swell-induced surface instabili in subs ate-confined hydrogel...
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Swell-Induced Surface Instability in
Substrate-Confined Hydrogel Layer
Rui Huang and Min K. Kang
Center for Mechanics of Solids, Structures and Materials
Department of Aerospace Engineering and Engineering Mechanics
The University of Texas at Austin
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Swelling of rubber and gels
• Southern and Thomas, 1965
• Tanaka et al, 1987
• Trujillo et al., 2008
� Critical condition for the onset of
surface instability?
� Any characteristic size?
� Effect of kinetics?
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A theoretical framework for gels
Nominal stress
),( CU F
iK
iKF
Us
∂
∂=
0=+∂
∂i
K
iK BX
s0=
∂
∂
KX
µ
0== iKiKi xNsT δor extµµ =
• Hong, Zhao, Zhou, and Suo, JMPS 2008
Free energy density function
Chemical potentialvC+=1)det(F
C
U
∂
∂=µ
Equilibrium equations
Boundary conditions
Volume change
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)()(),( CUUCU me += FF
++
+=
vC
vC
vC
vCvC
TkCU B
m11
ln)(χ
ν
NkBT : initial shear modulus of the polymer network
N : No. of polymer chains per unit volume
ν : Volume of a solvent molecule
χ : Enthalpy of mixing parameter
A specific material model
Free energy density function
Neo-Hookeon rubber elasticity:
( ) ( )( )[ ]FF detln232
1−−= iKiKBe FFTNkU
Flory-Huggins polymer solution theory:
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-0.5 -0.4 -0.3 -0.2 -0.1 0 0.11
2
3
4
5
6
Normalized chemical potential µ/kBT
De
gre
e o
f sw
elli
ng
λh
Nv=0.001 and χ=0.4
p0v/k
BT=0.000023 at T=25oC
Homogeneous swelling of a hydrogel layer
3
2
1 1
1
2
31
>=
==
hλλ
λλ
Tk
pvNv
B
ext
h
h
hhh
−=
−++−
−
µ
λλ
λ
χ
λλ
1111ln
2
vppext )( 0−=µ
( )0/ln ppTkBext =µ
External chemical potential:
0pp ≤
0pp ≥
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A linear perturbation analysis
∂
∂+
∂
∂
∂
∂
∂
∂+
=
100
01
01
~
2
2
1
2
2
1
1
1
x
u
x
u
x
u
x
u
h
h
λ
λ
F
=
100
00
001
hλF
Linear perturbation: ( ) ( )21222111 ,,, xxuuxxuu ==
Homogeneous swelling
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
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( )
( ) 0
01
21
1
2
2
2
2
2
2
1
2
2
21
2
2
2
2
1
22
2
1
1
2
=∂∂
∂+
∂
∂++
∂
∂
=∂∂
∂+
∂
∂+
∂
∂+
xx
u
x
u
x
u
xx
u
x
u
x
u
hhhhh
hhhhh
ξλλξλ
ξλλξλ
Linearized equilibrium equations
Solution by the method
of Fourier transform
( ) ( )
( ) ( )
=
=
∑
∑
=
=
4
1
2
)(
222
4
1
2
)(
121
exp;ˆ
exp;ˆ
n
n
n
n
n
n
n
n
xquAkxu
xquAkxu
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
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04
1
=∑=n
nmn AD
[ ]
−
+−
+
+
+
−−
=
−−
−−
khkhkh
h
h
kh
h
h
kh
h
h
kh
h
h
kh
h
kh
h
hh
mn
eeee
eeeeD
ββ
ββ
ββλ
λλ
λ
λλ
λλλλ
ββλλ
2211
1122
1111
00
00
[ ] ( ) 0,;,det 0 == χλ NvkhfD hmn
Critical Conditions for Surface Instability
===
=∂
∂=
=
∂
∂+−=
00
1
221
2
1
212
2
1
122
xuu
hxx
ups
hxx
ups
at
at
atBoundary conditions
Critical condition:
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
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Effect of perturbation wave number
Critical swelling ratio
Nv = 0.001
• Long wavelength perturbation is stabilized by the substrate effect.
• Short wavelength perturbation is unaffected.
• Thus the critical condition can be taken at the short-wavelength
limit.
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
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Short-wave limit (kh0 →∞)
βλλ
λ h
h
h 41
2
=
+
• The critical swelling ratio
depends on Nv and χ, ranging
between 2.5 and 3.4.
• For each χ, there exists a
critical value for Nv.
• For small Nv (< 10-4), the
critical swelling ratio is nearly
a constant (~3.4).
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
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A stability diagram
χc = 0.63
Soft network in good solvent
Poor solvent
Stiff network
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
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Critical linear strain
10-5
10-4
10-3
10-2
10-1
0.3
0.35
0.4
0.45
0.5
Nv
Critic
al str
ain
, εc
χ=0.0
χ=0.2
χ=0.4
χ=0.6
D
D
3
3 1
λ
λε
−=
Relative to the unconstrained,
free swelling in 3D:
• Trujillo et al.’s experiments for a swelling hydrogel
• Biot’s analysis for rubber under equi-biaxial compression
33.0=cε
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
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0 2 4 6 83
3.5
4
4.5
5
5.5
6
Normalized wavelength S
Critic
al s
we
llin
g r
atio
λc
h0=1µm
h0=10µm
h0=100µm
Nv = 0.001, χ = 0.4
Effect of surface tension
� Long wavelength perturbation
is stabilized by the substrate.
� Short wavelength
perturbation is stabilized by
surface tension.
�An intermediate characteristic
wavelength emerges.
� The minimum critical swelling
ratio depends on the layer
thickness.
Kang and Huang, Soft Matter 6, 5736-5742 (2010).
Nv
nm
TNkL
B
53.0~
γ=A length scale:
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10-1
100
101
102
103
100
101
102
103
Initial thickness [µm]
Cri
tica
l wa
ve
len
gth
S* [
µm
]
L=1µm
L=0.5µm
L=0.1µm
Nv = 0.001, χ = 0.4
Characteristic wavelength
1.09.0
0
* ~ LhS
Kang and Huang, Soft Matter 6, 5736-5742 (2010).
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10-1
100
101
102
103
3
3.5
4
4.5
5
5.5
6
Initial thickness [µm]
Critica
l sw
elli
ng
ra
tio λ
c
L=0.1µm
L=0.53µm
L=1µm
Nv = 0.001, χ = 0.4
Thickness-dependent stability
• The hydrogel layer becomes increasingly stable as the initial layer
decreases;
• Below a critical thickness (hc), the hydrogel is stable at the
equilibrium state.Kang and Huang, Soft Matter 6, 5736-5742 (2010).
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10-5
10-4
10-3
10-2
10-1
0
0.2
0.4
0.6
0.8
1
Nv
Critic
al t
hic
kn
ess h
c [
µm
]
χ = 0.0
χ = 0.2
χ = 0.4
χ = 0.5
Critical thickness
Nv
LL
nmTk
vL
B
'
53.0'
=
==γ
� The critical thickness is linearly proportional to L, with the
proportionality depending on Nv and χ.
Kang and Huang, Soft Matter 6, 5736-5742 (2010).
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Finite element simulation
Nv = 0.001, χ = 0.4
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
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Surface Evolution
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
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W/H=1
W/H=5
Volume ratio
Inhomogeneous Swelling of
Substrate-Supported Hydrogel Lines
W/HKang and Huang, J. Applied Mechanics 77, 061004 (2010).
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Spontaneous Formation of Creases
W/H=12
Kang and Huang, J. Applied Mechanics 77, 061004 (2010).
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Swell-induced bucklingTirumala et al., 2005
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Effect of material parameters
10-3
10-2
10-1
0
0.5
1
1.5
2
Nv
A/H
Kang and Huang, Int. J. Applied Mechanics, in press.
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Effect of geometry (constraint)
0 1 2 3 40
0.5
1
1.5
2
2.5
W/H
A/H
Kang and Huang, Int. J. Applied Mechanics, in press.
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Summary
• Opportunity: Within the general theoretical framework,
instability of hydrogel-like soft material can be understood
and exploited.
• Challenge: The highly nonlinear aspects in the material,
geometry, and instability mechanics pose serious challenges
for theoretical analysis and numerical simulations.
• Strategy: Collaborations between experimental and
theoretical studies will be most successful.