A hydrogel consists of a cross-linked polymer network and solvent molecules. Depending on its chemical and mechanical environment, the polymer network may undergo enormous volume change. The present work develops a general formulation based on a variational approach, which leads to a set of governing equations coupling mechanical and chemical equilibrium conditions along with proper boundary conditions. A specific material model is employed in a finite element implementation, for which the nonlinear constitutive behavior is derived from a free energy function, with explicit formula for the true stress and tangent modulus at the current state of deformation and chemical potential. Such implementation enables numerical simulations of hydrogels swelling under various constraints. Several examples are presented, with both homogeneous and inhomogeneous swelling deformation. In particular, the effect of geometric constraint is emphasized for the inhomogeneous swelling of surface-attached hydrogel lines of rectangular cross sections, which depends on the width-to-height aspect ratio of the line. The present numerical simulations show that, beyond a critical aspect ratio, creaselike surface instability occurs upon swelling.

1.
Galaev
,
I. Y.
, and
Mattiasson
,
B.
, 1999, “
‘Smart’ Polymers and What They Could Do in Biotechnology and Medicine
,”
Trends Biotechnol.
0167-7799,
17
(
8
), pp.
335
340
.
2.
Peppas
,
N. A.
,
Hilt
,
J. Z.
,
Khademhosseini
,
A.
, and
Langer
,
R.
, 2006, “
Hydrogels in Biology and Medicine: From Molecular Principles to Bionanotechnology
,”
Adv. Mater.
0935-9648,
18
(
11
), pp.
1345
1360
.
3.
Ulijn
,
R. V.
,
Bibi
,
N.
,
Jayawarna
,
V.
,
Thornton
,
P. D.
,
Todd
,
S. J.
,
Mart
,
R. J.
,
Smith
,
A. M.
, and
Gough
,
J. E.
, 2007, “
Bioresponsive Hydrogels
,”
Mater. Today
1369-7021,
10
(
4
), pp.
40
48
.
4.
Tokarev
,
I.
, and
Minko
,
S.
, 2009, “
Stimuli-Responsive Hydrogel Thin Films
,”
Soft Matter
1744-683X,
5
, pp.
511
524
.
5.
Beebe
,
D. J.
,
Moore
,
J. S.
,
Bauer
,
J. M.
,
Yu
,
Q.
,
Liu
,
R. H.
,
Devadoss
,
C.
, and
Jo
,
B. -H.
, 2000, “
Functional Hydrogel Structures for Autonomous Flow Control Inside Microfluidic Channels
,”
Nature (London)
0028-0836,
404
, pp.
588
590
.
6.
Dong
,
L.
,
Agarwal
,
A. K.
,
Beebe
,
D. J.
, and
Jiang
,
H.
, 2006, “
Adaptive Liquid Microlenses Activated by Stimuli-Responsive Hydrogels
,”
Nature (London)
0028-0836,
442
, pp.
551
554
.
7.
Sidorenko
,
A.
,
Krupenkin
,
T.
,
Taylor
,
A.
,
Fratzl
,
P.
, and
Aizenberg
,
J.
, 2007, “
Reversible Switching of Hydrogel-Actuated Nanostructures Into Complex Micropatterns
,”
Science
0036-8075,
315
, pp.
487
490
.
8.
Kuhn
,
W.
,
Hargitay
,
B.
,
Katchalsky
,
A.
, and
Eisenberg
,
H.
, 1950, “
Reversible Dilation and Contraction by Changing the State of Ionization of High-Polymer Acid Networks
,”
Nature (London)
0028-0836,
165
, pp.
514
516
.
9.
Southern
,
E.
, and
Thomas
,
A. G.
, 1965, “
Effect of Constraints on the Equilibrium Swelling of Rubber Vulcanizates
,”
J. Polym. Sci. A
0449-2951,
3
, pp.
641
646
.
10.
Tanaka
,
T.
,
Sun
,
S. -T.
,
Hirokawa
,
Y.
,
Katayama
,
S.
,
Kucera
,
J.
,
Hirose
,
Y.
, and
Amiya
,
T.
, 1987, “
Mechanical Instability of Gels at the Phase Transition
,”
Nature (London)
0028-0836,
325
, pp.
796
798
.
11.
Matsuo
,
E. S.
, and
Tanaka
,
T.
, 1992, “
Patterns in Shrinking Gels
,”
Nature (London)
0028-0836,
358
, pp.
482
485
.
12.
Tanaka
,
H.
, and
Sigehuzi
,
T.
, 1994, “
Surface-Pattern Evolution in a Swelling Gel Under a Geometrical Constraint: Direction Observation of Fold Structure and Its Coarsening Dynamics
,”
Phys. Rev. E
1063-651X,
49
, pp.
R39
R42
.
13.
Tirumala
,
V. R.
,
Divan
,
R.
,
Ocola
,
L. E.
, and
Mancini
,
D. C.
, 2005, “
Direct-Write E-Beam Patterning of Stimuli-Responsive Hydrogel Nanostructures
,”
J. Vac. Sci. Technol. B
1071-1023,
23
(
6
), pp.
3124
3128
.
14.
Mora
,
T.
, and
Boudaoud
,
A.
, 2006, “
Buckling of Swelling Gels
,”
Eur. Phys. J. E
1292-8941,
20
(
2
), pp.
119
124
.
15.
Sultan
,
E.
, and
Boudaoud
,
A.
, 2008, “
The Buckling of a Swollen Thin Gel Layer Bound to a Compliant Substrate
,”
ASME J. Appl. Mech.
0021-8936,
75
, p.
051002
.
16.
Trujillo
,
V.
,
Kim
,
J.
, and
Hayward
,
R. C.
, 2008, “
Creasing Instability of Surface-Attached Hydrogels
,”
Soft Matter
1744-683X,
4
, pp.
564
569
.
17.
Zhang
,
Y.
,
Matsumoto
,
E. A.
,
Peter
,
A.
,
Lin
,
P. -C.
,
Kamien
,
R. D.
, and
Yang
,
S.
, 2008, “
One-Step Nanoscale Assembly of Complex Structures Via Harnessing of an Elastic Instability
,”
Nano Lett.
1530-6984,
8
, pp.
1192
1196
.
18.
Li
,
Y.
, and
Tanaka
,
T.
, 1992, “
Phase Transitions of Gels
,”
Annu. Rev. Mater. Sci.
0084-6600,
22
, pp.
243
277
.
19.
Onuki
,
A.
, 1989, “
Theory of Pattern Formation in Gels: Surface Folding in Highly Compressible Elastic Bodies
,”
Phys. Rev. A
1050-2947,
39
(
11
), pp.
5932
5948
.
20.
Suematsu
,
N.
,
Sekimoto
,
N.
, and
Kawasaki
,
K.
, 1990, “
Three-Dimensional Computer Modeling for Pattern Formation on the Surface of an Expanding Polymer Gel
,”
Phys. Rev. A
1050-2947,
41
(
10
), pp.
5751
5754
.
21.
Durning
,
C. J.
, and
Morman
,
K. N.
, Jr.
, 1993, “
Nonlinear Swelling of Polymer Gels
,”
J. Chem. Phys.
0021-9606,
98
(
5
), pp.
4275
4293
.
22.
Wilder
,
J.
, and
Vilgis
,
T. A.
, 1998, “
Elasticity in Strongly Interacting Soft Solids: A Polyelectolyte Network
,”
Phys. Rev. E
1063-651X,
57
, pp.
6865
6874
.
23.
Maskawa
,
J.
,
Takeuchi
,
T.
,
Maki
,
K.
,
Tsujii
,
K.
, and
Tanaka
,
T.
, 1999, “
Theory and Numerical Calculation of Pattern Formation in Shrinking Gels
,”
J. Chem. Phys.
0021-9606,
110
(
22
), pp.
10993
10999
.
24.
Boudaoud
,
A.
, and
Chaieb
,
S.
, 2003, “
Mechanical Phase Diagram of Shrinking Cylindrical Gels
,”
Phys. Rev. E
1063-651X,
68
, p.
021801
.
25.
Dolbow
,
J.
,
Fried
,
E.
, and
Jia
,
H. D.
, 2004, “
Chemically Induced Swelling of Hydrogels
,”
J. Mech. Phys. Solids
0022-5096,
52
, pp.
51
84
.
26.
Li
,
H.
,
Luo
,
R.
,
Birgersson
,
E.
, and
Lam
,
K. Y.
, 2007, “
Modeling of Multiphase Smart Hydrogels Responding to pH and Electric Voltage Coupled Stimuli
,”
J. Appl. Phys.
0021-8979,
101
(
11
), p.
114905
.
27.
Hong
,
W.
,
Zhao
,
X.
,
Zhou
,
J.
, and
Suo
,
Z.
, 2008, “
A Theory of Coupled Diffusion and Large Deformation in Polymeric Gels
,”
J. Mech. Phys. Solids
0022-5096,
56
, pp.
1779
1793
.
28.
Zhao
,
X.
,
Hong
,
W.
, and
Suo
,
Z.
, 2008, “
Stretching and Polarizing a Dielectric Gel Immersed in a Solvent
,”
Int. J. Solids Struct.
0020-7683,
45
, pp.
4021
4031
.
29.
Hong
,
W.
,
Liu
,
Z.
, and
Suo
,
Z.
, 2009, “
Inhomogeneous Swelling of a Gel in Equilibrium With a Solvent and Mechanical Load
,”
Int. J. Solids Struct.
0020-7683,
46
, pp.
3282
3289
.
30.
Zhang
,
J.
,
Zhao
,
X.
,
Suo
,
Z.
, and
Jiang
,
H.
, 2009, “
A Finite Element Method for Transient Analysis of Concurrent Large Deformation and Mass Transport in Gels
,”
J. Appl. Phys.
0021-8979,
105
, p.
093522
.
31.
Gibbs
,
J. W.
, 1878,
The Scientific Papers of J. Willard Gibbs, Vol. 1: Thermodynamics
,
Ox Bow Press
,
Woodbridge, CT
, pp.
184
, 201, and 215.
32.
Biot
,
M. A.
, 1941, “
General Theory of Three-Dimensional Consolidation
,”
J. Appl. Phys.
0021-8979,
12
, pp.
155
164
.
33.
Biot
,
M. A.
, 1973, “
Nonlinear and Semilinear Rheology of Porous Solids
,”
J. Geophys. Res.
0148-0227,
78
, pp.
4924
4937
.
34.
Prigogine
,
I.
, 1967,
Introduction to Thermodynamics of Irreversible Processes
, 3rd ed.,
Wiley
,
New York
.
35.
Flory
,
P. J.
, 1953,
Principles of Polymer Chemistry
,
Cornell University Press
,
Ithaca, NY
.
36.
Flory
,
P. J.
, 1950, “
Statistical Mechanics of Swelling of Network Structures
,”
J. Chem. Phys.
0021-9606,
18
, pp.
108
111
.
37.
Treloar
,
L. R. G.
, 1975,
The Physics of Rubber Elasticity
,
Oxford University Press
,
Oxford
.
38.
Rivlin
,
R. S.
, 1948, “
Large Elastic Deformations of Isotropic Materials. I. Fundamental Concepts
,”
Philos. Trans. R. Soc. London, Ser. A
0962-8428,
240
, pp.
459
490
.
39.
Simo
,
J. C.
, and
Pister
,
K. S.
, 1984, “
Remarks on Rate Constitutive Equations for Finite Deformation Problems: Computational Implications
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
46
, pp.
201
215
.
40.
Ogden
,
R. W.
, 1976, “
Volume Changes Associated With the Deformation of Rubber-Like Solids
,”
J. Mech. Phys. Solids
0022-5096,
24
(
6
), pp.
323
338
.
41.
Bischoff
,
J. E.
,
Arruda
,
E. M.
, and
Grosh
,
K.
, 2001, “
A New Constitutive Model for the Compressibility of Elastomers at Finite Deformations
,”
Rubber Chem. Technol.
0035-9475,
74
(
4
), pp.
541
559
.
42.
Boyce
,
M. C.
, and
Arruda
,
E. M.
, 2000, “
Constitutive Models of Rubber Elasticity: A Review
,”
Rubber Chem. Technol.
0035-9475,
73
(
3
), pp.
504
523
.
43.
Kuhn
,
W.
,
Pasternak
,
R.
, and
Kuhn
,
H.
, 1947, “
Mechanische und optische eigenschaften von gequollenem kautschuk
,”
Helv. Chim. Acta
0018-019X,
30
, pp.
1705
1740
.
44.
Hermans
,
J. J.
, 1947, “
Deformation and Swelling of Polymer Networks Containing Comparatively Long Chains
,”
Trans. Faraday Soc.
0014-7672,
43
, pp.
591
600
.
45.
Huggins
,
M. L.
, 1941, “
Solutions of Long Chain Compounds
,”
J. Chem. Phys.
0021-9606,
9
(
5
), p.
440
.
46.
Belytschko
,
T.
,
Liu
,
W. K.
, and
Moran
,
B.
, 2000,
Nonlinear Finite Elements for Continua and Structures
,
Wiley
,
New York
.
47.
ABAQUS version 6.8, 2008, Dassault Systèmes Simulia Corp., Providence, RI.
48.
Hartschuh
,
R. D.
,
Kisliuk
,
A.
,
Novikov
,
V.
,
Sokolov
,
A. P.
,
Heyliger
,
P. R.
,
Flannery
,
C. M.
,
Johnson
,
W. L.
,
Soles
,
C. L.
, and
Wu
,
W. -L.
, 2005, “
Acoustic Modes and Elastic Properties of Polymeric Nanostructures
,”
Appl. Phys. Lett.
0003-6951,
87
, p.
173121
.
49.
Gent
,
A. N.
, and
Cho
,
I. S.
, 1999, “
Surface Instabilities in Compressed or Bent Rubber Blocks
,”
Rubber Chem. Technol.
0035-9475,
72
, pp.
253
262
.
50.
Ghatak
,
A.
, and
Das
,
A. L.
, 2007, “
Kink Instability of a Highly Deformable Elastic Cylinder
,”
Phys. Rev. Lett.
0031-9007,
99
, p.
076101
.
51.
Biot
,
M. A.
, 1963, “
Surface Instability of Rubber in Compression
,”
Appl. Sci. Res., Sect. A
0365-7132,
12
, pp.
168
182
.
52.
Hohlfeld
,
E. B.
, 2008, “
Creasing, Point-Bifurcations, and Spontaneous Breakdown of Scale-Invariance
,” Ph.D. thesis, Harvard University, Cambridge, MA.
53.
Hong
,
W.
,
Zhao
,
X.
, and
Suo
,
Z.
, 2009, “
Formation of Creases on the Surfaces of Elastomers and Gels
,”
Appl. Phys. Lett.
0003-6951,
95
, p.
111901
.
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