Mechanics of
Soft Active Materials (SAMs)
Zhigang Suo
Harvard University
Soft: large deformation in response to small forces
(e.g., rubbers, gels)
Active: large deformation in response to diverse stimuli
(e.g., electric field, temperature, pH, salt)
1
Dielectric elastomer:
electrostatics meets mechanics
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
2
Insertion reaction:
electrochemistry meets mechanics
electron
Positive
electrode
−
Load
Negative
electrode
Li-ion
+
Electrolyte
Lithium-ion battery
Conducting polymers
3
gel = network + solvent
solvent
reversible
gel
network
Solid-like: long polymers crosslink by strong bonds. Retain shape
Liquid like: polymers and solvent aggregate by weak bonds. Enable transport
4
Valve in microfluidics
David Beebe
Responsive to
Physiological variables:
•pH
•Salt
•Temperature
•light
5
Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)
Gels regulate flow in plants
Missy Holbrook
Zwieniecki, Melcher, Holbrook,
Hydrogel control of xylem hydraulic resistance in plants
Science 291, 1095 (2001)
6
Swelling packers in oil wells
7
A project with Schlumberger
Gels as transducers
• A swelling gel is blocked by a hard material.
• Blocking force.
• Inhomogeneous deformation.
Need to formulate boundary-value problems
8
2 ways of doing work
Reference state
Current state
M=0
Equilibrium condition
AL
=
Helmholtz free energy per volume
λ=
δM
l
L
P
s=
A
C=
gel
P δl
δM
+µ
AL
AL
δW = sδλ + µδC
W (λ , C )
l
δF
M
pump
µSolvent,
µ
L
δF = Pδl + µδM
A
M
AL
δl
P
weight
9
Inhomogeneous field
Deformation gradient
FiK =
Concentration
Free-energy function
C (X ,t )
W (F, C )
∂x i (X,t )
∂X K
Gibbs (1878)
Condition of equilibrium
∫ δWdV = ∫ Biδxi dV + ∫ Tiδxi dA + µ ∫ δCdV
µ = constant
solvent
Gel
1. How to solve boundary-value problems?
2. How to prescribe W (F, C ) ?
3. What boundary-value problems to solve?
Weight
10
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
Finite element method
Equilibrium condition
∫ δWdV = ∫ B δx dV + ∫ T δx dA + µ ∫ δCdV
Legendre transform
ˆ (F, µ ) = W − µC
W
i
i
i
i
ˆ dV = B δx dV + T δx dA
δ
W
∫
∫ i i
∫ i i
•A gel in equilibrium is analogous to an elastic solid
•ABAQUS UMAT
11
Hong, Liu, Suo, Int. J. Solids Structures 46, 3282 (2009)
Free-energy function
•Swelling decreases entropy by straightening polymers.
•Swelling increases entropy by mixing solvent and polymers.
Paul Flory
Free-energy function
Free energy of stretching
Free energy of mixing
Adding volume
Flory, Rehner, J. Chem. Phys., 11, 521 (1943)
W (F, C ) = W s (F ) + W m (C )
Ws (F ) = 21 NkT [FiK FiK − 3 − 2 log (det F )]
Wm (C ) = −
kT
v
1 + vC = det F
⎡
1 ⎞
χ ⎤
⎛
⎟+
⎢vC log⎜ 1 +
⎥
⎝ vC ⎠ 1 + vC ⎦
⎣
12
Stress-strain relation
σ1 =
σ2 =
σ3 =
µ
kT
Π= −
v v
NkT (λ21 − 1 )
−Π
NkT (λ22 − 1 )
−Π
λ2 λ3 λ3
λ2 λ3 λ3
NkT (λ23 − 1 )
λ2 λ 3 λ 3
−Π
⎤
⎡ ⎛
χ
1 ⎞⎟
1
⎜
+
+
⎢ log ⎜ 1 −
⎟ λ λ λ (λ λ λ )2 ⎥
λ
λ
λ
⎥⎦
⎢⎣ ⎝
2 3 3 ⎠
2 3 3
2 3 3
13
Anisotropic swelling
Free swelling
Unidirectional swelling
3.5
6
analytical
FEM
5.5
λ0 H
5
3
s
4
2.5
λ1 H
λ
λ0
4.5
3.5
L
λ0L
3
2
2.5
2
1.5
-0.05
-0.04
-0.03
-0.02
µ 0/kT
-0.01
0
1.5
-0.05
-0.04
-0.03
-0.02
µ/kT
-0.01
0
A gel imbibes a different amount of solvent under constraint.
(λfree )3 ≠ λunidirectional
Treloar, Trans. Faraday Soc. 46, 783 (1950).
14
Inhomogeneous swelling
Core
Empty space
R
Aλ0
r
A
B
Dry polymer
b
Gel
(a)
Reference State
(b)
Equilibrium State
„
Concentration is inhomogeneous even in equilibrium.
„
Stress is high near the interface (debond, cavitation).
Sternstein, J. Macromolol. Soc. Phys. B6, 243 (1972)
Zhao, Hong, Suo, APL 92, 051904 (2008 )
15
Swelling-induced buckling
Ri / H = 5
Ri / H = 10
RO / H = 15
Ri / H = 7
Ri / H = 12
16
Gel and nano-rods
Joanna Aizenberg
λ0 = 1.5
θ
-0.3
χ = 0.1
Nv = 10-3
RH = 80%
vG/kT
-0.4
-0.5
-0.6
-0.7
RH = 95%
-0.8
RH = 100%
-0.9
-1
-0.5
Experiment: Sidorenko, Krupenin, Taylor, Fratzl, Aizenberg, Science 315, 487 (2007).
Theory: Hong, Zhao, Suo, JAP104, 084905 (2008) .
0
θ (rad)
0.5
1
17
Critical humidity can be tuned
1.4
1.2
λ0 = 1.7
θ (rad)
1
0.8
0.6
1.2
0.4
0.2
0
0
1.1
χ = 0.1
Nv = 10-3
20
40
60
Relative humidity (%)
80
100
18
Swelling-induced bifurcation
Shu Yang
19
Zhang, Matsumoto, Peter, Lin, Kamien, Yang, Nano Lett. 8, 1192 (2008).
Swelling-induced bifurcation
Experiment: Zhang, Matsumoto, Peter, Lin, Kamien, Yang, Nano Lett. 8, 1192 (2008).
Simulation: Hong, Liu, Suo, Int. J. Solids Structures 46, 3282 (2009)
20
Crease
Liang Fen (凉粉), a starch gel
A food popular in northern China
21
Dough
Zhigang, I was making bread this weekend, and realized that when the rising dough was constrained by the bowl
it formed the creases that you were talking about in New Orleans.
-- An email from Michael Thouless
22
Brain
23
Face
24
Crease: theory and experiment
Biot, Appl. Sci. Res. A 12, 168 (1963).
Theory: linear perturbation analysis
ε Biot ≈ 0.46
Maurice Biot
Gent, Cho, Rubber Chemistry and Technology 72, 253 (1999)
Ghatak, Das, PRL 99, 076101 (2007)
Experiments: bending rods of rubber and gel
ε Gent ≈ 0.35
Alan Gent
25
What’s wrong with Biot’s theory?
(a)
L
A'
L
O
A
reference state
Homogeneous
deformation
(b)
Mahadevan
A' O A
ε
ε
homogeneous state
(c)
A/A'
ε
O
ε
creased state
Biot’s theory: infinitesimal strain
from the state of homogeneous deformation
Crease: Large strain
from the state of homogeneous deformation
Hohlfeld, Mahadevan (2008):
crease is an instability different from that analyzed by Biot.
26
An energetic model of crease
L
(a)
A'
∆U = L2Gf (ε )
L
O
A
reference state
0.8
(b)
0.6
A' O A
ε
•Neo-Hookean material
•Plane strain conditions
ε
∆U/GL
2
homogeneous state
(c)
0.4
A/A'
ε
O
ε
creased state
0.2
ε c ≈ 0.35
0
0.2
Hong, Zhao, Suo, http://imechanica.org/node/5999
0.25
ε
0.3
εc
0.35
27
e
Crease under general loading
cr
ea
s
Incompressibility
λ1 λ2 λ3 = 1
λ2
λ3
λ1
Critical condition for crease
λ3 / λ1 = 2.4
Biot
λ3 / λ1 = 3.4
28
Hong, Zhao, Suo, http://imechanica.org/node/5999
Crease of a gel during swelling
Toyoichi Tanaka
1946-2000
29
Tanaka et al, Nature 325, 796 (1987)
Crease of a swelling gel
Ryan Hayward
swell
1
gel
η
substrate
Theories
ηc = 2.4
η biot = 3.4
Hong, Zhao, Suo, http://imechanica.org/node/5999
Experimental data
Southern, Thomas, J. Polym. Sci., Part A, 3, 641 (1965)
Tanaka, PRL 68, 2794 (1992)
η exp = 2.4
η exp = 2.5 − 3.7
Trujillo, Kim, Hayward, Soft Matter 4, 564 (2008)
η exp = 2.0
30
Polyelectrolyte gels
A network of polymers
+
+
+
-
+
+
-
+
+
-
+
-
+
-
+
Gel
+
+
-
Fixed charges
+ Counterions
+
-
-
-
-
-
-
+
-
-
+
-
-
+
+
-
-
-
-
-
-
-
-
-
Crosslinks
-
-
+
Solvent (e.g., water)
+
Co-ions
-
External
solution
31
3 ways of doing work to a gel
x(X, t)
pump
µa µ a
δMa
gel
δq
δQ
battery
Φ
P
weight δx
a
a
δ
WdV
=
B
δ
x
dV
+
T
δ
x
dA
+
Φ
δ
qdV
+
Φ
δω
dA
+
µ
δ
C
∑ ∫ dV
∫
∫ i i
∫ i i
∫
∫
a
W=
free energy of gel
volume in reference state
C a (X,t ) =
# of ions of species a
32
volume in reference state
Swelling regulated by concentration of ions
70
Nv = 0.001
χ = 0.1
vC0 = 0.01
60
Concentration
of fixed ions
55
0.005
s
vC + 1
Swelling ratio
65
50
45
0.002
40
nonionic gel
s
vC* + 1
35
10
-6
10
-5
10
-4
10
-3
10
-2
vc0
Concentration of ions in external solution
33
Constrained swelling
42
Nv = 0.001
χ = 0.1
λ0 = 1.5
vC0 = 0.01
38
s
vC + 1
Swelling ratio
40
36
gel
Concentration
of fixed ions
0.005
34
External solution
32
0.002
30
vC* + 1
s
10
-6
10
-5
10
-4
10
-3
10
-2
vc0
Concentration of ions in external solution
Hong, Zhao, Suo, http://imechanica.org/node/5960
34
pH-sensitive hydrogel
[RCOO ][H ] = K
−
RCOOH ⇔ RCOO − + H +
fixed
mobile
+
[RCOOH]
a
35
[ ]
pH = − log 10 H +
Polymers are charged
Polymers are neutral
36
Valve in microfluidics
37
Crease
38
Time-dependent process
Shape change: short-range motion of solvent molecules, fast
Volume change: long-range motion of solvent molecules, slow
39
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
Concurrent deformation and migration
Deformation of network
∂xi (X, t )
FiK =
∂X K
Conservation of solvent
∂C (X , t ) ∂J K (X , t ) ∂r (X , t )
+
=
∂t
∂t
∂X K
JK N K
Maurice Biot
JAP 12, 155 (1941)
pump
∂i (X , t )
=−
∂t
Gel
Nonequilibrium thermodynamics
∫ δWdV ≤ ∫ B δx dV + ∫ T δx dA + ∫ µδrdV + ∫ µδidA
i
i
i
i
Rate process
Local equilibrium
∂W (F, C )
siK =
∂FiK
µ=
∂W (F, C )
∂C
Weight
∂s iK (X , t )
+ Bi = 0
∂X K
s iK N K = Ti
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
J K = − M KL
∂µ (X, t )
∂X L
40
ideal kinetic model
Solvent molecules migrate in a gel by self-diffusion
J K = − M KL
∂µ
∂X L
Diffusion in true quantities
ji = −
cD ∂µ
kT ∂x i
Conversion between true and nominal quantities
ji =
FiK
JK
det F
∂µ
∂µ
=
FiK
∂X K ∂xi
M KL =
D
H iK H iL (det F − 1 )
vkT
41
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
Finite element method for
concurrent deformation and migration
Dt L2 = 0
Dt L2 = 10
Dt L2 = ∞
42
Zhang, Zhao, Suo, Jiang, JAP 105, 093522 (2009)
Alginate Hydrogels
Ionic crosslinks
Ca++( )
Na+
Covalent crosslinks
AAD
Irreversible
43
strain
Stress-relaxation test
PBS
Gel disk
Impermeable
time
stress
Rigid plates
time
44
Zhao, Huebsch, Mooney, Suo, submitted.
Elasticity, plasticity, fracture
Ionic
crosslinks
10mm
Swollen state
45% compressive strain
50% compressive strain
Covalent
crosslinks
Swollen state
15% compressive strain
20% compressive strain
45
Stress Relaxation
4
stress (kPa)
3.5
Covalent
3
~102
2.5
Ionic
2
1.5
1
Strain: 15%
Radius: 6mm
0.5 -1
10
0
10
1
10
2
10
time (s)
3
10
4
10
46
Size effect
R
47
Gel with covalent crosslinks relaxes stress
by migration of water
⎛ t⎞
σ (t, R ) = f ⎜⎜ ⎟⎟
⎝ R⎠
D~
R2
trelax
~ 10 −8 m 2 /s
48
Outlook
• Soft Active Materials (SAMs) have many uses
(microfluidics, artificial muscles, drug delivery, tissue
engineering, water treatment, packers in oil wells).
• Mechanics of SAMs is interesting and challenging (large
deformation, mass transport, multiple thermodynamic
forces, many modes of instability).
• The field is wide open (fabrication, computation, devices,
phenomena).
3 lectures on SAMs: http://imechanica.org/node/3215
•Dielectric elastomers
•Gels
•Polyelectrolytes
49