Computational Modelling of Surfactant Liquid Crystal Structures

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Computational Modelling of Surfactant Liquid
Crystal Structures
J.D. Enlow, K.M. McGrath, R.L. Enlow, M.W. Tate
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Certain surfactant (surface-active agent)
liquid crystals are known to have intricate and
complex three dimensional structures, which are
conjectured to be based on minimal surfaces.The
most promising avenue for structure determination
involves X-ray diffraction, but analysis is
non-trivial and relies on mathematical and
computational models. This talk outlines the
development of recent models and associated
computation techniques, and demonstrates that the
conjecture is probably true for at least one
liquid crystal system.
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Seminar Outline

• Surfactant Liquid Crystals
• Minimal Surfaces
• X-ray Diffraction of Liquid Crystals
• Computational Methods
• Results

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Surface Active Agents
Surfactant is an abbreviation for surface
active agent.
Hydrophobic Tail Group
Hydrophilic Head Group
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Surface Active Agents
In solution, the surfactant molecules position
themselves to shield the tail groups from water.
They surround oil, forming an aggregate that is
soluble in water. They can also form liquid
crystals.
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Liquid Crystals
The liquid crystal phase has molecular ordering
somewhere between that of solid and that of a
liquid.
Liquid Crystal
Solid
Liquid
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Liquid Crystals
Liquid crystals have two basic classifications th
ermotropic and lyotropic.
Thermotropic liquid crystals melt from solid
to liquid phase without solvent.
Some thermotropics react to electric and magnetic
fields.
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Lyotropic Liquid Crystals
Lyotropic liquid crystals form in the presence of
a solvent. Formation depends on both temperature
and concentration.
Medium-range order arises from the orientational
preferences of the molecules.
Images from the Dept of Physics, Syracuse
University
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Lyotropic Liquid Crystals
They can form surprisingly complex structures,
some of which are not fully understood.
125 unit cells
One unit cell
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Surfactant
Oil
Water
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Lyotropic Liquid Crystals
It is not easy to fully determine the more
complex lyotropic liquid crystal structures. The
details are too small to observe directly, and
the available techniques are limited due to the
liquid crystals being in solution. X-ray
diffraction allows detailed structural analysis,
but deducing the details of the structure from
the diffraction patterns is difficult!
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Some Uses of Lyotropic LCs

• Detergency (soaps)
• Stabilizing hydrocarbon foams
• Removal of residual oil in wells

• Coating oral drugs to delay digestion
• Increasing drug concentration in solution

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Minimal Surfaces
Some surfactant liquid crystals are thought to
have structures based on minimal
surfaces. Minimal surfaces have minimal surface
area for a given perimeter.
Soap films are examples of minimal surfaces.
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Minimal Surfaces
Minimal surfaces have the defining property of
zero mean curvature everywhere, and hence
non-positive Gaussian curvature.
Mean curvature
Gaussian curvature
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Minimal Surfaces
Zero mean curvature forces every point on the
surface to be locally planar or a saddle point.
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Minimal Surfaces
The Double Diamond
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Minimal Surfaces
The Primitive
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Minimal Surfaces
The Gyroid
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Minimal Surfaces
The I-WP Surface
The S Surface
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Minimal Surface Construction
Some exact minimal surfaces can be constructed
using the Weierstraß equation, which maps a
region in the complex plane to a portion of the
minimal surface.
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Minimal Surface Construction
The Weierstraß equation
is the Bonnet angle. Any analytic Weierstraß
function produces a minimal surface.
Usually we seek additional attributes, such as
being free from self-intersection.
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Minimal Surface Construction
The Weierstraß equation represents the inverse of
the combined Gauss and stereographic projection
mappings.
Thus only a small patch of the surface can be
generated directly from the equation.
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Minimal Surface Construction
The full unit cell is then constructed using
symmetry properties of the particular surface.
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Minimal Surface Construction
The initial patches of the D, P and G
surfaces are generated using the following
parameters.
D P G
Bonnet Angle 0o 90o 38.015o
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Minimal Surface Approximations
Approximations to minimal surfaces can be
generated with Fourier series. These are often
used, and usually only the leading terms are
considered!
Primitive
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Minimal Surface Approximations
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Why Minimal Surfaces?

• Minimal surfaces allow the surfactant molecules
  to occupy regions with near-ideal geometry
  (surfactant parameter) throughout the bilayer.
• Bending frustration (curvature) analysis, phase
  models, and molecular simulations all point to
  minimal surfaces as the best known candidates for
  some liquid crystals.

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X-ray Diffraction of Liquid Crystals
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X-ray Diffraction of Liquid Crystals
Diffraction pattern from a single liquid crystal
thought to be related to the primitive minimal
surface.
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X-ray Diffraction of Liquid Crystals
Peaks can be indexed by their Bragg indices
(hkl values).
The 110 family of peaks is circled.
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X-ray Diffraction of Liquid Crystals
The 211 family is also present.
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X-ray Diffraction of Liquid Crystals
And the outer peaks are from the 220 family.
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X-ray Diffraction of Liquid Crystals
The position of the peaks allows determination of
most of the symmetry properties of the liquid
crystal.
Full space group determination requires
measurement of the intensities of the individual
peaks within each family (a single crystal
sample is necessary).
Comparison of the intensities of peaks from
different families reveals further details about
the liquid crystal structure.
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X-ray Diffraction of Liquid Crystals
Each unit cell contains millions of atoms, so
automated methods (such as Shake and Bake by
Hauptman) are not applicable.
Create a structural model
Calculate X-ray diffraction intensities
Refine model
Compare with experimental data
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X-ray Diffraction of Liquid Crystals
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X-ray Diffraction of Liquid Crystals
In order to remove unwanted experimental factors
such as beam intensity and exposure time,
experimental and theoretical relative integrated
intensities are compared.
The integrated intensity is the integral of the
intensity over the detector screen region and the
crystal rotations for which the intensity is
significantly greater than background noise.
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X-ray Diffraction of Liquid Crystals
At the peak with Bragg indices hkl, the
integrated intensity satisfies
represents the product of the standard
correction factors (Lorentz, polarization,
temperature). The structure factor is defined as
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Calculation of the Integrated Intensity
Direct calculation for arbitrary structures is
slow.
Existing work focuses on approximating bilayers
by surfaces decorated with step function electron
density profiles.
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Calculation of the Integrated Intensity
Clerc et al. (1994) approximate using isotropic
spherical electron density shells about every
point on the central surface.
Neglecting curvature effects, they find

• Easy to calculate
• Inaccurate if the curvature is large

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Calculation of the Integrated Intensity
Garstecki and Holyst (2002) extend Clercs model
by fitting correction factors to partially
compensate for curvature effects.
Despite using leading term nodal approximations,
their results correlate well with experimental
data.
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Calculation of the Integrated Intensity
Harper (2000) takes a better approach for step
profiles, converting the volume integral to a
surface integral via Gausss divergence theorem.
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Calculation of the Integrated Intensity
Harpers method yields good results, but still
requires prohibitively complex calculations to
cope with different bilayer widths.
A calculation method is needed which is

• accurate (uses exact minimal surfaces)
• easy to use (fast and simple)
• allows arbitrarily complex electron density
  profiles to be used

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Calculation of the Integrated Intensity
We seek a function such that, for an
arbitrary profile ,
where D is the maximum distance to the surface
within the unit cell, and k is some constant.
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Calculation of the Integrated Intensity
First the unit cells volume is divided into
bands, such that
where is the unit cell volume and is
the shortest distance from to the central
surface.
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Calculation of the Integrated Intensity
The contribution to the structure factor from
each band can then be numerically calculated
Then the structure factor can be expressed as
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Calculation of the Integrated Intensity
The accuracy of this method can be improved by
interpolating the values with some function
that is zero at the endpoints and preserves
area.
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Calculation of the Integrated Intensity
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Calculation of the Integrated Intensity
G surface, 441 peak.
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Calculation of the Integrated Intensity
D surface, 321 peak.
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Calculation of the Integrated Intensity
Garstecki and Holyst
D surface, 321 peak.
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Results
Double diamond minimal surface, decorated with a
step function profile.
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Results
The electron density profile is modelled using
Gaussian functions. The Gaussian positions are
chosen based on empirical data.
Pn3m (D) Im3m (P) Ia3d (G)
Lattice parameter (nm) 12.793 16.459 31.335
DDAB Weight 31.37 30.25 20.92
Dodecane Weight 5.49 6.30 2.88
D2O Weight 63.14 63.45 76.20
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Results
The remaining parameters are fitted to best match
the experimental diffraction data. The downhill
simplex method is used for optimization. The
function to be optimized is the sum of the
squares of the differences between experimental
and theoretical relative integrated intensities.
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Results
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Results
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Results
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Results
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Results
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Results
A similar analysis for the I-WP and S surfaces
produces a poor match to the experimental data.
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Conclusions

• A new method of structure factor calculation has
  been developed, which is convenient to use,
  accurate, and can accommodate arbitrary electron
  density profiles.
• The surfactant system under investigation forms
  structures that are consistent with the double
  diamond, primitive and gyroid minimal surfaces.
  The structures are not consistent with the I-WP
  or S minimal surfaces.
• A locally optimal electron density profile has
  been found, which gives insight into the detailed
  bilayer structure in the surfactant liquid
  crystal.