Page 1
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Chirality and Curvature in the Gyroid Mesophase
Jonathan Chin, Peter Coveney
August 2005
<[email protected] >
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 2
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
1 Amphiphile mesophases
2 Membrane model
3 Lattice Boltzmann modelling
4 Domain growth law
5 Chiral domains
6 Summary
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 3
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
What is an amphiphile?
Like a soap molecule
Composed of oil-loving and water-loving parts
Migrates to interfaces, reduces surface tension
Diblock copolymers are similar
Image from http://www.vcbio.sci.kun.nl/fesem/applets/amphiphiles/
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 4
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
What is an amphiphile?
Like a soap molecule
Composed of oil-loving and water-loving parts
Migrates to interfaces, reduces surface tension
Diblock copolymers are similar
Image from http://www.vcbio.sci.kun.nl/fesem/applets/amphiphiles/
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 5
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
What is an amphiphile?
Like a soap molecule
Composed of oil-loving and water-loving parts
Migrates to interfaces, reduces surface tension
Diblock copolymers are similar
Image from http://www.vcbio.sci.kun.nl/fesem/applets/amphiphiles/
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 6
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Bicontinuous phases
Continuous, interpenetrating networks of channels
Can self-assemble
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 7
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Bicontinuous phases
Continuous, interpenetrating networks of channels
Can self-assemble
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 8
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Membrane model
Surfactant interactions extremely complicated
Simplified model treats surfactant layer as curved membrane
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 9
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Membrane model
��
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Mean curvature H = 12(c1 + c2) ; Gaussian curvature K = c1c2
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 10
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Canham-Helfrich Hamiltonian
Ecurv =
∫ [2κ (H − H0)
2 + κ̄K]dS
Spontaneous curvature H0
Splay curvature modulus κ
Saddle splay modulus κ̄
Gaussian curvature term depends only on topology, due toGauss-Bonnet theorem:
κ̄
∫KdS = 2πχ
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 11
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Cubic TPMS phases
P surface D surface G surface
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 12
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Ternary amphiphilic LB
Model of Chen, Boghosian, Coveney, and Nekovee: Proc. R.Soc. London A 456, 2043 (2000)
Basically Shan-Chen immiscible fluid model modified toinclude amphiphiles.
Amphiphile particles modelled as dipoles, with orientationaldegrees of freedom.
Grid computing techniques used to run large simulations toavoid finite size effects
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 13
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Ternary amphiphilic LB
Model of Chen, Boghosian, Coveney, and Nekovee: Proc. R.Soc. London A 456, 2043 (2000)
Basically Shan-Chen immiscible fluid model modified toinclude amphiphiles.
Amphiphile particles modelled as dipoles, with orientationaldegrees of freedom.
Grid computing techniques used to run large simulations toavoid finite size effects
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 14
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Ternary amphiphilic LB
Model of Chen, Boghosian, Coveney, and Nekovee: Proc. R.Soc. London A 456, 2043 (2000)
Basically Shan-Chen immiscible fluid model modified toinclude amphiphiles.
Amphiphile particles modelled as dipoles, with orientationaldegrees of freedom.
Grid computing techniques used to run large simulations toavoid finite size effects
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 15
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Ternary amphiphilic LB
Model of Chen, Boghosian, Coveney, and Nekovee: Proc. R.Soc. London A 456, 2043 (2000)
Basically Shan-Chen immiscible fluid model modified toinclude amphiphiles.
Amphiphile particles modelled as dipoles, with orientationaldegrees of freedom.
Grid computing techniques used to run large simulations toavoid finite size effects
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 16
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Gyroid formation dynamics
Three stages:
Rapid phase segregation and formation of channels
Morphological ordering
Gyroid domain growth
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 17
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Gyroid formation dynamics
Three stages:
Rapid phase segregation and formation of channels
Morphological ordering
Gyroid domain growth
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 18
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Gyroid formation dynamics
Three stages:
Rapid phase segregation and formation of channels
Morphological ordering
Gyroid domain growth
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 19
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Rapid phase segregation
3.5
4
4.5
5
5.5
6
6.5
7
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
L 1 (l
attic
e un
its)
Timestep
643 set 81283 set 82563 set 8
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 20
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Rapid phase segregation
0 100 200
300 400 500
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 21
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Characterization of gyroid regions
Automatic identification of TPMS regions is highly nontrivial
Interfacial curvature identifies non-TPMS regions
Triangulate interface
Calculate curvature at each vertex
Bin results back to find interfacial curvature per lattice site
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 22
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Characterization of gyroid regions
Automatic identification of TPMS regions is highly nontrivial
Interfacial curvature identifies non-TPMS regions
Triangulate interface
Calculate curvature at each vertex
Bin results back to find interfacial curvature per lattice site
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 23
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Characterization of gyroid regions
Automatic identification of TPMS regions is highly nontrivial
Interfacial curvature identifies non-TPMS regions
Triangulate interface
Calculate curvature at each vertex
Bin results back to find interfacial curvature per lattice site
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 24
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Characterization of gyroid regions
Automatic identification of TPMS regions is highly nontrivial
Interfacial curvature identifies non-TPMS regions
Triangulate interface
Calculate curvature at each vertex
Bin results back to find interfacial curvature per lattice site
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 25
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Characterization of gyroid regions
Automatic identification of TPMS regions is highly nontrivial
Interfacial curvature identifies non-TPMS regions
Triangulate interface
Calculate curvature at each vertex
Bin results back to find interfacial curvature per lattice site
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 26
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Formation of domains
0 500 10000 250000
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 27
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Gyroid domains
TEM images a,b from Laurer et al, Macromolecules 30 3938(1997).
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 28
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Gyroid domains
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 29
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Domain growth law
10-3
10-2
104 105
⟨H2 ⟩ (
inve
rse
squa
re la
ttice
uni
ts)
Timestep
1283 set 82563 set 81283 set 92563 set 9
t-1/2 power law
〈H2〉 ' Atλ, where λ = −0.48± 0.04
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 30
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Diffusive growth?
Suppose the main contribution to 〈H2〉 is from domain wallsof similar thickness λ
Consider a system of volume V , containing many gyroiddomains of length scale L1(t) ∼ tn
Area of single domain is A1 ∼ t2n ; volume is V1 ∼ t3n
Total number of domains in volume is V /V1, so N ∼ t−3n
Total surface area of domains A ∼ N(t)A1(t) ∼ t−n
Volume of domain walls scales as Vdom ∼ λA ∼ t−n
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 31
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Domain wall
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 32
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Chiral domains
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 33
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Conclusions
Lattice Boltzmann permits a dynamical modelling of lyotropicmesophase formation.
Gyroid TPMS formation from a mixture appears to roughlyconsist of three phases:
Rapid oil/water separation and formation of channelsMorphological reorderingDomain growth
Surface-averaged mean curvature gives a measure of defectdensity
Curvature appears to scale as t−1/2 at late times, possiblyindicating diffusive behaviour
Domains may be differently oriented or have different chirality.
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 34
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Conclusions
Lattice Boltzmann permits a dynamical modelling of lyotropicmesophase formation.
Gyroid TPMS formation from a mixture appears to roughlyconsist of three phases:
Rapid oil/water separation and formation of channelsMorphological reorderingDomain growth
Surface-averaged mean curvature gives a measure of defectdensity
Curvature appears to scale as t−1/2 at late times, possiblyindicating diffusive behaviour
Domains may be differently oriented or have different chirality.
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 35
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Conclusions
Lattice Boltzmann permits a dynamical modelling of lyotropicmesophase formation.
Gyroid TPMS formation from a mixture appears to roughlyconsist of three phases:
Rapid oil/water separation and formation of channelsMorphological reorderingDomain growth
Surface-averaged mean curvature gives a measure of defectdensity
Curvature appears to scale as t−1/2 at late times, possiblyindicating diffusive behaviour
Domains may be differently oriented or have different chirality.
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 36
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Conclusions
Lattice Boltzmann permits a dynamical modelling of lyotropicmesophase formation.
Gyroid TPMS formation from a mixture appears to roughlyconsist of three phases:
Rapid oil/water separation and formation of channelsMorphological reorderingDomain growth
Surface-averaged mean curvature gives a measure of defectdensity
Curvature appears to scale as t−1/2 at late times, possiblyindicating diffusive behaviour
Domains may be differently oriented or have different chirality.
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 37
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Conclusions
Lattice Boltzmann permits a dynamical modelling of lyotropicmesophase formation.
Gyroid TPMS formation from a mixture appears to roughlyconsist of three phases:
Rapid oil/water separation and formation of channelsMorphological reorderingDomain growth
Surface-averaged mean curvature gives a measure of defectdensity
Curvature appears to scale as t−1/2 at late times, possiblyindicating diffusive behaviour
Domains may be differently oriented or have different chirality.
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase
Page 38
OverviewAmphiphile mesophases
Membrane modelLattice Boltzmann modelling
Domain growth lawChiral domains
Summary
Thanks
Jonathan Chin, Peter Coveney Chirality and Curvature in the Gyroid Mesophase