swell-induced surface instabili in subs ate-confined hydrogel...

Swell-Induced Surface Instability in

Substrate-Confined Hydrogel Layer

Rui Huang and Min K. Kang

Center for Mechanics of Solids, Structures and Materials

Department of Aerospace Engineering and Engineering Mechanics

The University of Texas at Austin

Swelling of rubber and gels

• Southern and Thomas, 1965

• Tanaka et al, 1987

• Trujillo et al., 2008

� Critical condition for the onset of

surface instability?

� Any characteristic size?

� Effect of kinetics?

A theoretical framework for gels

Nominal stress

),( CU F

iK

iKF

Us

∂

∂=

0=+∂

∂i

K

iK BX

s0=

∂

∂

KX

µ

0== iKiKi xNsT δor extµµ =

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

Free energy density function

Chemical potentialvC+=1)det(F

C

U

∂

∂=µ

Equilibrium equations

Boundary conditions

Volume change

)()(),( CUUCU me += FF

++

+=

vC

vC

vC

vCvC

TkCU B

m11

ln)(χ

ν

NkBT : initial shear modulus of the polymer network

N : No. of polymer chains per unit volume

ν : Volume of a solvent molecule

χ : Enthalpy of mixing parameter

A specific material model

Free energy density function

Neo-Hookeon rubber elasticity:

( ) ( )( )[ ]FF detln232

1−−= iKiKBe FFTNkU

Flory-Huggins polymer solution theory:

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.11

2

3

4

5

6

Normalized chemical potential µ/kBT

De

gre

e o

f sw

elli

ng

λh

Nv=0.001 and χ=0.4

p0v/k

BT=0.000023 at T=25oC

Homogeneous swelling of a hydrogel layer

3

2

1 1

1

2

31

>=

==

hλλ

λλ

Tk

pvNv

B

ext

h

h

hhh

−=

−++−

−

µ

λλ

λ

χ

λλ

1111ln

2

vppext )( 0−=µ

( )0/ln ppTkBext =µ

External chemical potential:

0pp ≤

0pp ≥

A linear perturbation analysis

∂

∂+

∂

∂

∂

∂

∂

∂+

=

100

01

01

~

2

2

1

2

2

1

1

1

x

u

x

u

x

u

x

u

h

h

λ

λ

F

=

100

00

001

hλF

Linear perturbation: ( ) ( )21222111 ,,, xxuuxxuu ==

Homogeneous swelling

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

( )

( ) 0

01

21

1

2

2

2

2

2

2

1

2

2

21

2

2

2

2

1

22

2

1

1

2

=∂∂

∂+

∂

∂++

∂

∂

=∂∂

∂+

∂

∂+

∂

∂+

xx

u

x

u

x

u

xx

u

x

u

x

u

hhhhh

hhhhh

ξλλξλ

ξλλξλ

Linearized equilibrium equations

Solution by the method

of Fourier transform

( ) ( )

( ) ( )

=

=

∑

∑

=

=

4

1

2

)(

222

4

1

2

)(

121

exp;ˆ

exp;ˆ

n

n

n

n

n

n

n

n

xquAkxu

xquAkxu


04

1

=∑=n

nmn AD

[ ]

−

+−

+

+

+

−−

=

−−

−−

khkhkh

h

h

kh

h

h

kh

h

h

kh

h

h

kh

h

kh

h

hh

mn

eeee

eeeeD

ββ

ββ

ββλ

λλ

λ

λλ

λλλλ

ββλλ

2211

1122

1111

00

00

[ ] ( ) 0,;,det 0 == χλ NvkhfD hmn

Critical Conditions for Surface Instability

===

=∂

∂=

=

∂

∂+−=

00

1

221

2

1

212

2

1

122

xuu

hxx

ups

hxx

ups

at

at

atBoundary conditions

Critical condition:


Effect of perturbation wave number

Critical swelling ratio

Nv = 0.001

• Long wavelength perturbation is stabilized by the substrate effect.

• Short wavelength perturbation is unaffected.

• Thus the critical condition can be taken at the short-wavelength

limit.


Short-wave limit (kh0 →∞)

βλλ

λ h

h

h 41

2

=

+

• The critical swelling ratio

depends on Nv and χ, ranging

between 2.5 and 3.4.

• For each χ, there exists a

critical value for Nv.

• For small Nv (< 10-4), the

critical swelling ratio is nearly

a constant (~3.4).


A stability diagram

χc = 0.63

Soft network in good solvent

Poor solvent

Stiff network


Critical linear strain

10-5

10-4

10-3

10-2

10-1

0.3

0.35

0.4

0.45

0.5

Nv

Critic

al str

ain

, εc

χ=0.0

χ=0.2

χ=0.4

χ=0.6

D

D

3

3 1

λ

λε

−=

Relative to the unconstrained,

free swelling in 3D:

• Trujillo et al.’s experiments for a swelling hydrogel

• Biot’s analysis for rubber under equi-biaxial compression

33.0=cε


0 2 4 6 83

3.5

4

4.5

5

5.5

6

Normalized wavelength S

Critic

al s

we

llin

g r

atio

λc

h0=1µm

h0=10µm

h0=100µm

Nv = 0.001, χ = 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).

Nv

nm

TNkL

B

53.0~

γ=A length scale:

10-1

100

101

102

103

100

101

102

103

Initial thickness [µm]

Cri

tica

l wa

ve

len

gth

S* [

µm

]

L=1µm

L=0.5µm

L=0.1µm

Nv = 0.001, χ = 0.4

Characteristic wavelength

1.09.0

0

* ~ LhS


10-1

100

101

102

103

3

3.5

4

4.5

5

5.5

6

Initial thickness [µm]

Critica

l sw

elli

ng

ra

tio λ

c

L=0.1µm

L=0.53µm

L=1µm

Nv = 0.001, χ = 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.Kang and Huang, Soft Matter 6, 5736-5742 (2010).

10-5

10-4

10-3

10-2

10-1

0

0.2

0.4

0.6

0.8

1

Nv

Critic

al t

hic

kn

ess h

c [

µm

]

χ = 0.0

χ = 0.2

χ = 0.4

χ = 0.5

Critical thickness

Nv

LL

nmTk

vL

B

'

53.0'

=

==γ

� The critical thickness is linearly proportional to L, with the

proportionality depending on Nv and χ.


Finite element simulation

Nv = 0.001, χ = 0.4


Surface Evolution


W/H=1

W/H=5

Volume ratio

Inhomogeneous Swelling of

Substrate-Supported Hydrogel Lines

W/HKang and Huang, J. Applied Mechanics 77, 061004 (2010).

Spontaneous Formation of Creases

W/H=12

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

Swell-induced bucklingTirumala et al., 2005

Effect of material parameters

10-3

10-2

10-1

0

0.5

1

1.5

2

Nv

A/H

Kang and Huang, Int. J. Applied Mechanics, in press.

Effect of geometry (constraint)

0 1 2 3 40

0.5

1

1.5

2

2.5

W/H

A/H

Kang and Huang, Int. J. Applied Mechanics, in press.

Summary

• Opportunity: Within the general theoretical framework,

instability of hydrogel-like soft material can be understood

and exploited.

• Challenge: The highly nonlinear aspects in the material,

geometry, and instability mechanics pose serious challenges

for theoretical analysis and numerical simulations.

• Strategy: Collaborations between experimental and

theoretical studies will be most successful.

swell-induced surface instabili in subs ate-confined hydrogel...

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