Soft Active Materials

11

Soft Active Materials

Zhigang Suo

Harvard University

II. Gels

2

gel = network + solvent

Solid-like: long polymers crosslinked by strong bonds. Retain shapeLiquid like: polymers and solvent aggregate by weak bonds. Enable transport

solvent

networkgel

•Imbibe solvent, and swell•Exude solvent, and collapse

Ono, Sugimoto, Shinkai, Sada, Nature Materials 6, 429 (2007)

3

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)

foodplants

4

Gels in engineering

Gate in microfluidicsBeebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000) packer in oil well

Shell, 2003

Responsive to physiological variables:•pH•concentration of ions•temperature•light

5

Chemical potential of a molecular species in a system

vapor

•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•Helmholtz free energy of a system, F(M,…)•Chemical potential of water in a system,•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.

MMF

,...

wine

cheese

6

Chemical potential of a species in a one-species system

vpp 0 0/log ppkT

0p

vapor

State of referenceliquid & vapor in equilibrium

0

0pp

liquid liquid 0pp

vapor

liquid only vapor only

0p Pressure of saturated vapor v volume per solvent molecule

piston

weights

Relative humidity, 0/ pp

7

The ascent of sap

Dixon and Joly, Phil. Trans. R. Soc. Lond. B 186, 563 (1895)

•The air is dry. •The soil is wet.•The gradient of humidity pumps water up.

0/log ppkT soil

ghvppkT air 0/log

Henry Horatio Dixon

Liquid water in tension

water

water-permeable solid

0

logpp

kT airChemical potential of water in the air

Chemical potential of water in the xylem

v

0

logpp

vkT air

When the liquid water in the voidequilibrate with the water vapor in the air, xylem

Liquid water in tension

Wheeler, Stroock. Nature 455, 208 (2008)

Abe Stroock

0

logpp

vkT air

(MPa) voidtheinwaterliquidinstresstensile 10 20 30 40

0/air,theinhumidityrelative ppair

•Axisymmertic deformation•Creasing•Cavitation

10

Osmosis in a liquid

kTVN

p

mem

bra

ne

solution solvent

kTVNv

osmotic pressure

Chemical potential of water in the solution due to osmosis

Number of solute molecules, NVolume of the solution, VVolume per solvent molecule, v

solute

h

Chemical potential of water in the solution due to gravity

ghvJacobus H. Van’t Hoff, Osmotic pressure and chemical equilibrium, Nobel Lecture, 13 December 1901

Chemical potential of water in the solvent

0

Van’t Hoff

11

Osmosis in a gel

mem

bra

ne

solution

solvent

Osmosis balanced by gravity

Osmosis balanced by elasticity

Crosslink

12

Two ways of doing work

Condition of equilibrium

Weight

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

lMlF

P

, MMlF

,

Equations of state

Chemical potential of solvent

P Force

13

Gels as transducers

• A swelling gel is blocked by a hard material.• Blocking force.• Inhomogeneous deformation.

Need to formulate boundary-value problems

14

2 ways of doing work

AP

sP

weightl

l ge

l

pumpMMlPF

ALM

Ll

AP

ALF

CsW Ll

ALM

C

A

L

Reference state Current state

CW , Helmholtz free energy per volume

Equilibrium condition

M=0

Pump,

M

15

Inhomogeneous field

tC ,X K

iiK X

txF

,X

Deformation gradient Concentration

Gibbs (1878)

CdVdAxTdVxBWdV iiii

Free-energy function

CW ,F

Condition of equilibrium

pump

Weight

Gel

constant

Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)

1. How to prescribe ?2. How to solve boundary-value problems?3. What boundary-value problems to solve?

CW ,F

16


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 decreases entropy by straightening polymers.•Swelling increases entropy by mixing solvent and polymers.

Free energy of mixing


Paul Flory

17

+ =

Vdry + Vsol = Vgel

Molecular incompressibility

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.)

Fdet1 vC

v – volume per solvent molecule

Wei Hong

18

Stress-strain relation

2

332332332

111log

vkT

v

332

21

1

1 NkT

332

22

2

1 NkT

332

23

3

1

NkT


19

Anisotropic swelling

-0.05 -0.04 -0.03 -0.02 -0.01 01.5

2

2.5

3

3.5

4

4.5

5

5.5

6

/kT

λ1H

λ0H

s

-0.05 -0.04 -0.03 -0.02 -0.01 01.5

2

2.5

3

3.5

0/kT

0

analyticalFEM

L λ0L

Free swelling Unidirectional swelling

A gel imbibes a different amount of solvent under constraint.

onalunidirecti3

free

Treloar, Trans. Faraday Soc. 46, 783 (1950).

20

Inhomogeneous swelling

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 (debond, cavitation).

Zhao, Hong, Suo, APL 92, 051904 (2008 )

Sternstein, J. Macromolol. Soc. Phys. B6, 243 (1972)

21

Finite element method

CdVdAxTdVxBWdV iiii

dAxTdVxBdVW iiii ˆ

Hong, Liu, Suo, Int. J. Solids Structures 46, 3282 (2009)ABAQUS UMAT posted online http://imechanica.org/node/3163

Equilibrium condition

Legendre transform

•A gel in equilibrium is analogous to an elastic solid•ABAQUS UMAT

CWW ,ˆ F

http://imechanica.org/node/3163

22

5/ HRi 7/ HRi

10/ HRi 12/ HRi

Swelling-induced buckling

Zishun Liu

Liu, Hong, Suo, Swaddiwudhipong, Zhang. Computational Materials Science. In press.

23

Gel and nano-rods

Experiment: Sidorenko, Krupenin, Taylor, Fratzl, Aizenberg, Science 315, 487 (2007).Theory: Hong, Zhao, Suo, JAP104, 084905 (2008) .

-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

Joanna Aizenberg

24

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

25

Swelling-induced bifurcation

Zhang, Matsumoto, Peter, Lin, Kamien, Yang, Nano Lett. 8, 1192 (2008).

Shu Yang

26

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)

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Time-dependent process

Shape change: short-range motion of solvent molecules, fastVolume change: long-range motion of solvent molecules, slow


28

Concurrent deformation and migration

K

iiK X

txF

,X

Deformation of network

Local equilibrium

0

,

iK

iK BXts X

Conservation of solvent

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

Maurice BiotJAP 12, 155 (1941)


29

ideal kinetic model

LKLK X

MJ

ii x

μkTcD

j

1det FHHvkTD

M iLiKKL

Diffusion in true quantities

KiK

i JF

Fj

det

iKiK

FxX

Solvent molecules migrate in a gel by self-diffusion

Conversion between true and nominal quantities


30

Finite element method for concurrent

deformation and migration

02 LDt

Z

X

Y

Frame 001 28 Jul 2008

Z

X

Y

Frame 001 28 Jul 2008 Z

X

Y

Frame 001 28 Jul 2008

102 LDt

2LDt

Zhang, Zhao, Suo, Jiang, JAP 105, 093522 (2009)

Hanqing Jiang

31

Alginate Hydrogels

Ca++( )

Na+

AAD

Irreversible

Ionic crosslinks

Covalent crosslinks

David Mooney

32

Stress-relaxation test

Impermeable

Rigid platesGel diskPBS

time

str

ain

str

ess

time

Zhao, Huebsch, Mooney, Suo, J. Appl. Phys. In press.

33

Elasticity, plasticity, fracture

Ionic crosslinks

Covalent crosslinks

Swollen state 45% compressive strain 50% compressive strain

Swollen state 15% compressive strain 20% compressive strain

10mm

3410-1

100

101

102

103

1040.5

1

1.5

2

2.5

3

3.5

4

time (s)

stre

ss (

kPa)

Stress Relaxation

Strain: 15%

Radius: 6mm

Covalent

Ionic

~102

35

Size effect

R

36

Gel with covalent crosslinks relaxes stress by migration of water

Rt

fRt, /sm10~~ 282

relaxtR

D

Indent a gel

37

F

water

indenter

h

gel

migrationof solvent

t

h

F(∞)

F

t

F(0)

Indent into a gel by a certain depth

Record force as a function of time

Yuhang Hu

Hu, Zhao, Vlassak, Suo. Applied Physics Letters 96, 121904 (2010)

Joost Vlassak

Linear poroelasticity

38

i

j

j

iij x

u

xu

21

0

k

k

xJ

tC 0/ jij x

CW ijij 0

)( 0CCkk

2

21 kkijijGW

ijijkkijij G

0

212

Small deformation

Force BalanceMass conservation

Kinetics

Incompressibility of solvent molecules and polymers

Free energy

Local equilibrium

Constitutive relations

ii x

kJ

2

Biot, JAP 12, 155 (1941). Originally developed for soil. Now adapted for gels.

PDEs

v

GkvD

2112

iik

k

kk

i

xxxu

xxu

G

22

211

kk xxC

DtC

2

)( 0CCkk

3 poroelastic constants: D

G ,

Poisson’s ratio of a gel

Initial state Instantaneous deformation Equilibrium state

0t 0t t

)1( LL

A

G3

)2

1(A

)1( L

)1( A

)1(2 G

covalently crosslinked alginate hydrogel

41

•viscoelasticity•Poroelasticity


42

From relaxation curve to poroelastic constants

2a

R

F

h

(a) Plane-strain cylindrical indenter

14

log24

2

aR

Ra

h

RGaF /0 2

95.0exp209.0

213.0exp791.0

g

FR

h

2a

(b) Cylindrical punch

Ra

GhaF 80

254.0exp304.0

exp304.1

g

2a

θ

F

h

(c) Conical indenter

tan/2 ha

GhaF 40

348.1exp507.0

822.0exp493.0

g

2a

F

h

R

(d) Spherical indenter

Rha

GhaF 3/160

679.1exp509.0

908.0exp491.0

g

Hui , Lin, Chuang, Shull, Ling. J. Polymer Sci B: Polymer Phys. 43, 359 (2006) Y.Y. Lin, B.-W. Hu, J. Non-Crystalline Solids 352 4034 (2006).

g

FFFtF

02aDt

120

FF


0F

Ft

43

12/0 FF

GahF 4/0

348.1exp507.0

822.0exp493.0

g

kPaG 5.32

22.0v

smD /106.6 29

Covalently crosslinked alginate gel

F(0)

F(∞)

44

Crease

Liang Fen (凉粉 ), a starch gel A food popular in northern China

45

DoughZhigang, 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

46

Brain

47

Face

48

Crease: theory and experiment

Gent, Cho, Rubber Chemistry and Technology 72, 253 (1999)Ghatak, Das, PRL 99, 076101 (2007)Experiments: bending rods of rubber and gel

35.0Gent

Biot, Appl. Sci. Res. A 12, 168 (1963).Theory: linear perturbation analysis

46.0Biot

Maurice Biot

Alan Gent

49

reference state

LL

homogeneous state

O AA'

O AA'

ε ε

creased state

O

A/A'

ε ε

(a)

(b)

(c)

What’s wrong with Biot’s theory?

Biot’s theory: infinitesimal strain from the state of homogeneous deformation

Hohlfeld, Mahadevan (2008): crease is an instability different from that analyzed by Biot.

Crease: Large strain

from the state of homogeneous deformation

Homogeneousdeformation Mahadevan

50

An energetic model of crease

reference state

LL

homogeneous state

O AA'

O AA'

ε ε

creased state

O

A/A'

ε ε

(a)

(b)

(c)

GfLU 2

0.2 0.25 0.3 0.350

0.2

0.4

0.6

0.8

U/G

L2

c

Hong, Zhao, Suo, Applied Physics Letters 95, 111901 (2009)

35.0c

•Neo-Hookean material•Plane strain conditions

51

Crease under general loading

4.2/ 13

4.3/ 13

Critical condition for crease

Biot

1

32

1321 crea

se Incompressibility

Hong, Zhao, Suo, Applied Physics Letters 95, 111901 (2009)

52

Crease of a gel during swelling

Tanaka et al, Nature 325, 796 (1987)

Toyoichi Tanaka1946-2000

53

Crease of a swelling gel

4.2c

4.3biot

substrate

gel1

Experimental data

Southern, Thomas, J. Polym. Sci., Part A, 3, 641 (1965)

Tanaka, PRL 68, 2794 (1992)

Trujillo, Kim, Hayward, Soft Matter 4, 564 (2008) 0.2exp

4.2exp

7.35.2exp

Theories

Ryan Hayward

swell

Hong, Zhao, Suo, Applied Physics Letters 95, 111901 (2009)Further development: Huang, http://imechanica.org/node/7587

Rui Huang

http://imechanica.org/node/7587

54

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).

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Soft Active Materials