Large deformation and instability in gels
Zhigang Suo
Harvard University
1
Gel = elastomer + solvent
Elasticity: long polymers are cross-linked by strong bonds.
Fluidity: polymers and solvent molecules aggregate by weak bonds.
reversible
Thermodynamics of swelling
•Entropy of mixing
•Entropy of contraction
•Enthalpy of mixing
2
Ono, Sugimoto, Shinkai, Sada, Nature Materials 6, 429, 2007
Gels in daily life
Jelly
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
Ion exchange for water treatment
Sealing in oil wells
Shell, 2003
Valves in fluidics
Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)
4
THB theory
ui - solid displacement
εij =
1
2
(u
i, j
+ u j ,i ) - strain
vi = 0 neglect
liquid motion
Equilibrium equation
∂ui
=0
σ ij , j − f
∂t
body force due to friction
Stress-strain relation
2Gνεkk
σ ij = 2Gεij +
δij
1 − 2ν
Limitations
„ No instant response to loads
„ Linear theory, small deformation
„ Basic principles are unclear,
hard to extend
5
Tanaka, Hocker, and Benedek, J. Chem. Phys. 59, 5151 (1973)
Multiphasic theory
biphasic theory
List of Equations
A nonionic polymeric gel is a mixture of two phases: network (n) and solvent (s).
Mass conservation
∂ρ n
+ ∇ ⋅ ρn v n = 0
∂t
( )
+ ∇ ⋅ (ρ v ) = 0
s
∂ρ
∂t
s s
Balance of linear momentum
Dn vn
= ∇ ⋅ σ n + ρ n bn + π n
Dt
ρn
Ds v s
= ∇ ⋅ σ s + ρs bs + π s
Dt
where the material derivative with respect to phase α
Helmholtz free energy for each species
ρs
α
D
∂
= + vα ⋅ ∇ ,
∂t
D
n
s
π +π =0
As = e s − Ts s
Energy conservation
ρn
(
)
= A (T ,C, ρ , ρ )
An = e n − Tsn = An T ,C, ρn , ρ s
and π α is the momentum supply to α component from the other phase
Dnen
= σ n : ∇v n − ∇ ⋅ q n + ρ n r n + ε n
Dt
s
n
s
Dses
= σ s : ∇v s − ∇ ⋅ q s + ρ s r s + ε s
Dt
Second law of thermodynamics
ρs
ρn
Dnsn
qn
rn
Dsss
qs
rs
+∇⋅
− ρn
+ ρs
+∇⋅
− ρs
≥0
T
T
T
T
Dt
Dt
where s α is the entropy density of α component, and the Helmholtz free energy density
(
)
A s = e s − Ts s = A s (T ,C , ρ n , ρ s )
φ ∇ ⋅ v + φ ∇ ⋅ v + (v − v )⋅ ∇φ = 0
A n = e n − Ts n = A n T ,C , ρ n , ρ s
Incompressibility
n
n
s
s
s
n
„ View solvent and solute as two phases
s
where the volume fraction φα is related to the density as ρ α = φα ρ0α .
The constitutive relations:
⎛
∂A s ⎞⎟ T
∂An
σ n = −φn λI + 2F ⋅ ⎜ ρ n
⋅F
+ ρs
⎜
C
∂C ⎟⎠
∂
⎝
⎛
∂A n
∂A s ⎞
σ s = −φ s λI − ρ s ⎜ ρ n
+ ρ s s ⎟I
s
⎜
∂ρ
∂ρ ⎟⎠
⎝
π n = − π s = λ∇φn − ρn
∂A n
∂ρ s
∇ρ s + ρ s ∇C :
Is the theory applicable to gels?
(
∂A s
+ f vs − vn
∂C
„ Unclear physical picture.
„ Unmeasurable quantities.
)
Bowen, Int. J. Eng. Sci. (1980)
Lai, Hou, Mow, J. Biomech. Eng. Trans. (1991)
6
Our preference: monophasic theory
• Regard a gel as a single phase.
• Start with thermodynamics.
Gibbs, The Scientific Papers of J. Willard Gibbs, pp. 184, 201, 215 (1878):
Derive equilibrium theory from thermodynamics. (This part of his work seems less known.)
Biot, JAP 12, 155 (1941): Use Darcy’s law to model migration. Poroelasticity.
……
Hong, Zhao, Zhou, Suo, JMPS (2007). http://dx.doi.org/10.1016/j.jmps.2007.11.010
7
System = gel + weight + solvent
µδM work done by the pump
Pump, µ
Pδl work done by the weight
Solvent
F (l , M ) Helmholtz free energy of gel
δM
Equilibrium condition
µ = ( p − p0 )v
µ = kT log ( p / p0 )
Gel = Network +
M solvent molecules
δF = Pδl + µδM
P=
∂F (l , M )
∂l
µ=
∂F (l , M )
∂M
δl
Weight, P
8
Field variables
Equilibrium condition
λ=
δF = Pδl + µδM
l
L
C=
M
AL
s=
P
A
Reference state
Current state
M=0
δW = sδλ + µδC
s=
∂W ( λ , C )
∂λ
µ=
A
M
gel
P δl
δM
+µ
AL
AL
l
AL
=
pump
µ
Pump, µ
L
δF
δM
∂W ( λ , C )
∂C
W (λ , C ) Helmholtz free energy per volume
δl
P
weight
9
3D inhomogeneous field
Deformation gradient
FiK =
Concentration
∂xi (X, t )
∂X K
Free-energy function
C (X , t )
W (F, C )
Equilibrium condition
∫ δWdV = ∫ B δx dV + ∫ T δx dA + µ ∫ δCdV
i
i
i
i
pump
Equivalent equilibrium condition
∂W (F, C )
siK =
∂FiK
∂siK
+ Bi = 0
∂X K
Gibbs (1878)
µ=
∂W (F, C )
∂C
Gel
Weight
s iK N K = Ti
10
Flory-Rehner free energy
Swelling increases entropy by mixing solvent and polymers,
but decreases entropy by straightening the polymers.
Free-energy function
Free energy of stretching
Free energy of mixing
W (F, C ) = W s (F ) + W m (C )
Ws (F ) = 21 NkT [FiK FiK − 3 − 2 log (det F )]
Wm (C ) = −
kT
v
⎡
1 ⎞
χ ⎤
⎛
⎟+
⎢vC log⎜ 1 +
⎥
⎝ vC ⎠ 1 + vC ⎦
⎣
11
Flory, Rehner, J. Chem. Phys., 11, 521 (1943)
Molecular incompressibility
+
=
Vdry +
Vsol =
Vgel
1 + vC = det F
v – volume per solvent molecule
Assumptions:
– Individual solvent molecule and polymer are incompressible.
– Gel has no voids. (a gel is different from a sponge.)
12
Equations of state
Enforce molecular incompressibility as a constraint by introducing a Lagrange multiplier Π
siK =
∂W (F, C )
− Π H iK det (F )
∂FiK
µ=
∂W (F, C )
+ Πv
∂C
Use the Flory-Rehner free energy
siK = NkT (FiK − H iK ) − Π H iK det F
(
= NkT (λ
⎡
⎤
χ
vC
1
µ = kT ⎢ log
+
+
+ Πv
2⎥
+
+
vC
vC
1
1
(1 + vC ) ⎦
⎣
)
)− Πλ λ
s1 = NkT λ1 − λ1−1 − Π λ2 λ3
s2
(
−1
−
λ
2
2
)
3 1
s3 = NkT λ3 − λ−31 − Π λ1 λ2
13
Inhomogeneous equilibrium state
Core
Empty space
R
Aλ0
r
A
B
Dry polymer
Gel
(a)
Reference State
b
(b)
Equilibrium State
„
Concentration is inhomogeneous even in equilibrium.
„
Stress is high near the interface (debonding, cavitation,
wrinkles).
14
Zhao, Hong, Suo, APL 92, 051904 (2008 )
Hydrogel-actuated nanostructure
λ0 = 1.5
-0.3
χ = 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, http://imechanica.org/node/2487
0
θ (rad)
0.5
1
15
Critical humidity can be tuned
1.4
1.2
λ0 = 1.7
θ (rad)
1
0.8
0.6
1.2
0.4
0.2
0
0
1.1
χ = 0.1
Nv = 10-3
20
40
60
Relative humidity (%)
80
100
16
Hong, Zhao, Suo, http://imechanica.org/node/2487
Crease
Tanaka et al, Nature 325, 796 (1987)
Denian
17
L
(a)
A'
Our model
L
O
A
reference state
∆U
∆U = L2 µf (ε )
(b)
A' O A
ε
ε
homogeneous state
∆U = µL2 f (ε )
(c)
A/A'
ε
O
εc
ε
creased state
Linear perturbation, Biot (1963),
ε
ε biot ≈ 0.46
Experiment, Gent, Cho, Rubber Chemistry and Technology 72, 253 (1999)
Our model
18
ε c ≈ 0.35
Elasticity with huge volumetric change
Equilibrium condition
Legendre transform
∫ δWdV = ∫ B δx dV + ∫ T δx dA + µ ∫ δCdV
i
i
i
i
∫ δ (W − µC )dV = ∫ B δx dV + ∫ T δx dA
i
An alternative free-energy function
Recall
i
i
i
ˆ = W − µC
W
δW = siK δFiK + µδC
ˆ = s δF − Cδµ
δW
iK
iK
ˆ (F, µ )
∂W
s iK =
∂FiK
ABAQUS UMAT
ˆ (F, µ )
∂W
C =−
∂µ
Liu, Hong, Suo, manuscript in preparation.
19
Time-dependent process
Shape change: short-range motion of solvent molecules, fast
Volume change: long-range motion of solvent molecules, slow
20
Hong, Zhao, Zhou, Suo, JMPS (2007). http://dx.doi.org/10.1016/j.jmps.2007.11.010
Coupled deformation and migration
pump
Deformation of network Conservation of solvent molecules
∂xi (X, t )
FiK =
∂X K
∂C (X , t ) ∂J K (X , t ) ∂r (X , t )
+
=
∂t
∂t
∂X K
JK N K =
Gel
∂i (X , t )
∂t
Weight
Nonequilibrium thermodynamics
∫ δWdV ≤ ∫ B δx dV + ∫ T δx dA + ∫ µδrdV + ∫ µδidA
i
Local equilibrium
∂W (F, C )
siK =
∂FiK
µ=
∂W (F, C )
∂C
Biot, JAP 12, 155 (1941)
i
i
i
Mechanical equilibrium
∂s iK (X , t )
+ Bi = 0
∂X K
s iK N K = Ti
Rate process
J K = − M KL
∂µ (X, t )
∂X L
21
Hong, Zhao, Zhou, Suo, JMPS (2007). http://dx.doi.org/10.1016/j.jmps.2007.11.010
Ideal kinetic model
Solvent molecules migrate in a gel by self-diffusion
J K = − M KL
∂µ
∂X L
Diffusion in true quantities
ji = −
cD ∂µ
kT ∂x i
Conversion between true and nominal quantities
ji =
FiK
JK
det F
∂µ
∂µ
=
FiK
∂X K ∂xi
M KL =
D
H iK H iL (det F − 1 )
vkT
22
Hong, Zhao, Zhou, Suo, JMPS (2007). http://dx.doi.org/10.1016/j.jmps.2007.11.010
Diffusion of labeled water
Stokes-Einstein formula
Dself = kT / (6πRη ) ≈ 8 ×10−10 m 2 /s
polymer volume fraction
Muhr and Blanshard, Polymer, 1982
23
Diffusion or convection?
~50 × Ds
D = 4×10-8 m2/s
Tokita, Tanaka, J. Chem. Phys, 1991
• Macroscopic pores?
• Convection?
24
Swelling of a thin layer
solvent
relative swelling
gel
dimensionless time
Fitting swelling experiment is hard:
„ Missing data
„ Fick’s law: Concentration is not
the only driving force
„ TBH theory also has problems
25
Zhou, Zhao, Hong, Suo, unpublished work (2007)
Summary
• Gels have many uses (soft robots, drug
delivery, tissue engineering, water
treatment, sealing in oil wells).
• Mechanics is interesting and challenging
(large deformation, mass transport,
thermodynamic forces, instabilities).
• The field is wide open.
26