Chirality and Curvature in the Gyroid Mesophasethe.earth.li/~jon/junk/dsfd-2k5-talk.pdf · Chirality and Curvature in the Gyroid Mesophase Jonathan Chin, Peter Coveney August 2005

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




Summary

1 Amphiphile mesophases

2 Membrane model

3 Lattice Boltzmann modelling

4 Domain growth law

5 Chiral domains

6 Summary





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/





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Bicontinuous phases

Continuous, interpenetrating networks of channels

Can self-assemble





Summary

Bicontinuous phases

Continuous, interpenetrating networks of channels

Can self-assemble





Summary

Membrane model

Surfactant interactions extremely complicated

Simplified model treats surfactant layer as curved membrane





Summary

Membrane model

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Mean curvature H = 12(c1 + c2) ; Gaussian curvature K = c1c2





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





Summary

Cubic TPMS phases

P surface D surface G surface





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





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Gyroid formation dynamics

Three stages:

Rapid phase segregation and formation of channels

Morphological ordering

Gyroid domain growth





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Three stages:








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Three stages:








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





Summary

Rapid phase segregation

0 100 200

300 400 500





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





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Formation of domains

0 500 10000 250000





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Gyroid domains

TEM images a,b from Laurer et al, Macromolecules 30 3938(1997).





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Gyroid 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





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





Summary

Domain wall





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Chiral domains





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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.





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Thanks


Documents

Chirality and Curvature in the Gyroid Mesophasethe.earth.li/~jon/junk/dsfd-2k5-talk.pdf · Chirality and Curvature in the Gyroid Mesophase Jonathan Chin, Peter Coveney August 2005