Abstract

Polyelectrolyte (PE) gels consist of crosslinked polymer networks that are grafted with ionizable groups and ionic solution. Many stimuli-responsive gels, including pH-responsive, electric-responsive, and light-responsive ones, are PE gels. Most soft biological components are also PE gels. Due to the increasing scientific interests and applications of PE gels, a comprehensive model is needed. In PE gels, not only solvent, but also ions and other small molecules all diffuse inside, and the flows of the different components are coupled. This phenomenon is called cross-diffusion, meaning the flow of one species is not only driven by its own chemical potential gradient, but also influenced by the flow of other species. In this work, we develop a rigorous nonequilibrium thermodynamics framework to study the coupled deformation and diffusion of the PE gels where cross-diffusion is emphasized and quantified. Specific forms of free energy and kinetic laws are proposed. A finite element method is developed and implemented into abaqus through a user element subroutine. The model is used to simulate the deformation of biological axon and PE gels.The numerical results are compared with experimental data. It is shown that cross-diffusion generates anomalous effects not only on the flux but also on the deformation of PE gels.

References

References
1.
Brannon-Peppas
,
L.
, and
Peppas
,
N. A.
,
1991
, “
Equilibrium Swelling Behavior of Ph-Sensitive Hydrogels
,”
Chem. Eng. Sci.
,
46
(
3
), pp.
715
722
. 10.1016/0009-2509(91)80177-Z
2.
Horta
,
A.
,
Molina
,
M. J.
,
Gómez-Antón
,
M. R.
, and
Piérola
,
I. F.
,
2009
, “
The Ph Inside a Ph-Sensitive Gel Swollen in Aqueous Salt Solutions: Poly (n-Vinylimidazole)
,”
Macromolecules
,
42
(
4
), pp.
1285
1292
. 10.1021/ma802204b
3.
Marek
,
S. R.
,
Conn
,
C. A.
, and
Peppas
,
N. A.
,
2010
, “
Cationic Nanogels Based on Diethylaminoethyl Methacrylate
,”
Polymer
,
51
(
6
), pp.
1237
1243
. 10.1016/j.polymer.2010.01.060
4.
Jabbari
,
E.
,
Tavakoli
,
J.
, and
Sarvestani
,
A. S.
,
2007
, “
Swelling Characteristics of Acrylic Acid Polyelectrolyte Hydrogel in a DC Electric Field
,”
Smart Mater. Struct.
,
16
(
5
), p.
1614
. 10.1088/0964-1726/16/5/015
5.
Lin
,
J.
,
Tang
,
Q.
,
Hu
,
D.
,
Sun
,
X.
,
Li
,
Q.
, and
Wu
,
J.
,
2009
, “
Electric Field Sensitivity of Conducting Hydrogels With Interpenetrating Polymer Network Structure
,”
Colloids. Surf., A
,
346
(
1–3
), pp.
177
183
. 10.1016/j.colsurfa.2009.06.011
6.
Sumaru
,
K.
,
Takagi
,
T.
,
Sugiura
,
S.
, and
Kanamori
,
T.
,
2014
,
Soft Actuators
,
K.
Asaka
and
H.
Okuzaki
, eds.,
Springer
,
New York
, pp.
219
229
.
7.
Zhang
,
Q. M.
,
Xu
,
W.
, and
Serpe
,
M. J.
,
2014
, “
Optical Devices Constructed From Multiresponsive Microgels
,”
Angew. Chem., Int. Ed.
,
53
(
19
), pp.
4827
4831
. 10.1002/anie.201402641
8.
Suzuki
,
A.
, and
Tanaka
,
T.
,
1990
, “
Phase Transition in Polymer Gels Induced by Visible Light
,”
Nature
,
346
(
6282
), p.
345
. 10.1038/346345a0
9.
Drury
,
J. L.
, and
Mooney
,
D. J.
,
2003
, “
Hydrogels for Tissue Engineering: Scaffold Design Variables and Applications
,”
Biomaterials
,
24
(
24
), pp.
4337
4351
. 10.1016/S0142-9612(03)00340-5
10.
Hu
,
Y.
,
You
,
J.-O.
, and
Aizenberg
,
J.
,
2016
, “
Micropatterned Hydrogel Surface With High-Aspect-Ratio Features for Cell Guidance and Tissue Growth
,”
ACS Appl. Mater. Interfaces
,
8
(
34
), pp.
21939
21945
. 10.1021/acsami.5b12268
11.
Jen
,
A. C.
,
Wake
,
M. C.
, and
Mikos
,
A. G.
,
1996
, “
Hydrogels for Cell Immobilization
,”
Biotechnol. Bioeng.
,
50
(
4
), pp.
357
364
. 10.1002/(SICI)1097-0290(19960520)50:4<357::AID-BIT2>3.0.CO;2-K
12.
Hoare
,
T. R.
, and
Kohane
,
D. S.
,
2008
, “
Hydrogels in Drug Delivery: Progress and Challenges
,”
Polymer
,
49
(
8
), pp.
1993
2007
. 10.1016/j.polymer.2008.01.027
13.
Li
,
J.
, and
Mooney
,
D. J.
,
2016
, “
Designing Hydrogels for Controlled Drug Delivery
,”
Nat. Rev. Mater.
,
1
(
12
), p.
16071
. 10.1038/natrevmats.2016.71
14.
Qiu
,
Y.
, and
Park
,
K.
,
2001
, “
Environment-Sensitive Hydrogels for Drug Delivery
,”
Adv. Drug Delivery Rev.
,
53
(
3
), pp.
321
339
. 10.1016/S0169-409X(01)00203-4
15.
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
,
404
(
6778
), p.
588
. 10.1038/35007047
16.
Dong
,
L.
,
Agarwal
,
A. K.
,
Beebe
,
D. J.
, and
Jiang
,
H.
,
2006
, “
Adaptive Liquid Microlenses Activated by Stimuli-Responsive Hydrogels
,”
Nature
,
442
(
7102
), p.
551
. 10.1038/nature05024
17.
Gerlach
,
G.
,
Guenther
,
M.
,
Sorber
,
J.
,
Suchaneck
,
G.
,
Arndt
,
K.-F.
, and
Richter
,
A.
,
2005
, “
Chemical and pH Sensors Based on the Swelling Behavior of Hydrogels
,”
Sens. Actuators, B
,
111
, pp.
555
561
. 10.1016/j.snb.2005.03.040
18.
Richter
,
A.
,
Paschew
,
G.
,
Klatt
,
S.
,
Lienig
,
J.
,
Arndt
,
K.-F.
, and
Adler
,
H.-J.
,
2008
, “
Review on Hydrogel-Based Ph Sensors and Microsensors
,”
Sensors
,
8
(
1
), pp.
561
581
. 10.3390/s8010561
19.
Maeda
,
S.
,
Hara
,
Y.
,
Sakai
,
T.
,
Yoshida
,
R.
, and
Hashimoto
,
S.
,
2007
, “
Self-Walking Gel
,”
Adv. Mater.
,
19
(
21
), pp.
3480
3484
. 10.1002/adma.200700625
20.
Kreuer
,
K.
,
2001
, “
On the Development of Proton Conducting Polymer Membranes for Hydrogen and Methanol Fuel Cells
,”
J. Membr. Sci.
,
185
(
1
), pp.
29
39
. 10.1016/S0376-7388(00)00632-3
21.
Keplinger
,
C.
,
Sun
,
J.-Y.
,
Foo
,
C. C.
,
Rothemund
,
P.
,
Whitesides
,
G. M.
, and
Suo
,
Z.
,
2013
, “
Stretchable, Transparent, Ionic Conductors
,”
Science
,
341
(
6149
), pp.
984
987
. 10.1126/science.1240228
22.
Yang
,
C. H.
,
Chen
,
B.
,
Lu
,
J. J.
,
Yang
,
J. H.
,
Zhou
,
J.
,
Chen
,
Y. M.
, and
Suo
,
Z.
,
2015
, “
Ionic Cable
,”
Extreme Mech. Lett.
,
3
, pp.
59
65
. 10.1016/j.eml.2015.03.001
23.
Kim
,
C.-C.
,
Lee
,
H.-H.
,
Oh
,
K. H.
, and
Sun
,
J.-Y.
,
2016
, “
Highly Stretchable, Transparent Ionic Touch Panel
,”
Science
,
353
(
6300
), pp.
682
687
. 10.1126/science.aaf8810
24.
Xue
,
S.-L.
,
Li
,
B.
,
Feng
,
X.-Q.
, and
Gao
,
H.
,
2017
, “
A Non-Equilibrium Thermodynamic Model for Tumor Extracellular Matrix With Enzymatic Degradation
,”
J. Mech. Phys. Solids
,
104
, pp.
32
56
. 10.1016/j.jmps.2017.04.002
25.
Swartz
,
M. A.
, and
Fleury
,
M. E.
,
2007
, “
Interstitial Flow and Its Effects in Soft Tissues
,”
Annu. Rev. Biomed. Eng.
,
9
, pp.
229
256
. 10.1146/annurev.bioeng.9.060906.151850
26.
Levick
,
J.
,
1987
, “
Flow Through Interstitium and Other Fibrous Matrices
,”
Q. J. Exp. Physiol.
,
72
(
4
), pp.
409
437
. 10.1113/expphysiol.1987.sp003085
27.
Frank
,
E. H.
, and
Grodzinsky
,
A. J.
,
1987
, “
Cartilage Electromechanics—I. Electrokinetic Transduction and the Effects of Electrolyte Ph and Ionic Strength
,”
J. Biomech.
,
20
(
6
), pp.
615
627
. 10.1016/0021-9290(87)90282-X
28.
Fraldi
,
M.
, and
Carotenuto
,
A. R.
,
2018
, “
Cells Competition in Tumor Growth Poroelasticity
,”
J. Mech. Phys. Solids
,
112
, pp.
345
367
. 10.1016/j.jmps.2017.12.015
29.
Krishna
,
R.
, and
Wesselingh
,
J.
,
1997
, “
The Maxwell-Stefan Approach to Mass Transfer
,”
Chem. Eng. Sci.
,
52
(
6
), pp.
861
911
. 10.1016/S0009-2509(96)00458-7
30.
Lai
,
W. M.
,
Hou
,
J.
, and
Mow
,
V. C.
,
1991
, “
A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage
,”
ASME J. Biomech. Eng.
,
113
(
3
), pp.
245
258
. 10.1115/1.2894880
31.
Maxwell
,
J. C.
,
1867
, “
IV. on the Dynamical Theory of Gases
,”
Philos. Trans. R. Soc. London
,
157
, pp.
49
88
. 10.1098/rstl.1867.0004
32.
Stefan
,
J.
,
1871
, “
Über Das Gleichgewicht Und Die Bewegung Insbesondere Die Diffusion Von Gasgemengen
,”
Sitzungsber. Akad. Wiss. Wien
,
63
(
2
), pp.
63
124
. Cited By 167.
33.
Vanag
,
V. K.
, and
Epstein
,
I. R.
,
2009
, “
Cross-Diffusion and Pattern Formation in Reaction–Diffusion Systems
,”
Phys. Chem. Chem. Phys.
,
11
(
6
), pp.
897
912
. 10.1039/B813825G
34.
Grim
,
E.
, and
Sollner
,
K.
,
1957
, “
The Contributions of Normal and Anomalous Osmosis to the Osmotic Effects Arising Across Charged Membranes With Solutions of Electrolytes
,”
J. Gen. Physiol.
,
40
(
6
), pp.
887
899
. 10.1085/jgp.40.6.887
35.
Toyoshima
,
Y.
,
Kobatake
,
Y.
, and
Fujita
,
H.
,
1967
, “
Studies of Membrane Phenomena. Part 5.—Bulk Flow Through Membrane
,”
Trans. Faraday Soc.
,
63
, pp.
2828
2838
. 10.1039/TF9676302828
36.
Sasidhar
,
V.
, and
Ruckenstein
,
E.
,
1982
, “
Anomalous Effects During Electrolyte Osmosis Across Charged Porous Membranes
,”
J. Colloid Interface Sci.
,
85
(
2
), pp.
332
362
. 10.1016/0021-9797(82)90003-0
37.
Gu
,
W.
,
Lai
,
W.
, and
Mow
,
V.
,
1997
, “
A Triphasic Analysis of Negative Osmotic Flows Through Charged Hydrated Soft Tissues
,”
J. Biomech.
,
30
(
1
), pp.
71
78
. 10.1016/S0021-9290(96)00099-1
38.
Gu
,
W.
,
Lai
,
W.
, and
Mow
,
V.
,
1998
, “
A Mixture Theory for Charged-Hydrated Soft Tissues Containing Multi-Electrolytes: Passive Transport and Swelling Behaviors
,”
ASME J. Biomech. Eng.
,
120
(
2
), pp.
169
180
. 10.1115/1.2798299
39.
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
,
56
(
5
), pp.
1779
1793
. 10.1016/j.jmps.2007.11.010
40.
Hong
,
W.
,
Zhao
,
X.
, and
Suo
,
Z.
,
2010
, “
Large Deformation and Electrochemistry of Polyelectrolyte Gels
,”
J. Mech. Phys. Solids
,
58
(
4
), pp.
558
577
. 10.1016/j.jmps.2010.01.005
41.
Beltran
,
S.
,
Baker
,
J. P.
,
Hooper
,
H. H.
,
Blanch
,
H. W.
, and
Prausnitz
,
J. M.
,
1991
, “
Swelling Equilibria for Weakly Ionizable, Temperature-Sensitive Hydrogels
,”
Macromolecules
,
24
(
2
), pp.
549
551
. 10.1021/ma00002a032
42.
Yadav
,
M.
, and
Rhee
,
K. Y.
,
2012
, “
Superabsorbent Nanocomposite (Alginate-g-Pamps/Mmt): Synthesis, Characterization and Swelling Behavior
,”
Carbohydr. Polym.
,
90
(
1
), pp.
165
173
. 10.1016/j.carbpol.2012.05.010
43.
Tanaka
,
T.
,
Fillmore
,
D.
,
Sun
,
S.-T.
,
Nishio
,
I.
,
Swislow
,
G.
, and
Shah
,
A.
,
1980
, “
Phase Transitions in Ionic Gels
,”
Phys. Rev. Lett.
,
45
(
20
), p.
1636
. 10.1103/PhysRevLett.45.1636
44.
Tanaka
,
T.
, and
Fillmore
,
D. J.
,
1979
, “
Kinetics of Swelling of Gels
,”
J. Chem. Phys.
,
70
(
3
), pp.
1214
1218
. 10.1063/1.437602
45.
Dehghany
,
M.
,
Zhang
,
H.
,
Naghdabadi
,
R.
, and
Hu
,
Y.
,
2018
, “
A Thermodynamically-Consistent Large Deformation Theory Coupling Photochemical Reaction and Electrochemistry for Light-Responsive Gels
,”
J. Mech. Phys. Solids
,
116
, pp.
239
266
. 10.1016/j.jmps.2018.03.018
46.
Coleman
,
B. D.
, and
Noll
,
W.
,
1963
, “
The Thermodynamics of Elastic Materials With Heat Conduction and Viscosity
,”
Arch. Ration. Mech. Anal.
,
13
(
1
), pp.
167
178
. 10.1007/BF01262690
47.
De Groot
,
S. R.
, and
Mazur
,
P.
,
2013
,
Non-Equilibrium Thermodynamics
,
Courier Corporation
,
New York
.
48.
Onsager
,
L.
,
1931
, “
Reciprocal Relations in Irreversible Processes. II.
,”
Phys. Rev.
,
38
(
12
), p.
2265
. 10.1103/PhysRev.38.2265
49.
Flory
,
P. J.
,
1942
, “
Thermodynamics of High Polymer Solutions
,”
J. Chem. Phys.
,
10
(
1
), pp.
51
61
. 10.1063/1.1723621
50.
Flory
,
P. J.
, and
Rehner, Jr.
,
J.
,
1943
, “
Statistical Mechanics of Cross-Linked Polymer Networks I. Rubberlike Elasticity
,”
J. Chem. Phys.
,
11
(
11
), pp.
512
520
. 10.1063/1.1723791
51.
Flory
,
P.
,
1953
,
Principles of Polymer Chemistry
,
Cornell University Press
,
Ithaca, NY
.
52.
Huggins
,
M. L.
,
1941
, “
Solutions of Long Chain Compounds
,”
J. Chem. Phys.
,
9
(
5
), pp.
440
440
. 10.1063/1.1750930
53.
Debanne
,
D.
,
2004
, “
Information Processing in the Axon
,”
Nat. Rev. Neurosci.
,
5
(
4
), p.
304
. 10.1038/nrn1397
54.
García-Grajales
,
J. A.
,
Jérusalem
,
A.
, and
Goriely
,
A.
,
2017
, “
Continuum Mechanical Modeling of Axonal Growth
,”
Comput. Methods Appl. Mech. Eng.
,
314
, pp.
147
163
. 10.1016/j.cma.2016.07.032
55.
Lodish
,
H.
,
Berk
,
A.
,
Kaiser
,
C. A.
,
Krieger
,
M.
,
Scott
,
M. P.
,
Bretscher
,
A.
,
Ploegh
,
H.
, and
Matsudaira
,
P.
,
2008
,
Molecular Cell Biology
,
Macmillan
,
New York
.
56.
Pinto
,
T. M.
,
Wedemann
,
R. S.
, and
Cortez
,
C. M.
,
2014
, “
Modeling the Electric Potential Across Neuronal Membranes: the Effect of Fixed Charges on Spinal Ganglion Neurons and Neuroblastoma Cells
,”
PLoS One
,
9
(
5
), p.
e96194
. 10.1371/journal.pone.0096194
57.
Sun
,
Y.
,
Liu
,
S.
,
Du
,
G.
,
Gao
,
G.
, and
Fu
,
J.
,
2015
, “
Multi-Responsive and Tough Hydrogels Based on Triblock Copolymer Micelles As Multi-Functional Macro-Crosslinkers
,”
Chem. Commun.
,
51
(
40
), pp.
8512
8515
. 10.1039/C4CC10094H
58.
Bouklas
,
N.
,
Landis
,
C. M.
, and
Huang
,
R.
,
2015
, “
A Nonlinear, Transient Finite Element Method for Coupled Solvent Diffusion and Large Deformation of Hydrogels
,”
J. Mech. Phys. Solids
,
79
, pp.
21
43
. 10.1016/j.jmps.2015.03.004
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