Soft Active Materials
II. Gels
Zhigang Suo
Harvard University
1
gel = network + solvent
solvent
gel
network
Solid-like: long polymers crosslinked by strong bonds. Retain shape
Liquid like: polymers and solvent aggregate by weak bonds. Enable transport
•Imbibe solvent, and swell
•Exude solvent, and collapse
Ono, Sugimoto, Shinkai, Sada, Nature Materials 6, 429 (2007)
2
Gels in daily life
food
plants
Contact lenses
Wichterie, Lim, Nature 185, 117 (1960)
Drug delivery
Ulijn et al. Materials Today 10, 40 (2007)
Tissues, natural or engineered
3
Gels in engineering
Responsive to physiological variables:
•pH
•concentration of ions
•temperature
•light
Gate in microfluidics
Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)
packer in oil well
Shell, 2003
4
Chemical potential
of a molecular species in a system
vapor
wine
cheese
•A molecular species: water, or a type of alcohol, or a type of protein
•A system: the wine, or the cheese, or the vapor, or the composite
•Number of water molecules in a system, M
F M ,...
•Helmholtz free energy of a system, F(M,…)

•Chemical potential of water in a system,
M
•In equilibrium, the chemical potential of water in the wine equals the chemical
potential of water in the cheese, and equals the chemical potential of water in
the vapor.
•Measure the chemical potential of water in one system by equilibrating this
system with another system in which the chemical potential of water is known.
5
Chemical potential
of a species in a one-species system
weights
piston
vapor
vapor
p  p0
p0
liquid
State of reference
liquid & vapor in equilibrium
 0
liquid
p  p0
liquid only
vapor only
   p  p0 v
  kT log  p / p0 
Relative humidity, p / p0
p0 Pressure of saturated vapor
v volume per solvent molecule
6
The ascent of sap
Henry Horatio Dixon
•The air is dry.
•The soil is wet.
•The gradient of humidity pumps water up.
  kT log pair / p0   vgh
  kT log psoil / p0 
7
Dixon and Joly, Phil. Trans. R. Soc. Lond. B 186, 563 (1895)
Liquid water in tension
Chemical potential
of water in the air
 pair
 p0
  kT log



water-permeable solid
xylem
 pair 
kT

s   log
v
 p0 
water
s
Chemical potential
of water in the xylem
When the liquid water in the void
equilibrate with
the water vapor in the air,
  vs
Liquid water in tension
tensile stress in liquid water in the void (MPa)
10
20
30
 pair 
kT

s   log
v
 p0 
•Axisymmertic deformation
•Creasing
•Cavitation
relative humidity in the air, pair / p0
Wheeler, Stroock. Nature 455, 208 (2008)
40
Abe Stroock
Osmosis in a liquid
solute
Number of solute molecules, N
Volume of the solution, V
Volume per solvent molecule, v
h
solution
membrane
osmotic pressure
N
kT
V
Van’t Hoff
solvent
Chemical potential
of water in the solution
due to osmosis
N
 v kT
V
p
Chemical potential
of water in the solution
due to gravity
 vgh
Chemical potential
of water in the solvent
0
10
Jacobus H. Van’t Hoff, Osmotic pressure and chemical equilibrium, Nobel Lecture, 13 December 1901
Osmosis in a gel
Crosslink
solution
membrane
solvent
Osmosis balanced by elasticity
Osmosis balanced by gravity
11
Two ways of doing work
M work done by the pump
Pump

Chemical potential of solvent
Pl work done by the weight
F l , M  Helmholtz free energy of gel
Condition of equilibrium
F  Pl  M
Solvent
M
Gel = Network +
M solvent molecules
Equations of state
P
F l , M 
l

F l , M 
M
l
Weight
P Force
12
Gels as transducers
• A swelling gel is blocked by a hard material.
• Blocking force.
• Inhomogeneous deformation.
Need to formulate boundary-value problems 13
2 ways of doing work
Reference state
Current state
M=0
Equilibrium condition
AL

Helmholtz free energy per volume

s
C
gel
P l
M

AL
AL
W  s  C
W  , C 
M
l
F
pump
Pump,

L
F  Pl  M
A
M
l
L
P
A
M
AL
l
P
weight
14
Inhomogeneous field
Deformation gradient
FiK 
Concentration
Free-energy function
C X,t 
W F, C 
x i X ,t 
X K
Gibbs (1878)
Condition of equilibrium
 WdV   Bixi dV   Tixi dA    CdV
  constant
pump
Gel
1. How to prescribe W F, C  ?
2. How to solve boundary-value problems?
3. What boundary-value problems to solve?
Weight
15
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
Free-energy function
•Swelling decreases entropy by straightening polymers.
•Swelling increases entropy by mixing solvent and polymers.
Paul Flory
Free-energy function
Free energy of stretching
Free energy of mixing
W F, C   Ws F   Wm C 
Ws F   21 NkTFiK FiK  3  2 logdet F 
Wm C   
kT
v

1 
χ 


vC log 1 

 vC  1  vC 

16
Flory, Rehner, J. Chem. Phys., 11, 521 (1943)
Molecular incompressibility
+
Vdry +
=
Vsol =
Vgel
1  vC  det F
Wei Hong
v – volume per solvent molecule
Assumptions:
– Individual solvent molecule and polymer are incompressible.
– Neglect volumetric change due to physical association
– Gel has no voids. (a gel is different from a sponge.)
17
Stress-strain relation
s1 
s2 
s3 

kT
 
v v
NkT 21  1 

NkT 22  1 

2 3 3
2 3 3
NkT 23  1
2 3 3

 

1 
1




 log 1 
       2 



 

2 3 3 
2 3 3
2 3 3
18
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
Anisotropic swelling
Free swelling
Unidirectional swelling
3.5
6
analytical
FEM
5.5
λ0 H
5
3
s
4
2.5
λ1 H

0
4.5
3.5
L
λ0L
3
2
2.5
2
1.5
-0.05
-0.04
-0.03
-0.02
 0/kT
-0.01
0
1.5
-0.05
-0.04
-0.03
-0.02
 /kT
-0.01
0
A gel imbibes a different amount of solvent under constraint.
fr ee3  u n idir ectoni a l
Treloar, Trans. Faraday Soc. 46, 783 (1950).
19
Inhomogeneous swelling
Core
Empty space
R
A0
r
A
B
Dry polymer
b
Gel
(a)
Reference State
(b)
Equilibrium State

Concentration is inhomogeneous even in equilibrium.

Stress is high near the interface (debond, cavitation).
Sternstein, J. Macromolol. Soc. Phys. B6, 243 (1972)
Zhao, Hong, Suo, APL 92, 051904 (2008 )
20
Finite element method
Equilibrium condition
 WdV   B x dV   T x dA    CdV
Legendre transform
ˆ F,    W  C
W
i
i
i
i
 WˆdV   Bixi dV   Tixi dA
•A gel in equilibrium is analogous to an elastic solid
•ABAQUS UMAT
Hong, Liu, Suo, Int. J. Solids Structures 46, 3282 (2009)
ABAQUS UMAT posted online http://imechanica.org/node/3163
21
Swelling-induced buckling
Ri / H  5
Ri / H  7
Zishun Liu
Ri / H  10
Liu, Hong, Suo, Swaddiwudhipong, Zhang.
Computational Materials Science. In press.
Ri / H  12
22
Gel and nano-rods
Joanna Aizenberg
 = 1.5
0

-0.3
 = 0.1
Nv = 10-3
RH = 80%
vG/kT
-0.4
-0.5
-0.6
-0.7
RH = 95%
-0.8
RH = 100%
-0.9
-1
-0.5
Experiment: Sidorenko, Krupenin, Taylor, Fratzl, Aizenberg, Science 315, 487 (2007).
Theory: Hong, Zhao, Suo, JAP104, 084905 (2008) .
0
 (rad)
0.5
1
23
Critical humidity can be tuned
1.4
1.2
 = 1.7
 (rad)
1
0
0.8
0.6
1.2
0.4
0.2
0
0
1.1
 = 0.1
Nv = 10-3
20
40
60
Relative humidity (%)
80
100
24
Swelling-induced bifurcation
Shu Yang
25
Zhang, Matsumoto, Peter, Lin, Kamien, Yang, Nano Lett. 8, 1192 (2008).
Swelling-induced bifurcation
Experiment: Zhang, Matsumoto, Peter, Lin, Kamien, Yang, Nano Lett. 8, 1192 (2008).
Simulation: Hong, Liu, Suo, Int. J. Solids Structures 46, 3282 (2009)
26
Time-dependent process
Shape change: short-range motion of solvent molecules, fast
Volume change: long-range motion of solvent molecules, slow
27
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
Concurrent deformation and migration
Deformation of network
xi X, t 
FiK 
X K
Conservation of solvent
C X, t  J K X, t  r X, t 


t
X K
t
Maurice Biot
JAP 12, 155 (1941)
pump
i X,t 
JK N K  
t
Gel
Nonequilibrium thermodynamics
 WdV   B x dV   T x dA   rdV   idA
i
i
i
i
Rate process
Local equilibrium
W F, C 
siK 
FiK

W F, C 
C
Weight
siK X, t 
 Bi  0
X K
siK N K  Ti
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
J K   M KL
 X, t 
X L
28
ideal kinetic model
Solvent molecules migrate in a gel by self-diffusion
J K   M KL

X L
Diffusion in true quantities
ji  
cD μ
kT x i
Conversion between true and nominal quantities
ji 



FiK
X K xi
FiK
JK
det F
M KL
D

H iK H iL det F  1
vkT
29
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
Finite element method
for concurrent
Z Y migration
deformation and
Frame 001  28 Jul 2008 
X
X
Hanqing Jiang
Z Y
X
Dt L2  0
Dt L2  10
Dt L2  
30
Zhang, Zhao, Suo, Jiang, JAP 105, 093522 (2009)
Alginate Hydrogels
David Mooney
Ionic crosslinks
Ca++( )
Na+
Covalent crosslinks
AAD
Irreversible
31
strain
Stress-relaxation test
PBS
Gel disk
Impermeable
time
stress
Rigid plates
time
32
Zhao, Huebsch, Mooney, Suo, J. Appl. Phys. In press.
Elasticity, plasticity, fracture
Ionic
crosslinks
10mm
Swollen state
45% compressive strain
50% compressive strain
Covalent
crosslinks
Swollen state
15% compressive strain
20% compressive strain
33
Stress Relaxation
4
stress (kPa)
3.5
Covalent
3
~102
2.5
Ionic
2
1.5
1
Strain: 15%
Radius: 6mm
0.5 -1
10
0
10
1
10
2
10
time (s)
3
10
4
10
34
Size effect
R
35
Gel with covalent crosslinks relaxes stress
by migration of water
 t
s t, R   f  
 R
D~
R2
trelax
~ 108 m 2 /s
36
Indent a gel
Yuhang Hu
F
Indent into a gel by a certain depth
indenter
h
gel
migration
of solvent
h
water
Joost Vlassak
t
Record force as a function of time
F
F(0)
F(∞)
t
Hu, Zhao, Vlassak, Suo. Applied Physics Letters 96, 121904 (2010)
37
Linear poroelasticity
Small
deformation
Mass
conservation
Kinetics
Local
equilibrium
1  u
u 
 ij   i  j 
2  x j xi 
C J k

0
t x k
Force Balance s ij / x j  0
 k  

J i  
2 
   x i
W  s ij ij    0 C
Incompressibility of solvent molecules and polymers  k k  (C  C 0 )
Free energy
Constitutive
relations


 k k 2 
W  G  ij ij 
1  2




s ij  2G  ij 

1  2
   0
 ij


 kk ij  
38
Biot, JAP 12, 155 (1941). Originally developed for soil. Now adapted for gels.
PDEs
  2ui
1
 2uk  
 
G


x

x
1

2


x

x
xi
k
i 
 k k
C
 2C
D
t
xk xk
D
 kk  (C  C0 )
21  v Gk
1  2v 
3 poroelastic constants:
G,
D
Poisson’s ratio of a gel
s  2G(1  )
s  3G
A

A(1  )
2
A(1  )
L(1   )
L(1   )
L
Initial state
t 0
Instantaneous deformation
t  0
Equilibrium state
t 
covalently crosslinked alginate hydrogel
•viscoelasticity
•Poroelasticity
41
Hu, Zhao, Vlassak, Suo. Applied Physics Letters 96, 121904 (2010)
From relaxation curve to poroelastic constants
F 0
F 0 
 21  
F  
F  
F t   F  
 g 
F 0  F  

Dt
a2
t
(a) Plane-strain cylindrical indenter
(b) Cylindrical punch
a2 
 4R  
h
2 log
  1

4R 
 a  
F
F 0  Ga / R
R
2
R

g   0.791 exp  0.213 
h
 0.209 exp 0.95 
2a
aR
F

F 0  8Gha

g   1.304 exp  
h
2a

 0.304 exp 0.254 
Y.Y. Lin, B.-W. Hu, J. Non-Crystalline Solids 352 4034 (2006).
Hui , Lin, Chuang, Shull, Ling. J. Polymer Sci B: Polymer Phys. 43, 359 (2006)
(d) Spherical indenter
(c) Conical indenter
F
F
h
θ
2a
a  2 /  h tan 
a  Rh
R
F 0   4Gha

g   0.493 exp  0.822 
 0.507 exp 1.348 
Hu, Zhao, Vlassak, Suo. Applied Physics Letters 96, 121904 (2010)

h
2a
F 0  16 / 3Gha

g   0.491 exp  0.908 
 0.509 exp 1.679 
42

Covalently crosslinked alginate gel
F(0)
F(∞)
F 0/ ah  4G
G  32.5kPa
F 0/ F    21  

g   0.493 exp  0.822 
 0.507 exp 1.348 
v  0.22

D  6.6  109 m2 / s
43
Crease
Liang Fen (凉粉), a starch gel
A food popular in northern China
44
Dough
Zhigang, I was making bread this weekend, and realized that when the rising dough was constrained by the bowl
it formed the creases that you were talking about in New Orleans.
-- An email from Michael Thouless
45
Brain
46
Face
47
Crease: theory and experiment
Biot, Appl. Sci. Res. A 12, 168 (1963).
Theory: linear perturbation analysis
 Biot  0.46
Maurice Biot
Gent, Cho, Rubber Chemistry and Technology 72, 253 (1999)
Ghatak, Das, PRL 99, 076101 (2007)
Experiments: bending rods of rubber and gel
 Gent  0.35
Alan Gent
48
What’s wrong with Biot’s theory?
L
(a)
A'
L
O
A
reference state
Homogeneous
deformation
(b)
Mahadevan
A' O A
ε
ε
homogeneous state
(c)
A/A'
ε
O
ε
creased state
Biot’s theory: infinitesimal strain
from the state of homogeneous deformation
Crease: Large strain
from the state of homogeneous deformation
Hohlfeld, Mahadevan (2008):
crease is an instability different from that analyzed by Biot.
49
An energetic model of crease
L
(a)
A'
U  L2Gf  
L
O
A
reference state
0.8
(b)
A' O A
ε
•Neo-Hookean material
•Plane strain conditions
0.6
ε
 U/GL
2
homogeneous state
(c)
0.4
A/A'
ε
O
creased state
ε
0.2
 c  0.35
0
0.2
0.25
0.3

Hong, Zhao, Suo, Applied Physics Letters 95, 111901 (2009)

c
0.35
50
Crease under general loading
Incompressibility
1 2 3  1
2
3
1
Critical condition for crease
3 / 1  2.4
Biot
3 / 1  3.4
51
Hong, Zhao, Suo, Applied Physics Letters 95, 111901 (2009)
Crease of a gel during swelling
Toyoichi Tanaka
1946-2000
52
Tanaka et al, Nature 325, 796 (1987)
Crease of a swelling gel
Ryan Hayward
swell
1

gel
substrate
Theories
c  2.4
 biot  3.4
Rui Huang
Experimental data
Southern, Thomas, J. Polym. Sci., Part A, 3, 641 (1965)
Tanaka, PRL 68, 2794 (1992)
ex p  2.5  3.7
Trujillo, Kim, Hayward, Soft Matter 4, 564 (2008)
Hong, Zhao, Suo, Applied Physics Letters 95, 111901 (2009)
Further development: Huang, http://imechanica.org/node/7587
ex p  2.4
 ex p  2.0
53
Outlook
• Gels are used in nature and engineering (tissues of
animals and plants, tissue engineering, drug delivery,
fluidics…).
• Deformation and transport are concurrent (The structure
of the theory is known. Application of the theory just
begins.)
• The field is wide open (characterization, computation,
devices, phenomena).
54