regulate flow in plants
Missy Holbrook
Zwieniecki, Melcher, Holbrook,
Hydrogel control of xylem hydraulic resistance in plants
Science 291, 1095 (2001)
5
Self-regulating fluidics
David Beebe
Responsive to
Physiological variables:
•pH
•Salt
•Temperature
•light
•Many stimuli cause deformation.
•Deformation regulates flow.
6
Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)
octopus
Mäthger, Hanlon, Kuzirian – Marine Biological Laboratory, Woods Hole MA
7
Squid changes color
Expand pigmented sacs by contracting muscles
8
Mathger, Denton, Marshall, Hanlon, J. R. Soc. Interface 6, S149 (2009)
Adaptive Optics
Ah, optics!
•Many stimuli cause deformation.
•Deformation affects optics.
Wilbur, Jackman, Whitesides, Cheung, Lee, Prentiss
Elastomeric optics
Chem. Mater. 1996, 8, 1380-1385
Aschwanden, Stemmer
Optics letters 31, 2610 (2006)
9
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.
Stimuli
pH, E, T, C…
Deformation provides a function.
SAM
deforms
Functions
optics, flow…
10
Very well, but how does stimulus X cause deformation?
11
Dielectric elastomer
Reference State
A
Current State
Dielectric
Elastomer
L
a
l
Compliant
Electrode
Pelrine, Kornbluh, Pei, Joseph
High-speed electrically actuated elastomers with strain greater than 100%.
Science 287, 836 (2000).
+Q
Φ
−Q
12
Parallel-plate capacitor
P
battery
a
Φ
−Q
electrode
l
+Q
vacuum
electrode
P
Electric field
Electric displacement field
stress field
force
E=
Φ
l
Q
D=
a
σ=
P
a
D = ε0E
ε0, permittivity of vacuum
1
2
σ = ε 0 E 2 Maxwell stress
13
Field equations in vacuum, Maxwell (1873)
∂Φ
Ei = −
∂x i
Electrostatic field
∂E i
q
=
∂x i ε 0
Fi = qE i
A field of forces maintain equilibrium of a field of charges
Fi =
−Q
∂
∂x j
ε0
⎛
⎞
ε
E
E
E
E
δ
−
⎜ 0 j i
k k ij ⎟
2
⎝
⎠
P
σ=
E
+Q
ε0
2
E2
σ ij = ε 0 E j E i −
Maxwell stress
P
ε0
2
E k E kδ ij
14
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
15
Trouble with Maxwell stress in dielectrics
-----------±
+++++++++
Maxwell stress
ε
σ 33 = − E 2
2
-----------+
-
+++++++++
Electrostriction
Our complaints:
•In general, ε varies with deformation.
•In general, E2 dependence has no special significance.
•Wrong sign?
16
Trouble with electric force in dielectrics
In a vacuum,
external force is needed to maintain equilibrium of charges
+Q
+Q
P
Fi = qE i
P
In a solid dielectric,
force between charges is NOT an operational concept
+Q
+Q
Fi = qE i
17
The Feynman Lectures on Physics
Volume 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
All troubles are gone if we use measurable quantities
Current State
Reference State
A
a
L
l
equilibrate elastomer and loads
+Q
Φ
−Q
P
δF = Pδl + ΦδQ
Nominal
δF
divide by volume
AL
name quantities
equations of state
=
Pδl ΦδQ
+
AL LA
~ ~
δW = sδλ + EδD
( )
~
∂W λ , D
s=
∂λ
( )
~
~ ∂W λ , D
E=
~
∂D
W = F /( AL )
λ = l/L
s = P/ A
~
E = Φ/ L
~
D = Q/ A
True
σ = P /a
E = Φ /l
D = Q/a
19
Fine, Zhigang, but what is
( )?
~
W λ, D
20
Dielectric constant is insensitive to stretch
VHB 4910
Kofod, Sommer-Larsen, Kornbluh, Pelrine
Journal of Intelligent Material Systems and Structures 14, 787 (2003).
21
Ideal dielectric elastomer
incompressibility
λ3
1
λ1 λ2 λ3 = 1
λ2
1
1
λ1
Dielectric behavior is liquid-like, unaffected by deformation.
(
)
~
D2
W λ1 , λ2 , D = Wstretch (λ1 , λ2 ) +
2ε
Elasticity
Polarization
For an ideal dielectric elastomer, electromechanical coupling is purely a geometric effect:
Q
D=
a
~ Q
D=
A
D=
~
D
λ1 λ2
22
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
Ideal dielectric elastomer
λ3 L3
λ1 L1
~2
~ µ 2
D
W λ1 , λ2 , D = (λ1 + λ22 + λ1−2 λ2−2 − 3) +
λ1−2 λ2−2
2
2ε
(
)
λ2 L2
Φ
Q
P2
P1
In terms of nominal quantities
(
~
∂W λ1 , λ2 , D
s1 =
∂λ1
)
(
~
∂W λ1 , λ2 , D
s2 =
∂λ2
)
In terms of true quantities
σ 1 = µ (λ21 − λ1−2 λ2−2 ) − εE 2
σ 2 = µ (λ22 − λ1−2 λ2−2 ) − εE 2
23
Young man, don’t be too fast. The theory sounds fine,
but can you relate the theory to experiments?
24
Two modes of failure
Electromechanical instability
Soft material
Φ
+Q
l
l
−Q
Q
Electrical breakdown
Hard material
polarizing
thinning
polarizing
Φ
Φ
Q
Q
25
Stark & Garton, Nature 176, 1225 (1955)
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?
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Electromechanical instability
limits deformation of actuation
1
λ
+Q
Φ
−Q
1
λ
λc = 21 / 3 ≈ 1.26
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
λ3 L3
λ1 L1
λ2 L2
Φ
Q
P2
Experiment: Pelrine, Kornbluh, Pei, Joseph
Science 287, 836 (2000).
P1
Theory: Zhao, Suo
APL 91, 061921 (2007)
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stiffening
Coexistent states
thinning
polarizing
+Q
l
Φ
Φ
−Q
thick
thin
Q
Top view
Cross section
Coexistent states: flat and wrinkled
Observation: Plante, Dubowsky,
Int. J. Solids and Structures 43, 7727 (2006)
Interpretation: Zhao, Hong, Suo
Physical Review B 76, 134113 (2007)
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When stretched, a polymer stiffens
n links
released
b
Langevin
b n
P
Gauss
stretched
P
λ
P
n
fully stretched
P
bn
P
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Stiffening stabilizes elastomer
µ
2
(λ
2
1
+ λ22 + λ23 − 3)
(
)
⎡1 2
(λ1 + λ22 + λ23 − 3)+ 1 (λ21 + λ22 + λ23 )2 − 9 + ...⎤⎥
20n
⎣2
⎦
µ⎢
Gauss
Langevin
Arruda, Boyce, J. Mech. Phys. Solids 41, 389 (1993)
Φ
+Q
l
Φ
−Q
Zhao, Hong, Suo
Physical Review B 76, 134113 (2007)
Q
31
Giant deformation of actuation?
Φ
1
λ
1
λ
+Q
−Q
EB
Φ
E
λ
λ
32
Interpenetrating networks
•Short chains provide a safety net.
•Long chains fill the space.
Experiment: Ha, Yuan, Pei, Pelrine
Adv. Mater. 18, 887 (2006).
Theory: Suo, Zhu. Unpublished
33
3D inhomogeneous field
Condition of equilibrium
∫ δWdV = ∫ Biδxi dV + ∫ Tiδxi dA + ∫ ΦδqdV + ∫ ΦδωdA
Need to specify a material model
( )
~
W λ, D
( ~)
( )
~
W F, D
~
~
δW F, D = siK δFiK + E K δDK
δQ
Φ
siK
( )
~
∂W F, D
=
∂FiK
(
~
∂W F, D
~
EK =
~
∂DK
)
P
δl
34
PDEs
FiK
∂x (X ,t )
= i
∂X K
∂siK (X ,t )
+ Bi (X ,t ) = 0
∂X K
siK
( )
~
∂W F, D
=
∂FiK
∂Φ (X ,t )
~
EK = −
∂X K
~
∂DK (X , t )
= q(X , t )
∂X K
(
~
∂W F, D
~
EK =
~
∂DK
)
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
A
L
Φ
+Q
a
l
−Q
P
s = P/ A
σ = P/a
~
E =Φ/L
E = Φ/l
~
D =Q/ A
D =Q/a
σ ij =
F jK
det(F )
siK
~
FiK E i = E K
Di =
FiK ~
DK
det (F )
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Ideal dielectric elastomers
Liquid-like dielectric behavior, unaffected by deformation
( )
~
D2
W F, D = Ws (F ) +
2ε
Stretch
Polarization
Ideal electromechanical coupling is purely a geometric effect:
Di =
( )
D i = εE i
( )
~
~ ∂W F, D
siK F, D =
∂FiK
~
~
~ ∂W F, D
E K F, D =
~
∂D K
( )
FiK ~
DK
det (F )
( )
σ ij =
FiK ∂Ws (F ) ⎛
1
⎞
+ ε ⎜ E i E j − E k E kδ ij ⎟
det(F ) ∂F jK
2
⎝
⎠
37
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
Programmable deformation
Design:
Kofod, Wirges, Paajanen, Bauer
APL 90, 081916 (2007)
Simulation:
Zhao, Suo
APL 93, 251902 (2008)
38
Electrostriction
-----------±
+++++++++
Maxwell stress
ε
σ 33 = − E 2
2
-----------+
-
+++++++++
Electrostriction
39
Non-ideal dielectric elastomer
Dielectric constant
ε = ε [1 + c (λ1 + λ2 + λ3 − 3)]
c=-0.062
Area ratio
deformation affects dielectric constant
40
Wissler, Mazza, Sens. Actuators, A 138, 384 (2007).
Quasi-linear dielectric elastomer
ε = ε (1 + c (λ1 + λ2 + λ3 − 3))
(
)
~ µ 2
D2
2
2
W λ1 , λ2 , D = (λ1 + λ2 + λ3 − 3) +
2
2ε
(
~
∂W λ1 , λ2 , D
s1 =
∂λ1
)
41
Zhao, Suo, JAP 104, 123530 (2008).
Energy harvesting
Generate electricity from walking
Generate electricity from waves
42
Pelrine, Kornbluh…
Maximal energy that can be converted
by a dielectric elastomer
6 J/g, Elastomer (theory)
0.4 J/g, Elastomer (experiment)
0.01 J/g ceramics
E=0
EMI
R
EB
LT
Really that large? Someone
43
should do more experiments.
Koh, Zhao, Suo, Appl. Phys. Lett. 94, 262902 (2009)
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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