1

Large deformation and instability in gels

Zhigang Suo

Harvard University

2

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.