Large deformation and instability in
swelling polymeric gels
Zhigang Suo
School of engineering and Applied Sciences
Harvard University
1
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
Ono, Sugimoto, Shinkai, Sada, Nature Materials 6, 429, 2007
2
Gels in daily life
food
Contact lenses
Wichterie, Lim, Nature 185, 117 (1960)
Drug delivery
Ulijn et al. Materials Today 10, 40 (2007)
Tissues, natural or engineered
3
Gels in engineering
Valve in fluidics
Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)
packer in oil well
Shell, 2003
4
Two ways of doing work to a gel
µδM work done by the pump
Pump, µ
Pδl work done by the weight
Solvent
F (l , M ) Helmholtz free energy of gel
δM
Equilibrium condition
µ = ( p − p0 )v
µ = kT log ( p / p0 )
Gel = Network +
M solvent molecules
δF = Pδl + µδM
P=
∂F (l , M )
∂l
µ=
∂F (l , M )
∂M
δl
Weight, P
5
Field variables
Equilibrium condition
Reference state
Current state
M=0
δF = Pδl + µδM
A
AL
=
P δl
δM
+µ
AL
AL
δW = sδλ + µδC
∂W ( λ , C )
s=
∂λ
∂W ( λ , C )
µ=
∂C
W (λ , C ) Helmholtz free energy per volume
λ=
δM
l
L
P
s=
A
C=
gel
δF
l
L
M
pump
µ
Pump, µ
δl
P
weight
M
AL
6
3D inhomogeneous field
Deformation gradient
FiK =
Concentration
∂xi (X, t )
∂X K
Free-energy function
C (X , t )
W (F, C )
Equilibrium condition
∫ δWdV = ∫ B δx dV + ∫ T δx dA + µ ∫ δCdV
i
i
i
i
pump
Equivalent equilibrium condition
siK =
∂W (F, C )
∂FiK
∂siK
+ Bi = 0
∂X K
∂W (F, C )
µ=
∂C
s iK N K = Ti
Gel
Weight
7
Gibbs (1878)
Flory-Rehner free energy
Swelling increases entropy by mixing solvent and polymers,
but decreases entropy by straightening polymers.
Free-energy function
Free energy of stretching
Free energy of mixing
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 ⎞
χ ⎤
⎛
⎟+
⎢vC log⎜ 1 +
⎥
⎝ vC ⎠ 1 + vC ⎦
⎣
8
Flory, Rehner, J. Chem. Phys., 11, 521 (1943)
Molecular incompressibility
+
=
Vdry +
Vsol =
Vgel
1 + vC = det F
v – volume per solvent molecule
Assumptions:
– Individual solvent molecule and polymer are incompressible.
– Neglect volumetric change due to physical association
– Gel has no voids. (a gel is different from a sponge.)
9
Equations of state
Enforce molecular incompressibility as a constraint by introducing a Lagrange multiplier Π
W = W (F, C ) + Π (1 + vC − det F )
Equations of state
siK =
∂W (F, C )
− Π H iK det F
∂FiK
µ=
∂W (F, C )
+ Πv
∂C
Use the Flory-Rehner free energy
⎡
siK = NkT (FiK − H iK ) − Π H iK det F
(
)
s1 = NkT λ1 − λ1−1 − Π λ2 λ3
µ = kT ⎢ log
⎣
(
)
s2 = NkT λ2 − λ2−1 − Π λ3 λ1
⎤
χ
vC
1
+
+
+ Πv
2⎥
1 + vC 1 + vC (1 + vC ) ⎦
(
)
s3 = NkT λ3 − λ3−1 − Π λ1 λ2
10
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
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 different amount of solvent under constraint.
(λfree )3 ≠ λuni
11
Inhomogeneous swelling
Core
Empty space
R
Aλ0
r
A
B
Dry polymer
Gel
(a)
Reference State
b
(b)
Equilibrium State
„
Concentration is inhomogeneous even in equilibrium.
„
Stress is high near the interface (debond, cavitation, crease).
12
Zhao, Hong, Suo, APL 92, 051904 (2008 )
Crease
Tanaka et al, Nature 325, 796 (1987)
Denian
13
Rising bread 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
14
The brain
15
16
The science of crease: a linearized history
Biot, Appl. Sci. Res. A 12, 168 (1963).
Theory: linear perturbation analysis
ε biot ≈ 0.46
Gent, Cho, Rubber Chemistry and Technology 72, 253 (1999)
Ghatak, Das, PRL 99, 076101 (2007)
Experiments: bending rods of rubber and gels
ε exp ≈ 0.35
Hohlfeld, Mahadevan, manuscript in preparation (2008)
A new theory: crease is a distinct instability different from that analyzed by Biot.
Hong, Zhao, Suo, manuscript in preparation (2008)
Crease under various conditions.
17
An energetic model
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
creased state
ε
0.2
ε c ≈ 0.35
0
0.2
0.25
Hong, Zhao, Suo, manuscript in preparation
ε
0.3
εc
0.35
18
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
Hong, Zhao, Suo, manuscript in preparation
19
Crease of a swelling gel
η
1
gel
substrate
Theories
ηc = 2.4
η biot = 3.4
Hong, Zhao, Suo, manuscript in preparation
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
20
Hydrogel-actuated nanostructure
λ0 = 1.5
-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, http://imechanica.org/node/2487
0
θ (rad)
0.5
1
21
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
22
Hong, Zhao, Suo, http://imechanica.org/node/2487
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 elasticity
•Treat the chemical potential like temperature
•ABAQUS UMAT
pump
Gel
Weight
ˆ (F, µ ) = W − µC = 1 NkT (I − 3 − 2 log J ) − kT ⎡(J − 1 ) log J + χ ⎤ − µ (J − 1 )
W
v ⎢⎣
J − 1 J ⎥⎦ v
2
23
Hong, Liu, Suo, http://imechanica.org/node/3163
Swelling-induced contact
uniaxial swelling
4.8
4.6
•Needle-like cylinder
•Pancake-like cylinder
•Contact
λ0D
4.2
λ0H
h/H
4.4
4
3.8
3.6
3.4
3.2
free swelling
0
1
10
10
2
10
D/H
D/H = 0.6
D/H = 2
D/H = 10
24
Swelling-induced bifurcation
Experiment: Zhang, Matsumoto, Peter, Lin, Kamien, Yang, Nano Lett. 8, 1192 (2008).
Simulation: Hong, Liu, Suo, http://imechanica.org/node/3163
25
Time-dependent process
Shape change: short-range motion of solvent molecules, fast
Volume change: long-range motion of solvent molecules, slow
26
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
0.5 0
10
1
10
2
10
time (s)
3
4
10
10
Size-dependent stress relaxation
8mm
3.5
6mm
3
2.5
2
5mm
4
3.8
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
0
Impermeable
Rigid plates
PBS
strain: 15%
alginate
1.5 0
10
stress (kPa)
stress (kPa)
4
Gel disk
(a)
strain
1
10
2
3
10
10
time (s)
⎛ t⎞
σ (t, R ) = f ⎜⎜ ⎟⎟
4
10
stress
time
⎝ R⎠
time (b)
0.5
1
1.5
2
2.5
4
0.5
0.5
time /R (s /m)
x 10
27
Zhao, Huebsch, Mooney, Suo, manuscript in preparation
Coupled deformation and migration
pump
Deformation of network Conservation of solvent molecules
∂xi (X, t )
FiK =
∂X K
∂C (X , t ) ∂J K (X , t ) ∂r (X , t )
+
=
∂t
∂t
∂X K
JK N K =
Gel
∂i (X , t )
∂t
Weight
Nonequilibrium thermodynamics
∫ δWdV ≤ ∫ B δx dV + ∫ T δx dA + ∫ µδrdV + ∫ µδidA
i
Local equilibrium
∂W (F, C )
siK =
∂FiK
µ=
∂W (F, C )
∂C
Biot, JAP 12, 155 (1941)
i
i
i
Mechanical equilibrium
∂s iK (X , t )
+ Bi = 0
∂X K
s iK N K = Ti
Rate process
J K = − M KL
∂µ (X, t )
∂X L
28
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
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
29
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
Diffusion of labeled water
Stokes-Einstein formula
Dself = kT / (6πRη ) ≈ 8 ×10−10 m 2 /s
polymer volume fraction
Muhr and Blanshard, Polymer, 1982
30
Diffusion or convection?
D = 4×10-8 m2/s
~50 × Ds
Tokita, Tanaka, J. Chem. Phys, 1991
• Macroscopic pores?
• Convection?
31
Swelling of a thin layer
solvent
relative swelling
gel
Fitting swelling experiment is hard:
„ Missing data
„ Fick’s law: Concentration is not
the only driving force
dimensionless time
32
Zhou, Zhao, Hong, Suo, unpublished work (2007)
Summary
• Gels have many uses (soft robots, drug delivery, tissue
engineering, water treatment, packers in oil wells).
• Mechanics is interesting and challenging (large
deformation, mass transport, multiple thermodynamic
forces, many modes of instability).
• The field is wide open (microfabrication, computation,
material models, experiments, bioinspiration,
imagination).
33