large deformation of hydrogels coupled with solvent diffusionruihuang/talks/hydrogel2015.pdf ·...

Large Deformation of Hydrogels Coupled with Solvent Diffusion

Rui Huang

Center for Mechanics of Solids, Structures and MaterialsDepartment of Aerospace Engineering and Engineering Mechanics

The University of Texas at Austin

July 2015

Ambrosio 2010

Crosslinkedpolymer

Solvent molecules

• Large and reversible deformation facilitated by solvent migration• Response to mechanical/chemical/electrical stimuli• Applications: biomedical, soft machines/robots

Hydrogel = polymer + water (solvent)

Transient behaviors of hydrogelsTanaka et al, 1987

• Swelling• Indentation• Fracture• …

Bonn et al, 1998

Hu et al, 2011

Linear Poroelasticity

3 material parameters: G, ν, k/η

i

j

j

iij x

uxu

21

0

ij

ij bx

kk xxpk

t

2

kk x

pkJ

0 kk

ijijkkijij pG

212

Biot, 1941; Scherer, 1989; Suo et al., 2010.

Linear constitutive relation

Linear kinematics Linear kinetics Inertia ignored

A Nonlinear Continuum Theory for Gels),( CU F

iKiK F

Us

0

iK

iK BXsCU

Diffusion kinetics:L

KLLk

L

k

K

BK X

MXx

XxX

TkDCJ

LKL

KK

K

XM

XXJ

tC Conservation of solvent

molecules:

Nonlinear constitutive relations

Nonlinear kinematics Geometrically nonlinear

kinetics Inertia ignored

Free energy density function:

Nominal stress:

Chemical potential:

Mechanical equilibrium:

Hong, Zhao, Zhou, and Suo, 2008.

A Material Model)()(),( CUUCU me FFFlory‐Rehner (1943) free energy

density function:

)det(ln2321 FF iKiKBe FFTNkU

CC

CCCTkCU B

m 11ln)(

F)FF det(1)()(),( CCUUCU me

Flory‐Huggins polymer solution theory:

Neo‐Hookean rubber elasticity:

C1)det(F

4 material parameters: NkBT, NΩ, χ, D

Volume of gel::

Linearization near a swollen state

Isotropic swollen state: 0321

1

50

20

30

20

2)1(

122

1

NN

0TNkG B

3

0060

30

30

300 1111

ln

N

TkB

ijijkkijij G

0

212

kkBk x

MxTk

Dcj

00

30

30

01

TkDMB

300 1 c 0cckk

kkk

k

xxM

xj

tc

2

0

0p c 30

30 1

TkDk

B

The linearized equations are identical to the theory of linear poroelasticity.

The 3 parameters in the linearized equations depend on the swelling ratio.

Bouklas and Huang, Soft Matter 2012.

Poisson’s ratioYoon et al., Soft Matter 6, 6004‐6012 (2010).

1

50

20

30

20

2)1(

122

1

NN

Instantaneously incompressible (ν0 = 0.5) The equilibrium Poisson’s ratio depends on

the initial state (λ0 or μ0) as well as the intrinsic material properties (χ and NΩ).

1 1.5 2 2.5 3 3.50.2

0.25

0.3

0.35

0.4

0.45

0.5

0

Poi

sson

's ra

tio,

N = 0.001

= 0.1-0.6

0 0.2 0.4 0.6 0.8 10.2

0.25

0.3

0.35

0.4

0.45

0.5

Poi

sson

's ra

tio

N = 10-4, 10-3, 10-2, 10-1

Bouklas and Huang, Soft Matter 2012.

poor solvent

good solvent

A nonlinear, transient finite element methodWeak form for the coupled fields:

0 0 0 0

, ,t

iJ i J K K i i i iV V S S

s u C tJ dV b u r t C dV T u dS i t dS

Discretizationu nu N un N μ

u nδu N δun N δμ

For numerical stability, the 2D Taylor‐Hood elements (8u4p) are implemented.

Bouklas et al., JMPS 2015CCUU -) (F,) (F, ˆLegendre transform

),()()(),( CFUCUUCU cme FF

A slightly different material model: 2

21)det(),( CKCUc

FF

Numerical stability for mixed finite element method• Nearly incompressible

behavior at the instantaneous response limit ⟶0 for K >> NkBT

• Discontinuity of the boundary condition at 0 with respect to the initial conditions

Choice of appropriate interpolations:LBB condition(Ladyzhenskaya‐Babuska‐Brezzi)

UNSTABLEEqual order8u8p

STABLETaylor Hood8u4p

• Incompressible linear elasticity• Stokes flow• Linear poroelasticity

Constrained swelling of hydrogel layersBC’sIC’s

0, 0 u X X X

0, 0 X

1 : 0S

2 2: , 0S J u 03 1 1: 0, 0S u J

UNSTABLEEqual order8u8μ element

μ/KBT

time

Oscillations in the early stage

Evolution of chemical potential oscillations

0

3

1.4, 0.001

0.4, 10 B

N

K Nk T

0 0.5 1-0.06

-0.04

-0.02

0

X2/H

/(k

BT)

0 0.5 1-0.06

-0.04

-0.02

0

X2/H

/(k

BT)

0 0.5 1-0.04

-0.03

-0.02

-0.01

0

X2/H

/(k

BT)

0 0.5 1-0.04

-0.03

-0.02

-0.01

0

X2/H

/(k

BT)

t/ = 10-3

t/ = 10-1

t/ = 10-4

t/ = 10-2

UNSTABLE8u8μ

2HD

Effect of compressibility

0 1.4, 0.0010.4

N

4/ 10t The oscillations decrease with increasing compressibility (K decreasing)

0 0.5 1-0.06

-0.04

-0.02

0

X2/H

/(k

BT)

0 0.5 1-0.06

-0.04

-0.02

0

X2/H

/(k

BT)

0 0.5 1-0.06

-0.04

-0.02

0

X2/H

/(k

BT)

0 0.5 1-0.06

-0.04

-0.02

0

X2/H

/(k

BT)

K =10NkBTK = 102NkBT

K = 103NkBTK = 104NkBTUNSTABLE8u8μ

Use of stable interpolationSTABLETaylor Hood8u4μ element

0.5 0.6 0.7 0.8 0.9 11-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

X2/H

/(K

BT)

t/ = 10-4

t/ = 10-3

t/ = 10-2

t/ = 10-1

K = 103NkBT

0.5 0.6 0.7 0.8 0.9 11-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

/(K

BT)

X2/H

K = 104 NkBT

K = 103 NkBT

K = 102 NkBT

K = 10 NkBT

t/ = 10-4

• Oscillations are significantly reduced and do not depend oncompressibility.

• The remaining oscillations are due to discontinuity of the boundary condition at 0 .

Bouklas et al., JMPS 2015

0

3

1.4, 0.001

0.4, 10 B

N

K Nk T

Swelling induced surface instability

10-3 10-2 10-1 100 101 102 103 1041

1.5

2

2.5

3

3.5

4

4.5

h/H

t/

ne = 50ne = 30ne = 20ne = 4

= 0.4

= 0.6

Swell induced compressive stress could lead to surface instability For a poor solvent (χ = 0.6), the surface remains flat and stable. For a good solvent (χ = 0.4), the homogeneous hydrogel layer is

instantaneously unstable upon swelling; numerically, divergence occurs at a finite time that depends on the mesh size.


Swell induced surface wrinkling

s11

10-2 10-1 100 101 102 103 1040

0.01

0.02

0.03

0.04

0.05

0.06

t/

A/h

0

• Regularization of the surface instability by using a stiff skin layer.

• Wrinkle amplitude grows over time, with a selected wrinkle wavelength.


Two types of surface instability

A B

Wu, Bouklas and Huang, IJSS 50, 578‐587 (2013).

Type A: soft-on-hard bilayer, critical condition at the short wave limit, forming surface creases;

Type B: hard-on-soft bilayer, critical condition at a finite wavelength, forming surface wrinkles first.

A soft‐on‐hard gel bilayer (Type A)

A hard‐on‐soft gel bilayer (Type B)

Flat Punch Indentation

S2x1

x2S5 hS3

S1

F (δ)

S4aa

b

2

D

10-2 100 102 104 1060

0.2

0.4

0.6

0.8

1

t/

(F(t)

-F(

)) /

(F(0

)-F(

))

/h = 10-3

/h = 10-2

/h = 0.1/h = 0.2fitting

F ( ( ) ( )) / ( (0) ( ))t F t F F F

σ/NkBT

σ/NkBT

σ/NkBT

σ/NkBT µ/kBT

µ/kBT

µ/kBT

µ/kBT

Fracture Properties of Hydrogels

Sun et al., 2012.Kwon et al., 2011.

• Reported values of fracture toughness range from 1 to 10000 J/m2

• Fracture mechanism may vary over different types of hydrogels• Potential rate dependence may relate to viscoelasticity or solvent diffusion

• Experimental methods to measure fracture properties of hydrogels are not well developed.

A center crack model

• Consider a center crack in a hydrogel plate under remote strain• A nonlinear, transient finite element method is employed• Sharp and notched crack geometries are used for small to large

deformationBouklas et al., JAM 2015

Immersed and not‐immersed cases

0t t Hydrogel is immersed in solvent Assume solvent‐permeable boundaries

and crack faces

Hydrogel is not immersed in solvent Assume solvent‐impermeable

boundaries and crack faces Overall volume is conserved, but

solvent redistribution within the gel may lead to local volume change.

1

2 3 2 50 0 0 0

1 1 22 2 ( 1)

N N

0 0.5

0t t

LEFM Predictions

0TNkG B

jiIij

Iij Tf

rKr 112

,

Crack‐tip stress field:

1

2G

aKI T

Crack opening: 21

212 2ˆ xaxu

Stress would relax over time

Crack opening would remain constant, independent of modulus or Poisson’s ratio

Crack‐tip stress fields 310

00.5

t

0.2415t

• The square‐root singularity is maintained at both the instantaneous and equilibrium states, but the stress intensity factor is lower in the equilibrium state due to poroelastic relaxation.

Evolution of crack opening

Immersed Not immersed

10-4 100 104 108

2

2.5

3

x 10-3

/a

t/

immersednot immersedLEFM

= 10-3 h/a

100

20

10

Opening at the center:

Bouklas et al., JAM 2015

Solvent diffusion around crack tip

Instantaneous limit:Homogeneous solvent concentration, but inhomogeneous chemical potential.

Equilibrium state:Homogeneous chemical potential, but inhomogeneous solvent concentration.


Evolution of Solvent Concentration Immersed

A singular concentration field is approached in the equilibrium state for both the immersed and not‐immersed cases.

10-3 10-2 10-1

10-4

10-2

100

(C-C

0)

r/a

t/ = 10-1

t/ = 1t/ = 102

t/ = 104 = 10-3

Not‐immersed

10-3 10-2 10-1

10-4

10-2

100

(C-C

0) r/a

t/ = 10-1

t/ = 1t/ = 102

t/=104 = 10-3


Energy release rate Rate of change of potential energy:

Total energy:

Rate of total energy change due to crack extension

Rate of dissipation due to diffusion:

Conserved throughout the transient evolution (without crack growth).

0 0 0 0 0

i ii iV V S V S

dx dxd dU dV b dV T dS rdV idSdt dt dt dt

0KV

K

d J dVdt X

0ddt

0 0

iK Ki KV S

K

dxdI dId dU dV T N dSda da X da da da

A modified J‐integral

• J* is path independent.• The second form is more convenient for numerical calculations.• A domain integral method is used to calculate J*.

1 0

*1

1 1

iiJ JS V

x CJ UN s N dS dVX X

1 0

*1

1 1

ˆ iiJ JS V

xJ UN s N dS CdVX X

* dJda

Define a nominal energy release rate: (per unit length of crack in the reference state)


Evolution of J‐integral

10-4 10-2 100 102 104 106 1080

1

2

3

4

5

6

7

8 x 10-5

J* /(aN

k BT)

t/

not immersedimmersedLEFM ( = 0.5)LEFM ( = 0.2415)

h/a = 10, 20, 100

= 10-3

aTNktJ B2

0* 40 aTNktJ B

20*

12

LEFM predictions at the instantaneous and equilibrium limits:

Dependence on material parameters

10-6 10-3 100 103 1060.9

1

1.1

1.2

1.3

1.4

1.5

J* /(4G

2

02 a

)

t/*

N = 10-2

N = 10-3

N = 10-4

= 0.2 = 10-3

10-6 10-3 100 103 1060.9

1

1.1

1.2

1.3

1.4

1.5

J* /(4G

2

02 a

)

t/*

= 0.2 = 0.4 = 0.6

N = 10-2

= 10-3

The J* is well predicted by LEFM at the instantaneous limit (t → 0) The equilibrium J* depends sensitively on χ (good/poor solvent) The time scale is captured effectively by the poroelastic diffusivity Delayed fracture possible.


Under moderately large remote strain

10-4 10-3 10-2 10-110-1

100

101

102

22/(N

k BT)

r/a

t/ = 10-4

t/ = 105 = 0.1

~ r-1/2

~ r-1

• Similar behavior for crack opening (and J* too, not shown)• Stress distributions show transition of singularity: with a stronger 1/r

singularity due to hyperelasticity and a weaker singularity closer to the crack tip due to solvent diffusion; the typical square‐root singularity is approached at the equilibrium limit.

1.0


Under large remote strainImmersed Not‐immersed

5.0

• The 1/r singularity becomes more prominent in both cases• The singularity weakens over time for the immersed case


J‐integral

10-4 100 104 1080

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t/

J* /(4G

2

02 a)

immersednot-immersed = 0.5

• The J* is considerably lower than the LEFM prediction, while the evolution trends are similar to the small‐strain behaviors.


Fracture Criterion

?**cJtJ

The right hand side of the fracture criterion has to be determined by well‐designed experiments.

Could the right‐hand side depend on time due to the evolving, inhomogeneous field of solvent concentration around the crack tip?

A nonlinear, transient finite element method was implemented for large deformation of hydrogels coupled with solvent diffusion.

The transient processes of constrained swelling (with surface instability) and flat punch indentation were studied in some details.

Effects of solvent diffusion on the crack‐tip fields and driving force for fracture of hydrogels were studied. A path‐independent, modified J‐integral was derived as the transient energy release rate of crack growth in hydrogels, although a complete fracture criterion is yet to be established.

Summary

Acknowledgments

• Min‐Kyoo Kang• Nikolas Bouklas• Zhigen Wu• Chad Landis

• Funding by NSF

large deformation of hydrogels coupled with solvent diffusionruihuang/talks/hydrogel2015.pdf ·...

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