institute of high performance computing (ihpc)xl/lzs-talk.pdf · triphasic model: lai, hou, mow, j....

Institute of High Performance Computing (IHPC) Computing (IHPC)

Dr Zishun Liu (Z S Liu)Dr Zishun Liu (Z S Liu)Dr. Zishun Liu (Z.S. Liu)Dr. Zishun Liu (Z.S. Liu)April 5, 2011 XJTUApril 5, 2011 XJTU

U i P l i G lU i P l i G l

Spring Workshop on Nonlinear Mechanics -XJTU

Using Polymeric Gel Using Polymeric Gel Theory to Explain Plant Theory to Explain Plant

PhyllotaxisPhyllotaxis

Z.S. LiuIHPC SingaporeIHPC, Singapore National University of Singapore

Collaborators:W H I St t U i itW. Hong: Iowa State UniversityZ.G. Suo: Harvard UniversityS. Swaddiwudhipong: National University of Singapore

Using Polymeric Gel Theory to Explain Plant Using Polymeric Gel Theory to Explain Plant PhyllotaxisPhyllotaxis

OUTLINE

• Introduction

OUTLINE

• Gel theories

• Modeling & Simulation of Gel Deformation

• Governing Equations of Film Gel Wrinkling

• Botanical plants modeling

• Future Study & Discussion

IntroductionIntroduction

Cactus

Fascinating comple patterns and shapes & interesting nd lating s rface morpholog

PumpkinFascinating complex patterns and shapes & interesting undulating surface morphology

In plants, interesting undulating surface morphology is often observed across a large scale

Phyllotaxis: (namely the arrangement of phylla) has intrigued natural scientists for over 400 years

Question:

IntroductionIntroductionQuestion:What are the physical mechanisms of this natural structure deformation? How to explain the mechanism of flower opening and closure? How to explain the morphogenesis of the natural growth of some leaves, flowersp p g g ,and vesicles?

The challenge:How to provide a rational explanation for the formation of such patterns, amechanism or combination of mechanisms which capture natural phenomena

Many natural tissues of plants and animals are to some extent polymeric gels.

PolymericPolymeric gel theorygel theory may be used to explain the formation of phyllotacticpatterns at plant

Various phyllotactic lattice and tiling patternsNewell et al, J. Theoy. Boil. 251, 2008

Introduction

Gel= elastomer + solvent•Elasticity: long polymers are cross-linked by strong bonds.

What is Gel?

ibl

y g y y g•Fluidity: polymers and solvent molecules aggregate by weak bonds.

reversible

+ Stimuli:Physical --T, electrical, light, pressure…

solvent

Chemical--pH, ions..

polymersgel

~ 10000% volumetric change

Jelly

Polyelectrolyte gels

Applications of Gels

IntroductionApplications of Gels

Soft robots; Drug delivery; Tissue engineering; Water treatment; Sealing in oil wells; Sensor & actuator…

Repair damaged heart muscle

Contact lensesContact lenses

Wichterie, Lim, Nature 185, 117 (1960)

Tissues, natural or engineered

Nanogel & Drug delivery

Ulijn et al. Materials Today 10, 40 (2007)Cartilage & Meniscus

S li i

IntroductionApplications of Gels

Sealing in oil wells

Shell, 2003

I h fIon exchange for water treatment

Hydrogel

Valves in fluidics

Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)

y g

gate valve

Gelling System in Oil Well

IntroductionIntroductionGelling System in Oil Well

1. Watered-out zone separated from an oil zone by anfrom an oil zone by an impermeable shale barrier-the solution is to cement in bottom zone.

2. Shale doesn't reach to production well. Solution is inject gel into the lower zoneinject gel into the lower zone.

3 Watered-out high permeability3. Watered out high permeability zone sandwiched between two oil zones-the solution is to isolate the zone and inject gel

Introduction

Challenges of Gel Applications

1 A lack of understanding of the relationship between gel composition and1. A lack of understanding of the relationship between gel composition and response kinetics

2. The majority of previous research efforts of gel are experimental-based

3. More complex shapes are required and the accurate dimensional measurements of their volume transition behaviors are not convenient in experimental analysisexperimental analysis

4. A lack of completed analytical theory for Gel

5 Th di i f l f b d li d i l i ill h b5. The prediction of gel performance by modeling and simulation will thus be critical for understanding the characteristics of gel

Theories of Gel Deformation

THB Theory: T k H k d B d k J Ch Ph (1973)THB Theory: Tanaka, Hocker, and Benedek, J. Chem. Phys. (1973)

Multiphasic theoryBiphasic theory: Bowen, Int. J. Eng. Sci. (1980)Biphasic theory: Bowen, Int. J. Eng. Sci. (1980)

Triphasic model: Lai, Hou, Mow, J. Biomech. Eng. Trans. (1991)

Mixture theory: Shi, Rajagopal, Wineman, Int. J. Eng. Sci. (1981)

No instant response to loads Linear theory, small deformation

Limitations View solvent and solute as two or three

phasesy, Basic principles are unclear,

hard to extend

Unclear physical picture. Unmeasurable quantities.

Monophasic theory: Regard a gel as a single phase Start with thermodynamics

Monophasic theory:0

CdVδCW

μdAxδNFW

TdVxδBFW

X iKiK

iiiiKK

Gibbs, The Scientific Papers of J. Willard Gibbs, 184, 201, 215 (1878): Derive equilibrium theory from thermodynamics.

Bi t JAP 12 155 (1941) U D ’ l t d l i ti Analogy to solid mechanics

Hong, Zhao, Zhou, Suo, JMPS (2007)

Biot, JAP 12, 155 (1941): Use Darcy’s law to model migration. ……Hong, Zhao, Zhou, Suo, JMPS (2008)Hong, Liu, Suo, IJSS (2009)Liu, Hong, Suo, Somsak, Zhang, COMMAT, (2010)Liu, Somsak, Cui, Hong, Suo, IJAM (2011)

3D inhomogeneous fields:

Theory of Gel Deformation

XXxF

3D inhomogeneous fields:

tC ,X Concentration of solvent

Deformation gradient

CW ,F

Chemical potential

In equilibrium, the change in the free energy of the gel, associated with arbitrary change in displacement field and concentration field , equals the work done by the mechanical loads and the environment

A gel is in contact with a solvent of a fixed chemical potential, and subject to a mechanical

load and geometric constraint

Free-energy of system changes by

CdVdAxTdVxBWdVG iiii

geometric constraint solvent

μ

iiii

Cδ

CWδ

CWδ

,, FF

Free-energy density changes gel

CW F

Gel

Cδ

CCW

FδFCW

Wδ iKiK

,, FF

Pmechanical

load

System = gel + load+ solvent

CW ,F


Chemical PotentialThe chemical potential definition:

NNpTG

NNVTF

NNVSU

,,,,,,

The chemical potential definition:

Introducing the energy, entropy and

The chemical potential of the solvent ( ) in general depends on theμ

g gy, pyvolume per molecular u, s, and v

pvTspvTsu

In equilibrium: the chemical potential inside the gel be a constant and equal to the chemical potential of the external solvent

The chemical potential of the solvent ( ) in general depends on the temperature (T) and pressure (p)

μ

μμ ˆ

),( pTμμ

the chemical potential of the external solvent

for incompressible liquid phase for ideal gas

))((),(ˆˆ 00 ppTvTp

)/l ()(ˆˆ TkT

μμ For constant Temperature

for ideal gas )/log(),(ˆˆ 00 ppTkTp B

00

00

)/log( )(

ˆppifppTkppifppv

B

where p0 is the equilibrium vapor pressure and depends on the temperature, At the equilibrium vapor pressure ( ), the external chemical potential:In a vacuum (p = 0):

0ˆ μμ 0pp


In equilibrium condition

0 CdVdAxTdVxBWdVG iiii

Cδ

CCW

FδFCW

Wδ iKiK

,, FF

Free-energy density changes

dAxNFWTdVxB

FW

X iKiii

Applying the divergence theorem AGG ddV

VA gel is in contact with a solvent of a fixed

chemical potential, and subject to a mechanical load and geometric constraint

0

CdVCW

FFX iKiK

iiiiKK

geometric constraint solvent

μ

0

,

iiKK

BFCW

XF

iK

iK

TNFCW

,F

gel

CW F

Gel

iK

CCW

,F P

mechanical load

System = gel + load+ solvent

CW ,F



Introduce a new free-energy function by a Legendre transform W

CμCWμW ,,ˆ FF

New Equilibrium condition: dAxδTdVxδBdVWδ iiiiˆ

Define nominal stress as the work conjugate to the deformation gradient

iK

iK FCWs

,F

CWCW FFThe change in free energy

CμδFδsWδ iKiK

Cδ

CCW

FδFCW

Wδ iKiK

,, FF

δμCFδsWδ iKiK ˆ iK

iK FWs

,ˆ F μμW

C

,ˆ F

iK

jK

iK

jKiK

jKij F

μWF

FCWF

sF

σ

,ˆ

det,

detdetF

FF

FFThe true stress

Molecular Incompressibility in Gels

+ =

Vdry + Vsol = Vgel

Fd tC

Assumptions:

Fdet1 vC v – volume per solvent molecule

– Individual solvent molecule and polymer are incompressible.– Gel has no voids.

This molecular incompressibility condition can be enforced as a constraint by introducing a Lagrange multiplier Π

FFF det1,, vCCWCW


Flory-Rehner Free Energy in Gels

Swelling increases entropy by mixing solvent and polymers, but decreases entropyby straightening the polymers.

Following Flory and Rehner, the free energy density can be written as

Free-energy function CWWCW ms FF,

Ws

W

Free energy of stretching

iKiK Fs

C

FF detlog2321

iKiKs FFNkTW

Flory, Rehner, J. Chem. Phys., 11, 521 (1943)

Free energy of mixing

vCχ

vCvC

vkT

CWm 11

1log

Flory-Huggins polymer theory


Flory-Rehner Free Energy in Gels

Consider the molecular incompressibility condition as a

Following Flory and Rehner, the free energy density can be written as

p yconstraint by introducing a Lagrange multiplier Π

),F(1 CFree-energy function FFF det1Π, vCCWWCW ms

g y gy y

FdetiKiK

s

iKiK H

FW

FW

s

v

CW

CW m

O S

s HWWs

1det F

Consider other energy terms

WW m

1

Open Space….

iKiK

iK

s

iKiK F

HFF

s

1det FC

vCCm

1


Free energy FFF det1Π, vCCWWCW ms

FF detlog2321

iKiKs FFNkTW

vCχ

vCvC

vkT

CWm 11

1log

FdetΠ iKiK

siK H

FW

s

vC

Wμ m Π

2 vCvCv

1

111 ss Assume the principal nominal stresses are

3

2

1

F

132

1111

)(

)(

NkTs

NkTsv

CCCvCkT

2)1(1

11

log

333 ss 222 ss

211

333

31222

)(

)(

NkTs

NkTs vCvCvC

2)1(11

g

111

vs

Constitutive equations:

32132122

1321

321321321

11

1

1111log1

1111log1

kTNv

kTvs

kTNv

kTvs

3321

321321321

33

3

23213212

1111log1

kTNv

kTvs

kTkT


General Form of Equations of State in GelsEnforce molecular incompressibility as a constraint by introducing a Lagrange multiplier P

FF

det,

iKiK

iK HFCW

s

vC

CW

,F

Lagrange multiplier P

iKF C

Use the Flory-Rehner free energy

FdetiKiKiKiK HHFNkTs v

vCvCvCvC

kT

2111

1log

iK HHFNvvs

FF det111logdet

Constitutive Equations

iKiKiK HkT

HFNvkT

F

FFF det

det1

det1logdet

FEM Implementation for Deformation in Gels

iK HHFNvvs

FF det111logdet

Constitutive equations (Equation of State)

We select new reference state of the gel by free swelling 0321 λλλλ

iKiKiKiK H

kTHFNv

kT

F

FFF det

det1

det1logdet

We select new reference state of the gel by free swelling 0321 λλλλ

0FFF

0

0

0

λ

λ

λ

F 'det' FJiKiK FFI ''' 0λ

1log1log6log2321

,ˆ 303

03

0

300

20

Jλvμ

Jλχ

λJJ

JλvkT

λJIλNkTμW F

iKiKiKiK H

kTμ

λχ

λλλHFλNv

kTsv

FFF

F detd

1d1

1logdet 63330

20

Constitutive equations (Equation of state)

iKiKiK kTλλλkT

FF detdet

g 60

30

30

00

ABAQUS UMAT ABAQUS UHYPERHong, Liu, Suo: IJSS (2009)

Liu, Hong, Suo, Somsak, Zhang, Com. Mat. Sci. (2010)

geometric

Theory of GelNeutral Gel Theory: geometric

constraint solventμ

gel

XXxF

tC ,X Concentration of solvent

Deformation gradient

Chemical potential

y

v – volume per solvent molecule

Fdet1 vCincompressible constraint

Gel

The change of free-energy of system

CdVdAxTdVxBWdVG iiii Pmechanical

load

CW ,F

Chemical potential molecule

System = gel + load+ solvent0

CdVCWdAxN

FWTdVxB

FW

X iKiK

iiiiKK

CW F


Cδ

CCW

FδFCW

Wδ iKiK

,, FF

0,

iiKK

BFCW

XF

iK

iK

TNFCW

,F

CCW

,F

iK

iK FCWs

,F

Define nominal stress

FFF det1, vCCWWCW solnet

FF detlog2321

iKiKnet FFNkTW

vCkTCW 11log

Flory, Rehner, J. Chem. Phys., (1943)

Constitutive Equations

iKiKiKiK H

kTHFNv

kTvs

F

FFF det

det1

det11logdet

vCvCvC

vCWsol 1

1log

Fdet1 vC

Hong, Liu, Suo, IJSS, (2009)

Analytical solution of 1-D beam gel buckling

Incremental Modulus of the GelFor column or bean case

Incremental Modulus of the Gel

221 11

11log

1

Nvvs

01

11

1log1 22

Nv

1

1

21221

221

21

1

1 11log

kT

NvkT

2

1

221~

NvE

kTv

011log2

2212

21221

221

2

2

kTNv

1111 //~ ssE 1

1.0 001.0/ kT

Incremental modulus of gel varying with stretch for a) various initial chemical potentials b) various dimensionless measures of the enthalpy of mixing

Liu, Somsak, Cui, Hong, Suo, Zhang, IJAM, (2011)

Critical values of beam gel under buckling

Analytical solution of 1-D beam gel buckling

Critical values of beam gel under bucklingthe critical stress of a column with hinged ends (Timoshenko and Gere)

2 ~E 22111 NkTkT2)/(

)(rlE

s cr

2

1

22

21

2212

21221

221

11 1

)/(1111log1

rlNkT

kTNv

vkT

Stability diagram showing normalized critical stress ofDifferent stress curves varying with stretch Stability diagram showing normalized critical stress of gel beam varying with its slenderness ratio of beam for

different chemical potentials


Thin Film Gel with Wrinkle Mode

In-plane Incremental Modulus of thin film gelIn-plane Incremental Modulus of thin film gel

2

1

231~NvE

kTv 21~ NvE

kTvk

In plane incremental modulus of thin film gel varying with chemical potentialIn-plane incremental modulus of thin film gel varying with chemical potential for a) different initial chemical potentials, b) different dimensionless

measures of the enthalpy of mixing


The buckling stress and wave length of thin film gelThe buckling stress and wave length of thin film gel

Schematics of thin gel film before and after bucking

Wrinkling of a sol-gel-derived thin film

bqwNwD 21

4 )(lxww m2sin

lxw

lAq m

2sin2

2

2

3

2

2

2

03

2

1 214

24 A

bhD

hlA

lh

bhD

000 22 bhhlbh

0 1 To minimize

3/1324

0 3)1(4

ANkT

hl

crcr

3/1324

2

3/2324322

1 3)1(4

23)1(4

3)1(

ANkTA

ANkTNkT

cr



The buckling stress and wave length of thin film gelThe buckling stress and wave length of thin film gel

Normalized critical stress, normalized wavelength and critical chemical potential of thin film gel varying

Normalized stresses varying with chemical potential for different gel film and substrate stiffness factor of NkT/Es and critical chemical potential of thin film gel varying

with gel film and substrate stiffness factor of NkT/Esdifferent gel film and substrate stiffness factor of NkT/Es


PDMS membrane gel on substrate in swelling

Swelling-induced bifurcation

Zhang Y et. al. Nano Letter 2008

The pattern of PDMS membrane gel with a square lattice of holes before and after swellingA square lattice of cylindrical holes bifurcates into a periodic structure of ellipses with

alternating directions Hong, Liu, Suo, IJSS, (2009)

Rectangular Strip Membrane Gel

50Mora & Boudaoud (2006)Fix one edge

http://www.lmm.jussieu.fr/~audoly/research/ssw/index.htm

20

30

40

λ /H

Fix one edgeFix two edges

0

10

20

0 2 4 6 8 10 12 14

The instability wavelength λ as a function of the width w of the swollen rectangular membrane gel strip for constraining one edge and two edges

0 2 4 6 8 10 12 14w/H

Liu, Hong, Suo, Somsak, Zhang, Com. Mat. Sci. (2010)

A l f i d d i i l ti

How To Explain Various Phyllotactic Patterns

A leaf growing and drying simulation

The deformation pattern of a leaf when drying by using membrane gel

Swelling of Corona membrane Gel in Equilibrium

14/ HRi

16/ HRi

18/ HRi

The buckling shapes of corona membrane gel for different inner radius (the ratio of outer radius and thickness , initial chemical potential, )20/0 HR 001394203.00

kT


The deformation pattern of a leaf is drying by using membrane gel deswelling

The deformation pattern of a leaf is growing by using membrane gel swelling

A l f i d d i i l ti



Episcia

The deformation pattern of a leaf when drying by using membrane gel


The deformation pattern of a leaf when drying by using membrane gel pH change

A Swelling of Spherical Shape Shell GelPhyllotaxis:Phyllotaxis:How to provide a rational explanation for the formation of friut patterns, a mechanism or combination of mechanisms which capture natural phenomena?

A swelling of spheroidal thin layer of gel to represent the growing process of plant (pH effect)

A S lli f S h i l A l G l Sh ll


A Swelling of Spherical Annulus Gel Shell z

Corpus (softerx

Corpus (softer gel)Shell

(stiffer gel)

Schematic of the geometry of spheroid model

Phyllotactic pattern lattice and tiling pattern of planttiling pattern of plant

A swelling of spheroidal thin film/substrate system of gel to represent the growing process of plants

How To explain Various Phyllotactic Patterns

Cucumber growing can be simulated by gel swelling with changing the chemical potential

How To explain Various Phyllotactic Patterns

Apple growing can be simulated by gel swelling with changing the chemical potential

Concluding remarks & Future Work

• Modeling and simulation results are inspiring

• Some natural phenomena can be explained by gel theory

• Acidic Leaves (actual dimension and components)

• Acidic fruits (actual dimension and components)

• Natural or engineered tissues; bio material• Natural or engineered tissues; bio-material

Various phyllotactic lattice and tiling patternsNewell et al, J. Theoy. Boil. 251, 2008

Thank You

institute of high performance computing (ihpc)xl/lzs-talk.pdf · triphasic model: lai, hou, mow, j....

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