Download - Large deformation and instability in gels
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Large deformation and instability in gels
Zhigang Suo
Harvard University
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Ono, Sugimoto, Shinkai, Sada, Nature Materials 6, 429, 2007
reversible
Gel = elastomer + solvent
•Entropy of mixing•Entropy of contraction•Enthalpy of mixing
Thermodynamics of swelling
Elasticity: long polymers are cross-linked by strong bonds.Fluidity: polymers and solvent molecules aggregate by weak bonds.
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Gels in daily life
Tissues, natural or engineered
Contact lenses
Wichterie, Lim, Nature 185, 117 (1960)
Drug delivery
Ulijn et al. Materials Today 10, 40 (2007)
Jelly
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Gels in engineering
Valves in fluidics
Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)
Ion exchange for water treatment
Sealing in oil wells Shell, 2003
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THB theory
iu - solid displacement
ijkk
ijij δν
νεGεGσ
212
2
0iv neglect liquid motion
ijjiij uuε ,,21 - strain
Stress-strain relation
Equilibrium equation
0,
tu
fσ ijij
No instant response to loads Linear theory, small deformation Basic principles are unclear,
hard to extend
Limitations
body force due to friction
Tanaka, Hocker, and Benedek, J. Chem. Phys. 59, 5151 (1973)
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Multiphasic theoryList of Equations A nonionic polymeric gel is a mixture of two phases: network (n) and solvent (s). Mass conservation
0 nn
nρ
tρ
v
0 ss
sρ
tρ
v
Balance of linear momentum
nnnnnn
n ρDt
Dρ πbσ
v
ssssss
s ρDtD
ρ πbσv
where the material derivative with respect to phase α
αα
tDD
v ,
and απ is the momentum supply to α component from the other phase
0ππ sn Energy conservation
nnnnnnnn
n εrρDteD
ρ qvσ :
ssssssss
s εrρDteD
ρ qvσ :
Second law of thermodynamics
0Tr
ρTDt
sDρ
Tr
ρTDt
sDρ
ss
ssss
nn
nnnn qq
where αs is the entropy density of α component, and the Helmholtz free energy density
snnnnn ρρTATseA ,,,C
snssss ρρTATseA ,,,C Incompressibility
0 snsssnn φφφ vvvv
where the volume fraction αφ is related to the density as ααα ρφρ 0 . The constitutive relations:
Ts
sn
nnn Aρ
Aρλφ F
CCFIσ
2
IIσ
s
ss
s
nnsss
ρ
Aρ
ρ
Aρρλφ
nss
sss
nnnsn A
ρρρ
Aρφλ vvf
CCππ
:
snnnnn ρρTATseA ,,,C
snssss ρρTATseA ,,,C
Helmholtz free energy for each species
View solvent and solute as two phases Unclear physical picture. Unmeasurable quantities.
Bowen, Int. J. Eng. Sci. (1980)Lai, Hou, Mow, J. Biomech. Eng. Trans. (1991)
biphasic theory
Is the theory applicable to gels?
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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
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System = gel + weight + solvent
Equilibrium condition
Weight, P
Gel = Network +M solvent molecules
MlPF
Solvent
Pump,
l
M MlF , Helmholtz free energy of gel
lP work done by the weight
M work done by the pump vpp 0 0/log ppkT
lMlF
P
, MMlF
,
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Field variables
AP
s
Pweightl
l ge
l
pumpM
MlPF
ALM
Ll
AP
ALF
CsW
λCλW
s
, CCλW
μ
,
Ll
ALM
C
A
L
Reference state Current state
CW , Helmholtz free energy per volume
Equilibrium condition
M=0
Pump,
M
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3D inhomogeneous field
tC ,X K
iiK X
txF
,X
Deformation gradient
0
iK
iK BX
s
Concentration
iK
iK F
CWs
,F
C
CW
,F
iKiK TNs
Gibbs (1878)
CdVdAxTdVxBWdV iiii
Free-energy function
CW ,F
Equilibrium condition
Equivalent equilibrium condition
pump
Weight
Gel
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Flory-Rehner free energy
Free energy of stretching
CWWCW ms FF,
FF detlog2321 iKiKs FFNkTW
Flory, Rehner, J. Chem. Phys., 11, 521 (1943)
vCχ
vCvC
vkT
CWm 11
1log
Swelling increases entropy by mixing solvent and polymers,but decreases entropy by straightening the polymers.
Free energy of mixing
Free-energy function
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+ =
Vdry + Vsol = Vg
el
Molecular incompressibility
Assumptions:– Individual solvent molecule and polymer are incompressible.
– Gel has no voids. (a gel is different from a sponge.)
Fdet1 vC
v – volume per solvent molecule
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Equations of state
FF
det,
iKiK
iK HF
CWs
v
CCW
,F
FdetiKiKiKiK HHFNkTs
131
222 NkTs
211
333 NkTs
v
vCvCvCvC
kT
2111
1log
Enforce molecular incompressibility as a constraint by introducing a Lagrange multiplier
Use the Flory-Rehner free energy
321
111 NkTs
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Inhomogeneous equilibrium state
Equilibrium StateReference State
A
B
Rr
b
0A
(b)(a)
Empty space
Dry polymer
Core
Gel
Equilibrium StateReference State
A
B
Rr
b
0A
(b)(a)
Empty space
Dry polymer
Core
Gel
Reference State
A
B
Rr
b
0A
(b)(a)
Empty space
Dry polymer
Core
Gel
Concentration is inhomogeneous even in equilibrium.
Stress is high near the interface (debonding, cavitation,
wrinkles).
Zhao, Hong, Suo, APL 92, 051904 (2008 )
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Hydrogel-actuated nanostructure
Experiment: Sidorenko, Krupenin, Taylor, Fratzl, Aizenberg, Science 315, 487 (2007).
Theory: Hong, Zhao, Suo, http://imechanica.org/node/2487
-1 -0.5 0 0.5 1-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
(rad)
vG/k
T
RH = 80%
RH = 95%
RH = 100%
0 = 1.5
= 0.1Nv = 10-3
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Critical humidity can be tuned
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
1.2
1.4
(ra
d)
Relative humidity (%)
0 = 1.7
1.2
1.1 = 0.1Nv = 10-3
Hong, Zhao, Suo, http://imechanica.org/node/2487
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Crease
Tanaka et al, Nature 325, 796 (1987)
Denian
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Our model
fLU 2
reference state
LL
homogeneous state
O AA'
O AA'
ε ε
creased state
O
A/A'
ε ε
(a)
(b)
(c)
U
c
Experiment, Gent, Cho, Rubber Chemistry and Technology 72, 253 (1999)Our model 35.0c
Linear perturbation, Biot (1963), 46.0biot
fLU 2
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Elasticity with huge volumetric change
CdVdAxTdVxBWdV iiii
ABAQUS UMAT
dAxTdVxBdVCW iiii
CWW ˆAn alternative free-energy function
Liu, Hong, Suo, manuscript in preparation.
CFsW iKiK ˆ
,ˆ,ˆ FF W
CF
Ws
iKiK
Equilibrium condition
Legendre transform
CFsW iKiK Recall
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Time-dependent process
Hong, Zhao, Zhou, Suo, JMPS (2007). http://dx.doi.org/10.1016/j.jmps.2007.11.010
Shape change: short-range motion of solvent molecules, fastVolume change: long-range motion of solvent molecules, slow
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Coupled deformation and migration
K
iiK X
txF
,X
Deformation of network
Local equilibrium
0
,
iK
iK BX
ts X
Mechanical equilibrium
Conservation of solvent molecules
ttr
XtJ
ttC
K
K
,,, XXX
Rate process
L
KLK X
tMJ
,X
iK
iK F
CWs
,F
C
CW
,F
tti
NJ KK
,X
iKiK TNs
dAidVrdAxTdVxBWdV iiii
pump
Weight
Gel
Nonequilibrium thermodynamics
Hong, Zhao, Zhou, Suo, JMPS (2007). http://dx.doi.org/10.1016/j.jmps.2007.11.010Biot, JAP 12, 155 (1941)
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Ideal kinetic model
LKLK X
MJ
ii x
μkTcD
j
1det FHHvkTD
M iLiKKL
Diffusion in true quantities
Conversion between true and nominal quantities
KiK
i JF
Fj
det
iKiK
FxX
Hong, Zhao, Zhou, Suo, JMPS (2007). http://dx.doi.org/10.1016/j.jmps.2007.11.010
Solvent molecules migrate in a gel by self-diffusion
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Diffusion of labeled water
Muhr and Blanshard, Polymer, 1982
/sm1086/ 210self
RkTD
polymer volume fraction
Stokes-Einstein formula
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Diffusion or convection?
Tokita, Tanaka, J. Chem. Phys, 1991
D = 410-8 m2/s
~50 Ds
• Macroscopic pores?
• Convection?
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Swelling of a thin layer
Fitting swelling experiment is hard: Missing data Fick’s law: Concentration is not
the only driving force TBH theory also has problemsdimensionless time
rela
tive
swel
ling
Zhou, Zhao, Hong, Suo, unpublished work (2007)
gel
solvent
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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.