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
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?
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
)()(),( 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:
-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 ≥
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).
( )
( ) 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).
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).
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).
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).
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).
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).
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:
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).
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).
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).
Finite element simulation
Nv = 0.001, χ = 0.4
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
Surface Evolution
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
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).
Spontaneous Formation of Creases
W/H=12
Kang and Huang, J. Applied Mechanics 77, 061004 (2010).
Swell-induced bucklingTirumala et al., 2005
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.
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.
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.
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