Post on 11-Feb-2022
Swell Induced Deformation and Surface Instability of Polymer Gels
Rui Huang
Center for Mechanics of Solids, Structures and MaterialsDepartment of Aerospace Engineering and Engineering Mechanics
The University of Texas at Austin
October 4, 2012
Gel: polymer network + solvent
Ono et al, Nature Materials, 6, 429, 2007.
• Large and reversible deformation
• Stimuli responsive: temperature, pH, UV, electric field, etc.
• Soft, bio‐compatible
Suo, 2008
Swell‐Induced Instability
Trujillo et al, 2008.Zhang et al, 2008.
DuPont et al., 2010
Kinetics of Swelling
Tanaka et al, 1987
Guvendiren et al., 2010.
A theoretical framework for gels
Nominal stress
),( CU F
iKiK F
Us
0
iK
iK BXs 0
KX
0 iKiKi xNsT or ext
Hong, Zhao, Zhou, and Suo, JMPS 2008
Free energy density function
Chemical potentialCU
Equilibrium equations
Boundary conditions
C1)det(F
Volume change
)()(),( CUUCU me FF
C
CC
CCTkCU Bm 11
ln)(
NkBT : initial shear modulus of the polymer network (entropic elasticity);
N : No. of polymer chains per unit volume;Ω : Volume of a solvent molecule; : polymer‐solvent interaction parameter (enthalpy of mixing).
A specific material modelFree energy density function
Neo‐Hookeon rubber elasticity:
FF detln2321
iKiKBe FFTNkU
Flory‐Huggins polymer solution theory:
Kinetics of Solvent Diffusion
k
L
LBkBk x
XX
CTk
DxTk
cDj
)det(FRate of flux (current configuration):
LKLk
k
KK X
MjxXJ
)det(FNominal rate of flux
(reference configuration):
LKL
KK
K
XM
XXJ
tC Nominal concentration:
A nonlinear diffusion equation, coupled with mechanical equilibrium equations (nonlinear elasticity).
Hong, Zhao, Zhou, and Suo, JMPS 2008
k
L
k
K
BKL x
XxX
TkDCMMobility tensor:
A Finite Element Method for Equilibrium Analysis
• Tangent modulus
klijijklBijkl JJJJ
NHJTNkC
231
11
1ln1
jkiljkiljlikjlikijkl BBBBH 21
• True stress jKiKiK
ij FFJ
JU
FI
IU
J
ˆˆ1
FFFFB3/1
J
T
CCUU ),(,ˆ FF• Legendre transformation
dAxTdVxBdVU iiii ˆ• Weak form Chemical potential is specified as a constant.
Kang and Huang, J. Applied Mechanics 77, 061004 (2010).
Implementation in ABAQUS (UMAT)• The chemical potential is mimicked by temperature.• Due to singularity of the chemical potential at the dry state,
FEM handles deformation after homogeneous swelling.
Analytical method
(1)F (2)F
(1)(2)FFF
Dry state Initial state Final deformed state
Homogeneousswelling
FEM simulation
= 0 > >
= 0
)1(
3
)1(2
)1(1
(1)F
Kang and Huang, J. Applied Mechanics 77, 061004 (2010).
Analytical method
1‐D homogeneous swellingSide wall relaxation Final deformed state
FEM simulation
Constrained Swelling of Hydrogel Lines
Kang and Huang, J. Applied Mechanics 77, 061004 (2010).
W/H=1 W/H=5
Volume ratio
Effect of constraint on swelling
Kang and Huang, J. Applied Mechanics 77, 061004 (2010).
W/H=10
Surface instability: creases
W/H=12
Kang and Huang, J. Applied Mechanics 77, 061004 (2010).
A rubber block under compression
• Surface wrinkling (Biot, 1961)
• Surface creasing (Hong, Zhao and Suo, 2009)
295.0321
417.0321
54.01 01 08.3 GP
65.01 01 34.2 GP
rubber
A linear perturbation analysis
100
01
01
~
2
2
1
2
2
1
1
1
xu
xu
xu
xu
h
h
F
10000001
hF
Linear perturbation: 21222111 ,,, xxuuxxuu
Homogeneous swelling
Kang and Huang, J. Mech. Phys. Solids 58, 1582‐1598 (2010).
0
01
21
12
22
22
21
22
21
22
22
12
221
12
xxu
xu
xu
xxu
xu
xu
hhhhh
hhhhh
Linearized equilibrium equations
Solution by the method of Fourier transform
4
12
)(222
4
12
)(121
exp;ˆ
exp;ˆ
nn
nn
nn
nn
xquAkxu
xquAkxu
Kang and Huang, J. Mech. Phys. Solids 58, 1582‐1598 (2010).
04
1
nnmn AD
khkhkh
hh
kh
hh
kh
hh
kh
hh
khh
khh
hh
mn
eeee
eeeeD
2211
1122
1111
00
00
0,;,det 0 NkhfD hmn
Critical Condition for Surface Instability
00
1
221
21
212
21
122
xuu
hxxups
hxxups
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
NΩ = 0.001
• Long wavelength perturbation is stabilized by the substrate effect.• Short wavelength perturbation is independent of substrate.• The critical condition is set at the short‐wavelength limit.
Kang and Huang, J. Mech. Phys. Solids 58, 1582‐1598 (2010).
Short‐wave limit (kh0 )
hh
h 412
• The critical swelling ratio depends on NΩ and , ranging between 2.5 and 3.4.
• For each , there exists a critical value for NΩ.
• For small NΩ (< 10‐4), the critical swelling ratio is nearly a constant (~3.4).
Kang and Huang, J. Mech. Phys. Solids 58, 1582‐1598 (2010).
NΩ
A stability diagram
c = 0.63
Soft network in good solvent
Poor solvent
Stiff network
Kang and Huang, J. Mech. Phys. Solids 58, 1582‐1598 (2010).
NΩ
Finite element simulationNΩ = 0.001, = 0.4
Kang and Huang, J. Mech. Phys. Solids 58, 1582‐1598 (2010).
21
Nv = 0.001, = 0.4
Surface Evolution
The numerical simulations are often unstable due to the highly nonlinear behavior and large distortion.
Quasi‐static equilibrium simulation
0 2 4 6 83
3.5
4
4.5
5
5.5
6
Normalized wavelength S
Crit
ical
sw
ellin
g ra
tio c
h0=1m
h0=10m
h0=100m
NΩ = 0.001, = 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).
Nnm
TNkL
B
53.0~A length scale:
10-1 100 101 102 103
100
101
102
103
Initial thickness [m]
Crit
ical
wav
elen
gth
S* [ m
]
L=1mL=0.5mL=0.1m
Nv = 0.001, = 0.4
Characteristic wavelength
1.09.00
* ~ LhS
Kang and Huang, Soft Matter 6, 5736‐5742 (2010).
10-1 100 101 102 1033
3.5
4
4.5
5
5.5
6
Initial thickness [m]
Crit
ical
sw
ellin
g ra
tio c
L=0.1mL=0.53mL=1m
Nv = 0.001, = 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 10-10
0.2
0.4
0.6
0.8
1
Nv
Crit
ical
thic
knes
s h c [
m]
= 0.0 = 0.2 = 0.4 = 0.5
Critical thickness
NL
TNkL
nmTk
L
B
B
'
53.0'
The critical thickness is linearly proportional to L, with the proportionality depending on NΩ and .
Kang and Huang, Soft Matter 6, 5736‐5742 (2010).
Bilayer gels: two types of instability
0 10 20 30 40 50-0.1
-0.08
-0.06
-0.04
-0.02
0
Perturbation wave number H
Crit
ical
che
mic
al p
oten
tial
c / (k
T)
Bilayer A: N2 < N1
Homogeneous layer
Bilayer B: N2 > N1
A
B
Wu, Bouklas and Huang, submitted.
Type A: soft-on-hard bilayer, critical condition at the short wave limit, forming surface creases;
Type B: hard-on-soft bilayer, critical condition at a finite wavelength, forming surface wrinkles first.
Swelling of a gel bilayer (Type A)
Swelling of a gel bilayer (Type B)
Kinetics of Surface Instability
Tanaka et al, 1987
Guvendiren et al., 2010.
iKiK F
Us
0
iK
iK BXs
CU
Diffusion kinetics:
k
L
k
K
BKL x
XxX
TkDCM
LKL
KK
K
XM
XXJ
tC
Nonlinear constitutive relations
Nonlinear kinematics Geometrically nonlinear
kinetics
Nominal stress:
Chemical potential:
Mechanical equilibrium:
Hong, Zhao, Zhou, and Suo, JMPS 2008.
Coupling deformation with kinetics
C1)det(F
Volume change
Linear Poroelasticity
3 material parameters: G, ν, k/η
i
j
j
iij x
uxu
21
0
ij
ij bx
kk xxpk
t
2
kk x
pkJ
0 kk
ijijkkijij pG
212
Biot, 1941; Yoon et al., 2010.
A Transient Finite Element MethoddVrdV
XJdV
tC
K
K
siJ
XJ xidV Bi xidV 0
dStixTdVtrxBC
dVtJCxstt
itt
itt
itt
it
Ktt
Ktt
Jitt
iJ
,,
Weak form:
),()()(),( CFUCUUCU cme FF
A slightly different material model:
21)det(21),( CKCUc FF
CCUU -) (F,) (F, ˆiJ
iJ FUs
ˆ
F;CfCU
F;1 fCL
KLK XMJ
k
L
k
K
BKL x
XxX
TkDCM
Constrained swelling with no instability
FEM vs FDM
Effect of compressibility
As K increases we approach the incompressible case.
4.0,001.0,4.10 N
21)det(21),( CKCUc FF
Numerical convergenceEffect of time stepEffect of element size
4.0,001.0,4.10 N
41 401
Surface instability
By condition of local equilibrium, the compressive stress near the surface becomes very high instantaneously, causing surface instability.
2.0,001.0,4.10 N
Summary
Swell Induced Deformation and Surface Instability of Polymer Gels by Modeling and Simulations
Large deformation: nonlinear constitutive model and nonlinear geometry
Kinetics: geometrically nonlinear diffusion
Instabilities: abundant physical phenomena and numerical challenges (work in progress)
Acknowledgments
• Chad Landis• Min‐Kyoo Kang• Nikolas Bouklas• Zhigen Wu
• Funding by NSF