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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
Hong, Zhao, Zhou, Suo, JMPS (2008)
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
Theory of Gel Deformation
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
Hong, Zhao, Zhou, Suo, JMPS (2008)
Theory of Gel Deformation
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
Theory of Gel Deformation
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
Theory of Gel Deformation
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
Theory of Gel Deformation
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
Theory of Gel Deformation
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
Hong, Zhao, Zhou, Suo, JMPS (2007)
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
Liu, Somsak, Cui, Hong, Suo, Zhang, IJAM, (2011)
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
Thin Film Gel with Wrinkle Mode
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
Liu, Somsak, Cui, Hong, Suo, Zhang, IJAM, (2011)
Thin Film Gel with Wrinkle Mode
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
Liu, Somsak, Cui, Hong, Suo, Zhang, IJAM, (2011)
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
A leaf growing and drying simulation
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
How To Explain Various Phyllotactic Patterns
A leaf growing and drying simulation
Episcia
The deformation pattern of a leaf when drying by using membrane gel
A leaf growing and drying simulation
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
How To Explain Various Phyllotactic Patterns
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