induced deformation and surface instability of polymer gels

Swell Induced Deformation and Surface Instability of Polymer Gels

Rui Huang

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

The University of Texas at Austin

October 4, 2012

Gel: polymer network + solvent

Ono et al, Nature Materials, 6, 429, 2007.

• Large and reversible deformation

• Stimuli responsive: temperature, pH, UV, electric field, etc.

• Soft, bio‐compatible

Suo, 2008

Swell‐Induced Instability

Trujillo et al, 2008.Zhang et al, 2008.

DuPont et al., 2010

Kinetics of Swelling

Tanaka et al, 1987

Guvendiren et al., 2010.

A theoretical framework for gels

Nominal stress

),( CU F

iKiK F

Us

0

iK

iK BXs 0

KX

0 iKiKi xNsT or ext

Hong, Zhao, Zhou, and Suo, JMPS 2008

Free energy density function

Chemical potentialCU

Equilibrium equations

Boundary conditions

C1)det(F

Volume change

)()(),( CUUCU me FF

C

CC

CCTkCU Bm 11

ln)(

NkBT : initial shear modulus of the polymer network (entropic elasticity);

N : No. of polymer chains per unit volume;Ω : Volume of a solvent molecule; : polymer‐solvent interaction parameter (enthalpy of mixing).

A specific material modelFree energy density function

Neo‐Hookeon rubber elasticity:

FF detln2321

iKiKBe FFTNkU

Flory‐Huggins polymer solution theory:

Kinetics of Solvent Diffusion

k

L

LBkBk x

XX

CTk

DxTk

cDj

)det(FRate of flux (current configuration):

LKLk

k

KK X

MjxXJ

)det(FNominal rate of flux

(reference configuration):

LKL

KK

K

XM

XXJ

tC Nominal concentration:

A nonlinear diffusion equation, coupled with mechanical equilibrium equations (nonlinear elasticity).

Hong, Zhao, Zhou, and Suo, JMPS 2008

k

L

k

K

BKL x

XxX

TkDCMMobility tensor:

A Finite Element Method for Equilibrium Analysis

• Tangent modulus

klijijklBijkl JJJJ

NHJTNkC

231

11

1ln1

jkiljkiljlikjlikijkl BBBBH 21

• True stress jKiKiK

ij FFJ

JU

FI

IU

J

ˆˆ1

FFFFB3/1

J

T

CCUU ),(,ˆ FF• Legendre transformation

dAxTdVxBdVU iiii ˆ• Weak form Chemical potential is specified as a constant.

Kang and Huang, J. Applied Mechanics 77, 061004 (2010).

Implementation in ABAQUS (UMAT)• The chemical potential is mimicked by temperature.• Due to singularity of the chemical potential at the dry state,

FEM handles deformation after homogeneous swelling.

Analytical method

(1)F (2)F

(1)(2)FFF

Dry state Initial state Final deformed state

Homogeneousswelling

FEM simulation

= 0 > >

= 0

)1(

3

)1(2

)1(1

(1)F


Analytical method

1‐D homogeneous swellingSide wall relaxation Final deformed state

FEM simulation

Constrained Swelling of Hydrogel Lines


W/H=1 W/H=5

Volume ratio

Effect of constraint on swelling


W/H=10

Surface instability: creases

W/H=12


A rubber block under compression

• Surface wrinkling (Biot, 1961)

• Surface creasing (Hong, Zhao and Suo, 2009)

295.0321

417.0321

54.01 01 08.3 GP

65.01 01 34.2 GP

rubber

A linear perturbation analysis

100

01

01

~

2

2

1

2

2

1

1

1

xu

xu

xu

xu

h

h

F

10000001

hF

Linear perturbation: 21222111 ,,, xxuuxxuu

Homogeneous swelling

Kang and Huang, J. Mech. Phys. Solids 58, 1582‐1598 (2010).

0

01

21

12

22

22

21

22

21

22

22

12

221

12

xxu

xu

xu

xxu

xu

xu

hhhhh

hhhhh

Linearized equilibrium equations

Solution by the method of Fourier transform

4

12

)(222

4

12

)(121

exp;ˆ

exp;ˆ

nn

nn

nn

nn

xquAkxu

xquAkxu


04

1

nnmn AD

khkhkh

hh

kh

hh

kh

hh

kh

hh

khh

khh

hh

mn

eeee

eeeeD

2211

1122

1111

00

00

0,;,det 0 NkhfD hmn

Critical Condition for Surface Instability

00

1

221

21

212

21

122

xuu

hxxups

hxxups

at

at

atBoundary conditions

Critical condition:


Effect of perturbation wave number

Critical swelling ratio

NΩ = 0.001

• Long wavelength perturbation is stabilized by the substrate effect.• Short wavelength perturbation is independent of substrate.• The critical condition is set at the short‐wavelength limit.


Short‐wave limit (kh0 )

hh

h 412

• The critical swelling ratio depends on NΩ and , ranging between 2.5 and 3.4.

• For each , there exists a critical value for NΩ.

• For small NΩ (< 10‐4), the critical swelling ratio is nearly a constant (~3.4).


NΩ

A stability diagram

c = 0.63

Soft network in good solvent

Poor solvent

Stiff network


NΩ

Finite element simulationNΩ = 0.001, = 0.4


21

Nv = 0.001, = 0.4

Surface Evolution

The numerical simulations are often unstable due to the highly nonlinear behavior and large distortion.

Quasi‐static equilibrium simulation

0 2 4 6 83

3.5

4

4.5

5

5.5

6

Normalized wavelength S

Crit

ical

sw

ellin

g ra

tio c

h0=1m

h0=10m

h0=100m

NΩ = 0.001, = 0.4

Effect of surface tension

Long wavelength perturbation is stabilized by the substrate.

Short wavelength perturbation is stabilized by surface tension.

An intermediate characteristic wavelength emerges.

The minimum critical swelling ratio depends on the layer thickness.

Kang and Huang, Soft Matter 6, 5736‐5742 (2010).

Nnm

TNkL

B

53.0~A length scale:

10-1 100 101 102 103

100

101

102

103

Initial thickness [m]

Crit

ical

wav

elen

gth

S* [ m

]

L=1mL=0.5mL=0.1m

Nv = 0.001, = 0.4

Characteristic wavelength

1.09.00

* ~ LhS


10-1 100 101 102 1033

3.5

4

4.5

5

5.5

6

Initial thickness [m]

Crit

ical

sw

ellin

g ra

tio c

L=0.1mL=0.53mL=1m

Nv = 0.001, = 0.4

Thickness‐dependent stability

• The hydrogel layer becomes increasingly stable as the initial layer decreases;

• Below a critical thickness (hc), the hydrogel is stable at the equilibrium state.


10-5 10-4 10-3 10-2 10-10

0.2

0.4

0.6

0.8

1

Nv

Crit

ical

thic

knes

s h c [

m]

= 0.0 = 0.2 = 0.4 = 0.5

Critical thickness

NL

TNkL

nmTk

L

B

B

'

53.0'

The critical thickness is linearly proportional to L, with the proportionality depending on NΩ and .


Bilayer gels: two types of instability

0 10 20 30 40 50-0.1

-0.08

-0.06

-0.04

-0.02

0

Perturbation wave number H

Crit

ical

che

mic

al p

oten

tial

c / (k

T)

Bilayer A: N2 < N1

Homogeneous layer

Bilayer B: N2 > N1

A

B

Wu, Bouklas and Huang, submitted.

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.

Swelling of a gel bilayer (Type A)

Swelling of a gel bilayer (Type B)

Kinetics of Surface Instability

Tanaka et al, 1987

Guvendiren et al., 2010.

iKiK F

Us

0

iK

iK BXs

CU

Diffusion kinetics:

k

L

k

K

BKL x

XxX

TkDCM

LKL

KK

K

XM

XXJ

tC

Nonlinear constitutive relations

Nonlinear kinematics Geometrically nonlinear

kinetics

Nominal stress:

Chemical potential:

Mechanical equilibrium:

Hong, Zhao, Zhou, and Suo, JMPS 2008.

Coupling deformation with kinetics

C1)det(F

Volume change

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; Yoon et al., 2010.

A Transient Finite Element MethoddVrdV

XJdV

tC

K

K

siJ

XJ xidV Bi xidV 0

dStixTdVtrxBC

dVtJCxstt

itt

itt

itt

it

Ktt

Ktt

Jitt

iJ

,,

Weak form:

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

A slightly different material model:

21)det(21),( CKCUc FF

CCUU -) (F,) (F, ˆiJ

iJ FUs

ˆ

F;CfCU

F;1 fCL

KLK XMJ

k

L

k

K

BKL x

XxX

TkDCM

Constrained swelling with no instability

FEM vs FDM

Effect of compressibility

As K increases we approach the incompressible case.

4.0,001.0,4.10 N

21)det(21),( CKCUc FF

Numerical convergenceEffect of time stepEffect of element size

4.0,001.0,4.10 N

41 401

Surface instability

By condition of local equilibrium, the compressive stress near the surface becomes very high instantaneously, causing surface instability.

2.0,001.0,4.10 N

Summary

Swell Induced Deformation and Surface Instability of Polymer Gels by Modeling and Simulations

Large deformation: nonlinear constitutive model and nonlinear geometry

Kinetics: geometrically nonlinear diffusion

Instabilities: abundant physical phenomena and numerical challenges (work in progress)

Acknowledgments

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

• Funding by NSF

induced deformation and surface instability of polymer gels

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